Proof. By [45], \(\operatorname{rank}E(\mathbb Q)=r:=s_p(E)\) and the whole group \(\mathop{\mathrm{Sha}}(E/\mathbb Q)\) is finite. The Kummer exact sequence at two is \[0\longrightarrow E(\mathbb Q)\otimes(\mathbb Q_2/\mathbb Z_2) \longrightarrow\mathop{\mathrm{Sel}}_{2^\infty}(E/\mathbb Q) \longrightarrow\mathop{\mathrm{Sha}}(E/\mathbb Q)[2^\infty]\longrightarrow0.\] Taking coranks gives \(s_2(E)=\operatorname{rank}E(\mathbb Q)=r\le1\), so Theorem 1 applies. ◻

Proof. Kramer’s quadratic-extension parity comparison [34] and the Kramer–Tunnell local root-number/norm-index theorem [35], including the dyadic results of [21, 22, 17], give by the Hilbert product formula \[(-1)^{s_2(A)+s_2(A^b)}=w(A)w(A^b).\] Parity computed by finite two-descent agrees with Selmer-corank parity after subtracting the rational two-torsion dimension. Indeed, the Cassels–Tate pairing modulo divisibles is alternating, since the elliptic polarization is represented by the rational divisor \([O]\) [41] [48].

Calibrate the displayed relative parity equality using a coprime twist of sign plus with nonzero central value, supplied by the nonvanishing theorem. When the sign must change, take a negative discriminant split at the level. The analytic rank-zero theorem gives \(s_2=0\) for this twist, so its Selmer parity equals its functional sign. The relative equality then proves the first assertion for \(A\).

Restriction and induction give the rational decomposition over a quadratic field. The associated isogeny preserves the rational Kummer Selmer conditions, proving the displayed corank identity. Finally the same alternating pairing gives the stated even difference between finite Selmer dimension and Selmer corank. ◻

Proof. Apply [eq:GZ-E], then divide the analytic leading term by the arithmetic volume just computed. The squared full index converts the point height to the full Mordell–Weil regulator and retains the torsion factor. Taking \(v_2\) gives [eq:G]. ◻

Proof. Retain the model inclusions into the actual global cochains. Evaluations on every sequence of group elements, pairs of sequences, and higher tuples have simultaneous matrix limits. They preserve the cochain equations, so degree-one cycles give crossed homomorphisms on \(\mathcal G_F\).

It remains to show that evaluation detects their cohomology classes after specialization. At each residual stage a bounded finite list of group elements detects invariants and whether a model cocycle is a single coboundary. These are linear equations on bounded-dimensional spaces. The resulting evaluation map to \[E=[M\longrightarrow M^{\rm list}] \qquad\text{in degrees }0,1\] is therefore an isomorphism on residual \(H^0\) and injective on residual \(H^1\). For its unshifted mapping cone \(K\), writing \(k\) for the residue field, the long exact sequence gives \[H^{-1}(K\otimes k)=H^0(K\otimes k)=0.\] The complexes start in degree zero, so a degreewise minimal model of \(K\) has no terms in degrees at most zero. This remains true after every field base change. The same long exact sequence now proves the required \(H^1\)-injection at the specialized field. No upper boundedness of the ordinary real cochains is needed for this low-degree argument.

Conjugation and comparison identities on evaluations hold on classes by the transferred homotopies. A Selmer \(H^1\) likewise injects whenever its local maps include full \(H^0\) and inject in \(H^1\) at the field being tested. ◻

Proof. For \(x=\partial(e^*)\in H^2(C_0)\) by the second triangle, \(x\) pairs after projection with \(Y\) by \(p\), by the same adjunction. Thus up to signs the two tensors compared via \(C_0\) give \(p^2 x^*\otimes e,\ x^*\otimes e^*\), respectively; the two local lattice volumes are dual. The computation works also for rationally imposed local conditions in compatible triangle volumes defined by assigning a DVR determinant lattice on \(U_w\) and the dual one on \(Q_{\bar w}\). ◻

Proof. Then the generic finite evaluation is nonzero and \(C_-\) generically acyclic. On the fiber no transverse contribution can enter (it must annihilate the nonzero finite opposite evaluation by reciprocity), so \(H^1(C_s)\) has dimension \(h-1\) there by the triangles. Generically the triangle from lower gives again a line in \(C_s\), evaluating isomorphically on the local singular line.

If classes \(Y,Y'\) in \(C_f,C_s\) have \(s(Y')\) equal to \(f(Y)\) via a unit-volume line isomorphism, then \[d(C_f,Y)=d(C_s,Y')\] by the two triangles from \(C_-\) (same tensor 1 there). Extra unchanged terms with a fixed trivialization need not be included in this notation.

Thus in a sequence of such switches with integral classes at the test DVR and nonzero generic evaluations, when \(h=1\) the coordinate is integral (the complex is then just a free line by the minimal fiber model), and gives nonnegativity also backwards. ◻

Proof. We have \(H^2(C_-)=k_{\rm gen}Y^*\oplus k_{\rm gen}(Y')^*\) using upper/lower duality over the fraction field \(k_{\rm gen}\). The finite coordinate boundary pairs to zero against \(Y\); take a complement \(x\) of \(v_0\) mapping to \((Y')^*\). Then \(v_0\wedge x\) has unit plane volume by the cross pairing and the local identity. Likewise on the transverse plane interchanging the two classes. Thus the two triangles compare the class-functional tensors to the same lower tensor \((Y^*\wedge (Y')^*)^*\) times unit plane volumes.

This comparison requires compatible duality adjunctions, as in the preceding diagram discussion. At a split-in-cohomology switch place with conditions in degrees 0,1 they may be used simultaneously: isotropic restrictions admit compatible nullhomotopies on the lower comparisons (there is no ambiguity of a morphism into scalars in degree three on those tensor products). Unchanged local conditions retain the same choices. The comparison also permits volume changes from disjoint rational local modifications when these are defined by compatible triangles.

If at a surplus-dimension fiber one arranges rank two on the old finite plane (including a nonzero evaluation of the primitive reduction), the upper localization image there is the finite plane itself. Thus the switch drops \(h=\dim H^1(C_f)\) by two, while generically the two classes give the determinant comparison just stated.

Iteration to \(h=1\) gives nonnegative valuation for the class-functional tensor over the DVR if exact self-duality and integral classes hold there: at that fiber minimality and generic dimensions give two free lines with zero differential, paired integrally perfectly. In these switches a primitive reduction exists in a two-term DVR model before switching whenever there is a generic line: its kernel is saturated and reduction gives a nonzero fiber class. ◻

Proof. Specialize a line \(1+u_j=(1+x)^{n_j}\), one \(n_j=1\) and others successively very highly 2-divisible to separate a lexicographically leading monomial on reduction (assign the dominant lex coordinates largest \(2^{v_2(n_j)}\) in its weighted order, large relative to the lower-coordinate orders of the chosen monomial). This works simultaneously for a finite list by using the product. The line extends to a coordinate change. To retain the indicated characteristic-zero generic ranks and nonvanishings at \(t=0\), in formal log coordinates use leading homogeneous polynomials, choosing the free integers \(n_j\) avoiding finitely many proper conditions compatibly with the prescribed valuations. The same regularity test applies over a finite local coefficient order by its closed-point reduction. If a bounded \(B\) is divisible by \(L\) after inverting 2 at every sufficiently high \(2\)-power character in \(t\), then Weierstrass division in that tame variable proves \(L\mid B\) integrally. At those specializations the divisor remains regular of the same degree (evaluation always means the parameter is generator value minus one); the monic polynomial quotient has no 2-torsion, so the integral remainder vanishes there. Its bounded coefficients vanish identically by one-variable Weierstrass. One may check vanishing componentwise where the coefficient order embeds in a product of DVRs preserving the residual regularity. In general bounded nonzero series conditions in several variables can be avoided by finite-order tests by iterating the one-variable assertion (also with remaining unspecialized variables). For a nonzero bounded one-variable series the valuation at high 2-power characters is eventually the minimum coefficient valuation plus the first degree attaining it times the valuation of the parameter, by unique dominance. ◻

Proof. The common integral coefficient bound makes coefficientwise convergence uniform on every smaller closed disk. Let \(\alpha\) be a zero of \(f\) of multiplicity \(m\), and pass to a finite scalar extension containing \(\alpha\). If \(g=0\), there is nothing to prove. Otherwise choose, after a further finite extension if needed, a sufficiently small closed disk about \(\alpha\) containing no other zero of \(f\) or \(g\), and with neither series vanishing on its boundary. After translation and rescaling, the dominant coefficient computes the Weierstrass degree on that disk. Uniform convergence preserves this degree for both series on a filter-large set. Thus \(f_i\) has exactly \(m\) zeros there, while the number of zeros of \(g_i\) equals the multiplicity of \(g\) at \(\alpha\). Stage divisibility forces the latter number to be at least \(m\), proving the assertion. ◻

Proof. Pullback by the map of pairs \(\Pi\) is integral. Relative Poincaré duality gives the displayed map; no rational Hecke projector is involved. Its differential coordinate for the dual exponential, with the Tate-twist de Rham comparison, is paired by the cup invariant with the ordinary finite Kummer logarithm on \(E'\).

For the lattice assertion, write \(E_0(\mathbb C)=\mathbb C/\Lambda_0\). Connectedness of the kernel of \(J_1(N)\to E_0\) makes the map on integral first homology onto \(\Lambda_0\). The quotient by \(C_{\mathrm{cusp}}\) replaces \(\Lambda_0\) by the lattice \(\Lambda_B\) generated by \(\Lambda_0\) and lifts of the cusp images. Relative cusp paths supply exactly these additional generators. Hence \[H_1(X_1(N)(\mathbb C),D_N;\mathbb Z)\longrightarrow\Lambda_B\] is surjective. The Farey triangulation generates the relative group by Manin edges \(\xi\{0,\infty\}\). Their images therefore generate \(\Lambda_B\) integrally.

The invariant sublattice is the kernel of complex conjugation minus one on a free integral lattice and is saturated; its generator \(a\) is primitive. Unimodularity of the intersection pairing on \(\Lambda_B\) then makes evaluation on \(a\) surjective on \(H^1(E'(\mathbb C),\mathbb Z)\). An integral combination of the Poincaré duals of the Manin edges consequently has evaluation one. ◻

Proof. A further dyadic division gives a Cartesian product of the two sets of division lifts. Independence remains automatic, since the old tuple is already a basis at 2. At the old precision the inverse-root twist agrees. Base change and the projection formula identify the trace of the cup on this product with the cup of the two traces. Siegel-unit distribution then recovers the old class.

The evaluated units and their inverses are integral away from the level. Horizontal valuations follow from their divisors. For vertical valuations, use smooth full-level models with cusps, on which every good geometric fiber component meets a cusp. The Tate expansion has leading factors consisting of signs, roots of unity, and powers of \(1-\zeta\) for nontrivial level roots; all are units away from the level. A quotient of level can be checked after pullback to such a model.

The smooth proper pair and its residue sequence show that geometric cohomology of the relative open curve is lisse away from this support and vanishes above degree one. The arithmetic degree-two to degree-one edge map therefore lands integrally in \[H^1\!\left(G_{\mathbb Q,S}, H^1(\overline{Y(n_1,n_2)},\mathbb Z/2^j)(1)\right).\] Here degree-one cohomology of a lisse sheaf on the integer base is fundamental-group cohomology; over a field this is the Hochschild–Serre edge map. Edge maps, maps of pairs, and traces commute. Passage to compact degree-one classes uses finiteness, or the Mittag–Leffler property in degree zero. The lattice push and the Shapiro coefficient push are integral, so neither requires a Hecke-idempotent or group-projector denominator.

The same Cartesian-lift calculation proves compatibility when the base dyadic part of \(M\) increases: the lower mark and determinant root agree. For the positive lift, first retain the Shapiro coefficient at the full cyclotomic level. This field is imaginary, and the induced coefficient has zero ordinary real \(H^1\). After pushing coefficients, the global class consequently has zero real restriction and lifts to the mapping fiber defining positive cohomology. Compatible lifts in a tower can be chosen by compactness of the finite-degree cohomology groups at finite precision. Rational wedge with the real boundary basis is independent of the choice of lift. ◻

Proof. The explicit reciprocity input. Use the full-level product formula of [30], specialized to \(k=2\) and \(r=r'=1\). The containment of prime supports required there is automatic at square level. The condition \(1\leq r\leq k-1\), with one of \(r,r'\) equal to \(k-1\), is satisfied. We are in its case \(p\mid M\), so no additional Euler operator occurs. The theorem applies at \(p=2\). This is the full-level theorem; the projected theorem [30] is not needed to obtain [eq:K2].

Initially take \(c,d\equiv1\pmod M\). The symmetric-power coefficient has degree zero. The étale two-unit symbol regulator is the Kummer cup, and the negative Tate specialization over the full dyadic division tuples uses \[e_{2^j}(MP',MQ')^\vee.\] It is the determinant-to-the-power-\(-1\) moment. Thus the test is a full-level basis coset: every lift remains a basis tuple at 2. On the differential side no raising or polynomial-coefficient operator remains. At weight one the shifted and symplectic Fourier-dual Eisenstein forms agree under the consistent conventions; see [30], or the Kronecker functional equation. Both labels are nonzero. The smoothing on the first row is \(c^2-c\,\mathrm{mult}_c^*\), and similarly on the second row. For the initial smoothing choices the two brackets in [eq:K2] are scalar multiples of the individual forms. Only this proportionality is needed from explicit reciprocity; we now determine its scalar.

For completeness, the matrix form of this specialization uses a residue coset \[\begin{pmatrix}a&b\\ d'&e\end{pmatrix}\pmod M\] with primitive determinant. Its two arguments are \[\frac{a\tau+b}{M},\qquad \frac{d'\tau+e}{M}.\] At precision \(2^j\), the matrices lift this coset modulo \(2^jM\). In the compatible cyclotomic basis, the inverse twist in [eq:K1] is the inverse determinant of the lift modulo \(2^j\), up to the fixed orientation. The determinant need not be restricted to a single lift. This agrees with the distribution formula [60] at \((k,j)=(2,1)\); its depth condition \(v_p(M)\geq v_p(2p)\) is satisfied here. The Bloch–Kato dual exponential identification used in the present proof is the original one in Kato’s theorem.

The normalized theta function. Put \[\Theta(t,q)=(1-t)\prod_{h\geq1}(1-q^ht)(1-q^h/t), \qquad \Theta(qt,q)=-t^{-1}\Theta(t,q).\] The normalized function is \[{}_c\theta_E(t)=(-1)^{(c-1)/2}q^{(c^2-1)/12} t^{-c(c-1)/2}\frac{\Theta(t,q)^{c^2}}{\Theta(t^c,q)}.\] It is periodic and has the required divisor. To check its norm normalization, take the \([s]\)-preimages \(t^{1/s}q^{h/s}\zeta_s^j\), \(0\leq h,j<s\), with \((s,c)=1\). The numerator theta factors multiply, before taking the \(c^2\)-th power, to \(\Theta(t,q)\). After taking the product over \(j\), the denominator factors are \(\Theta(q^{ch}t^c,q^s)\), \(0\leq h<s\). Write \[ch=h'+sk_h,\qquad 0\leq h'<s.\] The inverse quasiperiod factors have \(t^c\)-exponent and sign exponent \[\sum_h k_h=(c-1)(s-1)/2,\] and their \(q\)-exponent is \[\sum_h\left(h'k_h+\frac{s k_h(k_h-1)}2\right) =\frac{(c^2-1)(s-1)(2s-1)}{12} -\frac{(c-1)s(s-1)}4.\] Together with the prefactors, this proves exact norm invariance. The divisor and these normalizing norms determine the function uniquely: their quotient is constant, and the norms for \([2]\) and \([3]\) force that constant to have both its third and eighth powers equal to one.

For \(0<x<1\), \(cx\notin\mathbb Z\), set \[J_c(x)=c\lfloor cx\rfloor-\frac{c(c-1)}2 =c^2B_1(x)-cB_1(\{cx\}), \qquad B_1(x)=x-\tfrac12.\] At \(t=q^x\zeta\), the leading coefficient is \(\pm\zeta^{J_c(x)}\), and the \(q\)-order is \[xJ_c(x)+\frac{c^2-1}{12} -\frac{\lfloor cx\rfloor(\lfloor cx\rfloor+1)}2.\] The final two terms are integers.

Residues and field transfer. On any determinant component choose the cusp represented in Tate notation by \[P=q^{1/M},\qquad Q=q^{1/M}\zeta_M^u, \qquad u\in(\mathbb Z/M)^\times.\] Extend scalars to \(K_0=\mathbb Q_2(\mu_M)\). Its width is \(M\). Put \(M_j=2^jM\) and \(K_j=K_0(\mu_{M_j})\). For each \[a_1,a_2\in[0,M_j)\cap(1+M\mathbb Z),\] there is one cusp orbit over \(K_0\) with representative \[P'=q^{a_1/M_j},\qquad Q'=q^{a_2/M_j}\zeta_{M_j}^u,\] width \(M_j\), and residue field \(K_j\). Indeed, once the old labels are fixed, translations by the old width identify the second-index lifts above \(0,u\) in orbits of size \(2^j\). Determinant primitivity makes cyclotomic Galois act freely on the resulting orbits with order \(2^j\). The odd-order roots contribute only the fixed unramified data. There is no additional choice of simultaneous sign after fixing the old tuple. As a degree check, the \(2^{2j}\) cusp orbits, residue degrees \(2^j\), and ramification degrees \(2^j\) account for all \(2^{4j}\) division tuples.

Set \(x=a_1/M_j\), \(y=a_2/M_j\). Use the width-order \(M_j\operatorname{ord}_q\). The residue of the untwisted cup is the Kummer class of the tame symbol, hence of \(\zeta_{M_j}^{h}\), where \[h\equiv\pm uJ_d(y)a_1J_c(x)\pmod{2^j}.\] The coefficient signs have exponents from \(-1\) divisible by \(2^j\). The inverse twist uses the root \(\zeta_{2^j}^{\pm a_1u}\). Consequently the twisted residue on \(G_{K_j}\) is the additive character \[\pm h(a_1u)^{-1}\frac{\chi_{\mathrm{cy}}-1}{M_j} \pmod{2^j}.\] Here \(\chi_{\mathrm{cy}}\) is the dyadic cyclotomic character; the odd part of the roots is already fixed. Transfer to \(G_{K_0}\) replaces the cyclotomic expression by \[\frac{\chi_{\mathrm{cy}}^{\,2^j}-1}{M_j} \equiv \frac{\log\chi_{\mathrm{cy}}}{M}\pmod{2^j}.\] The equality on the cyclotomic quotient is the abelian transfer formula, and the congruence follows from \(v_2(M)\geq2\). Residue trace sums these residue-field transfers without a ramification-index multiplier. Bernoulli distribution gives \[\sum_{a_1}J_c(a_1/M_j)=J_c(1/M),\qquad \sum_{a_2}J_d(a_2/M_j)=J_d(1/M).\] The limiting residue, also the residue of the edge class by localization, is therefore \[\pm J_c(1/M)J_d(1/M) \frac{\log\chi_{\mathrm{cy}}}{M}.\]

Étale and de Rham residues for \(H^1(1)\) compare by sending the logarithmic differential of a parameter to residue one. The dual exponential on \(H^1(K_0,\mathbb Q_2)\) sends \(\log\chi_{\mathrm{cy}}\) to \(\pm1\). This follows by pairing with Kummer classes of exponential units and local reciprocity, with field trace in the de Rham pairing.

On the Eisenstein side, the constant term of \(E_{[x\tau+y]}\), for \(0<x<1\), is \(-B_1(x)\). One can obtain it by integrating in the second summation coordinate before the prefactor: the constant coefficient is the value at \(w=0\) of \[-i\frac{\sqrt\pi\,\Gamma(1/2+w)}{\Gamma(1+w)} (\operatorname{Im}\tau)^{-w} \bigl[\zeta(2w,x)-\zeta(2w,1-x)\bigr],\] where \(\zeta(s,x)\) is the Hurwitz zeta function. The two brackets in [eq:K2] thus have constant terms \(-J_c(1/M)\) and \(-J_d(1/M)\), both nonzero for the initial choices. Since \(dq/q\) has residue \(M\) at this cusp, the proportionality factor is exactly \(\pm M^{-2}\). The orientation is common to all components.

Arbitrary smoothing integers. In multiplicative group notation, divisor and norm normalization give \[(c_0^2-[c_0]^*)\,{}_c\theta_E =(c^2-[c]^*)\,{}_{c_0}\theta_E.\] Multiplying the first label by \(c\) in [eq:K1] changes the inverse twist basis by \(c^{-1}\). The induced operator is therefore \(c^2-c\,\mathrm{mult}_c^*\). Apply the two smoothing operators and cancel the nonzero scalar \(c_0^2-c_0\) for \(c_0\equiv1\pmod M\); do the same on the other row. This is a cancellation in the rational differential identity. It makes no assertion of integral divisibility by that scalar. Row multiplication may permute components, which is why a common calibration was used. The formula follows for every \(c,d\) prime to \(6M\). ◻

Proof. The fresh good prime. Before requiring independence at \(\ell\), the two division sets are Cartesian. Their norm is the old cup [eq:K1], with mark, inverse twist, and lower root agreeing, also through the dyadic tower. Subtract the lifts whose \(\ell\)-parts both lie in a fixed order-\(\ell\) line \(I_0\), sum over these lines, and add back \(\ell\) times the double-prime-to-\(\ell\) lift. This is inclusion–exclusion: the zero pair belongs to all \(\ell+1\) lines.

For a subtracted line put \(\rho:E\to\widetilde E=E/I_0\). Distribution identifies the summed cup with that on \(\widetilde E\) at the unique prime-to-\(\ell\) points \(\widetilde P,\widetilde Q\) mapping to the old points by \(\rho^\vee\); the relevant lifts were their \(\rho\)-preimages. Reversing the description gives old valid tuples on \(\widetilde E\), with arbitrary line \(\ker\rho^\vee\), mapping to \(E\) by quotient and pushing the mark forward. Relative to the source, the Weil roots downstairs are raised to \(\ell\). This gives the field translation \([\ell]\) and the inverse-twist scalar \(1/\ell\).

This is the good Hecke correspondence, and it acts on the marked elliptic push by \(a_\ell(f)\). Indeed, pull back the relative forms \(\Pi^*\omega,\Pi^*\bar\omega\) along the quotient and sum on the source; \(F,\bar F\) are the corresponding Hecke eigenforms. Equality holds also for integrals along the lifted paths, with zero endpoint-function coordinates. It is consequently an equality of rational relative Betti maps, and then of rational étale maps. Thus no additional boundary projection is needed.

The added-back lifts use multiplication by \(\ell\), of degree \(\ell^2\), and roots raised to \(\ell^2\). A possible diamond operator on the mark acts trivially on this push, by the same relative-form calculation. The resulting multiplier is the one in the statement. All maps used before and after the edge map are ordinary pullbacks and unnormalized traces on fiber products. At complete splitting, \([\ell]=1\), and \[1-\frac{a_\ell(f)}{\ell}+\frac1\ell =\frac{\det(1-\rho_T(\mathrm{Fr}_\ell))}{\ell}.\]

When integral derivative comparisons require removal of an error that vanishes rationally, one common nonzero integer suffices in the non-CM case. Indeed, over the abelian fields in question, the torsion of \(H^1(T)\) has bounded exponent. By the open Tate image [54], the commutator image contains a fixed deep special-linear congruence subgroup; its invariants on \((T\otimes\mathbb Q_2)/T\) have bounded exponent. The usual cohomology sequence gives the asserted bound on \(H^1(T)\)-torsion.

Filling a bad-prime row. Keep \(Q\) and take the \(q_0^a\)-division lifts of \(P\). The mark, the determinant map to the \(M\)-th roots, and the inverse-root twist agree. Without the rank condition at \(q_0\), their sum is the old cup. A discarded lift is one whose \(q_0\)-part has its \([q_0^{a-1}]\)-image in the order-\(q_0\) line \(I_Q\) of the \(Q\)-subgroup. Put \(\rho:E\to\widetilde E=E/I_Q\). These discarded lifts are exactly the preimages under \(\rho[q_0^{a-1}]\) of the unique prime-to-\(q_0\) point \(\widetilde P\) satisfying \(\rho^\vee\widetilde P=P\). Their norm is \({}_c g_{\widetilde P}\).

Express \({}_d g_Q\) by the Kummer sum over \(\rho^\vee\widetilde Q=Q\). Each solution has order \(n_2\). Indeed, by counting, \(\rho^\vee\widetilde E[q_0^a]\) is the subgroup of \(E[q_0^a]\) whose \([q_0^{a-1}]\)-image belongs to \(I_Q\). Primitivity of \(Q\) also gives \(I_{\widetilde Q}\ne\ker\rho^\vee\). These descriptions continue to hold at the upper dyadic stages in [eq:K1].

In reverse, the discarded sum is therefore indexed by old valid tuples on \(\widetilde E\) and all lines \(H_0\ne I_{\widetilde Q}\), with quotient by \(H_0\). Because \(l=(Ns,1)\), the line \(I_{\widetilde Q}\) is also the order-\(q_0\) line of the source mark. The roots are raised to \(q_0\), which is a unit on the shared field and twist indices. This marked correspondence is \(U_{q_0}\): at mark \(1/L\), its branches are \(\tau\mapsto(\tau+i)/q_0\), \(i\bmod q_0\), omitting exactly the line of the mark. It acts on \(F\) by \(a_{q_0}(f)\), since \(A\) is a power of 2. The translation of the root and the inverse twist thus give the stated factor \(a_{q_0}(f)\nu(q_0)/q_0\) for the discarded sum. Subtracting it proves the filling identity. ◻

Proof. The square-order trace. By Proposition 22, after tracing to the marked level the coefficient of \(dq/q\) is \[\frac{\pm\Delta_\nu}{(2\pi i)^2} \sum_{\substack{ U=\left(\begin{smallmatrix}h&j\\m&n\end{smallmatrix}\right) \in M_2(\mathbb Z),\;(\det U,M)=1\\ lU\equiv(0,1)\bmod L}} \frac{\nu(\det U)}{(h\tau+j)(m\tau+n)}. \tag{K5}\] Regularize by multiplying the summand by \[y_\tau^{u+v}|h\tau+j|^{-2u}|m\tau+n|^{-2v}, \qquad y_z=\operatorname{Im}z,\] and continuing to \((u,v)=(0,0)\). The row labels in a fiber over the mark \(1/L\) satisfy exactly these congruences, for every embedding of the determinant root; the target mark stays the same as that embedding varies. The Weil pairing is the determinant up to the common orientation, which has no effect for even \(\nu\). There is no simultaneous minus identification within this marked fiber. The two rescaled lattice sums contribute \(M^2\), canceling the \(M^{-2}\) in [eq:K3]. Multiplication of a row label by \(c\) or \(d\) preserves the mark and, after reindexing its determinant weight, produces the corresponding factor of \(\Delta_\nu\).

Hecke unfolding. Pair the sum with \(\bar F\) in \(dx\,dy\). For positive determinant \(b_0\), unfold by the right \(\Gamma_1(L)\)-action. The substitution \(z=U\tau\) contributes \(b_0^{-1-u-v}\) times \[\mathcal J_{F|_2U^{-1}}(u,v)= \int_{\mathcal H}\overline{(F|_2U^{-1})(z)} \frac{y_z^{u+v}}{z|z|^{2u}}\,dx_z\,dy_z,\] where the weight-two slash operator includes the determinant factor. The matrices \(\operatorname{adj}(\xi U)\) give the left \(\Gamma_1(L)\)-cosets of determinant \(b_0\) with first column \((1,0)^{\mathrm{tr}}\bmod L\). These map bijectively to the usual \(\Gamma_0(L)\) Hecke cosets, where the first entry is allowed to be any unit: because \((b_0,L)=1\), a left \(\Gamma_0(L)\)-matrix can normalize that entry to 1, uniquely modulo \(\Gamma_1(L)\). The slash sum is therefore \(a_{b_0}(f)F|_2\xi\). There is no extra \(-I\) stabilizer in \(\Gamma_1(L)\).

For negative determinants, flip the first row. This gives the subtraction with \[\xi'=J\xi J,\qquad J=\operatorname{diag}(-1,1).\] Thus, without the prefactor in [eq:K5], the paired expression is \[L^{(M)}(f,\nu,1+u+v) \bigl(\mathcal J_{F|_2\xi} -\mathcal J_{F|_2\xi'}\bigr)(u,v).\]

Continuation and the archimedean factor. Start in the region \(v>u>1\). Each Kronecker series has moderate-growth meromorphic continuation and is regular at zero in this situation; pairing with a cusp form therefore permits continuation. In the unfolded expression, first hold \(v>3\) fixed and continue to \(u=0\). For \(G=F|_2\xi\) or \(F|_2\xi'\), \[|G(z)|\leq C/y_z,\] and \(G\) has zero periodic mean and decays at infinity. The integral converges for \(0<\operatorname{Re}u<v\). On horizontal tails, integrate by parts using a bounded periodic primitive, which is also \(O(1/y_z)\) at small height. This permits the limit \(u=0\). On bounded horizontal intervals, the displayed growth bound, multiplied by \(y_z^{u+v}\), gives local domination.

At \(u=0\), the integral in \(x_z\), apart from \(y_z^v\), is \(-2\pi i\,\overline{G(2iy_z)}\): this follows termwise from the positive Fourier modes. Cuspidality then permits continuation of the remaining integral to \(v=0\), yielding \[\mathcal J_G(0,0) =\pi\,\overline{\int_0^{i\infty}G(z)\,dz}.\] Write \(I=\int_0^{i\infty}(F|_2\xi)(z)\,dz\). The integral for \(\xi'\) is \(-\bar I\), since the coefficients of \(f\) are real. Since \[\frac{dq}{q}\wedge\frac{d\bar q}{\bar q} =-8\pi^2i\,dx\,dy,\] the wedge pairing of the traced differential with \(\Pi^*\bar\omega\) is \[\pm2i\,\operatorname{Im}(A_\Pi)\, \Delta_\nu L^{(M)}(f,\nu,1), \qquad A_\Pi=c_F(2\pi i)I.\]

Projection to the elliptic differential. The relative pullback has zero cusp-function coordinates. Cutoff primitives vanishing at the cusps introduce no boundary term when paired with a differential having at worst logarithmic poles. The push is dual to this pullback, with the same trace convention on both curves. Write \(\delta_\xi=s_0\omega+\bar s_0\bar\omega\). Then \[\int_{E'}\omega\wedge\delta_\xi=A_\Pi,\qquad 2i\,\operatorname{Im}(A_\Pi) =(s_0+\bar s_0)\int_{E'}\omega\wedge\bar\omega,\qquad s_0+\bar s_0=\frac{\delta_\xi(a)}{\Omega_0}.\] The projected differential is on the Hodge line. These identities prove [eq:K4] in square order, with consistent sign for all symbols.

This comparison of complex and dyadic scalars is algebraic: the modular differentials with the specified weights are algebraic, and trace and push are algebraic de Rham operations. After restriction to a field trivializing the character, dual exponential uses the same weighted sum. Its local duality uses field trace, not the average of that trace.

Rectangular order. Apply the filling identity of Proposition 23 one full prime power \(q_0^a\parallel b\) at a time. This fills the first row to square level \(bM\), with determinant weight inflated to that level. Each factor \[1-a_{q_0}(f)\nu(q_0)/q_0\] is nonzero at a finite character, since \(|a_{q_0}(f)|\leq1\). It is exactly the extra omitted bad-prime Euler factor at \(q_0\). Canceling these factors in the rational period identity for square level gives [eq:K4] with omissions at \(M\), as asserted. This cancellation proves a rational differential formula. The integral fresh-prime trace for the rectangular system was constructed separately in Proposition 23. ◻

Proof. Take the upper row to be \((1,0)\). After the degeneracy, the path runs from \(0\) to \(2^k/(Ns)\). The Fricke relation for \(f\) identifies the integral of \(2\pi i f\) along it, up to sign, with the integral from \(\infty\) to \(-s/2^k\). For some \(k\), these latter integrals cannot all be real. Indeed, Rohrlich’s finite-support twist nonvanishing [50], in the form [30], applies with prime support \(\{2\}\). One can equivalently fix the odd quadratic character \(\eta=\chi_{-4}\) and vary primitive even characters of the real cyclotomic tower: multiplying a character of conductor \(2^k\), \(k\geq3\), by \(\eta\) preserves that conductor and makes it odd. Thus there is a primitive odd character \(\nu\) of sufficiently high dyadic conductor with \(L(f,\bar\nu,1)\ne0\).

Set \(\lambda(r)=\int_\infty^r2\pi i f(z)\,dz\). The primitive additive Gauss–Mellin formula is \[\sum_{s\bmod 2^k}\nu(s)\lambda(-s/2^k) =\nu(-1)\tau(\nu)L(f,\bar\nu,1),\qquad \tau(\nu)=\sum_{s\bmod2^k}\nu(s)e^{2\pi i s/2^k}.\] Its right-hand side is nonzero. If all the indicated integrals were real, the identity \(\lambda(-r)=\overline{\lambda(r)}\), which follows from the real Fourier coefficients, would make them unchanged under \(s\mapsto-s\). Their sum against the odd character \(\nu\) would then vanish. This contradiction gives a nonzero imaginary part. The path pairing in the proof of Proposition 24 then gives \(\delta_\xi(a)\ne0\). ◻

Proof. The first assertion is Lemma 20 and linearity. Include the conductor of \(\chi_h\) in \(M\). The weighted coefficient with this fixed factor identifies integrally with the coefficient for \(E'^{\,h}\) by the geometric twist isomorphism. Choose this isomorphism compatibly over \(\mathbb R\); positivity of \(h\) makes the primitive real-invariant cycles correspond up to sign. If \(\omega\) pulls back to \(b_h\omega_h\), then the transported exponential coefficient relative to \(\omega_h\) is multiplied by \(b_h\), while \[\Omega_{0,h}=\Omega_0/|b_h|.\] The omitted-support \(L\)-value on the twist uses the remaining character. Thus both sides of [eq:K4] transport with the same factor, up to the common sign. The coefficient construction is the weighted integral trace, so no division by a quadratic projector is made. ◻

Proof. All indicated fixed polynomials and smoothing factors are integral, nonzero modulo 2 even upon \(u_1=\cdots=u_k=0\). Indeed each of the real cyclotomic exponents of \(q,c,d\) here is nonzero, and \(D_q,P_q\) use the same type of elliptic polynomial in possibly opposite orientations (good quadratic, signed multiplicative linear, or trivial additive). This follows from the ordinary local Tate description: unramified action at good primes, quotient coinvariant unramified sign line at multiplicative primes, no rational coinvariant at additive primes. It uses the already twisted curve if testing that branch. Palindromicity gives the asserted residual unit, and the two orientations agree at the central specialization, giving ratio one. ◻

Proof. Required local tests.

All added primes split on \(W\) and in the fixed twisting field, and their arithmetic Tate Frobenius matrices have ultrafilter limits \(g_j\) with determinant 1 and trace \(\ne2\) (use chosen local embeddings). This includes the required hypothesis for the distinguished prime.

Use Frobenius lifts there and at the other moving primes trivial in their own tame cyclic field, and inertia generators taking the chosen generator values. Write \(e_{jl}\in\mathbb F_2\) for the bit of the Legendre symbol \((r_{j,i}/r_{l,i})\) for \(j\ne l\) (stagewise), symmetric by reciprocity. We arrange the following two properties.

1. For each \(q\in S_f\), the limiting unramified tame-scalar exponent vector at \(q\) is nonzero.

2. If \(W\) is irreducible, at the stages on the filter, for common generic \(\lambda_j\in\overline{\mathbb F}_2^\times\), the following residual \(W\otimes\overline{\mathbb F}_2\) Selmer problem is zero: unramified at fixed finite and other nonmoving places, unrestricted at infinity, and at the moving places \[f_j=b_j s_j,\qquad b_j=\lambda_j^{-1}\sum_{l\ne j}e_{jl}\lambda_l , \qquad f_j=x({\rm Fr}_j),\quad s_j=x(\sigma_j), \tag{L2}\] with \(\sigma_j\) the indicated tame generator. If \(W\) is reducible use just one dummy prime in addition to the distinguished prime if used; when there are two, require their mutual bit to be one.

The dummy prime and the fixed-place exponents. First add the dummy prime. To achieve (i), prescribe over the field of \(W,\chi_h\) (where used) and the \(2\)-cyclotomic tower simultaneous nonzero Kummer exponents on compatible \(2\)-power roots of each \(q\in S_f\), using accuracy tending to infinity. Each coordinate can indeed be nonzero: otherwise the pure cyclotomic tower would have only bounded-degree further radical extensions for that \(q\), hence none (the bounded image on the compatible roots there is a finite subgroup of \(\mathbb Z_2(1)\)). Even all positive radicals of \(q\) cannot lie in the abelian tower, by Eisenstein and nonreal conjugates.

Joint nonzero values now follow by additivity, and the bit with the distinguished prime remains free by its separate new ramification (take bit zero when \(W\) irreducible). In tame residue characters at the chosen primes the power residue exponents thus have bounded valuations (normalization of a cyclic generator only changes by units).

Trace nonexceptionality can simultaneously be kept with \(v_2(2-\operatorname{tr} g_j)\) bounded: within the kernel of such bounded-derived-length constraints the Tate image contains a fixed deep \({\rm SL}_2\) by open image [54] and closed iterated commutators as in the evaluation construction of Section 3. Its perturbations can keep the trace off 2 uniformly for a compact set of starting matrices. Chebotarev now gives stage choices to increasing precision, split also to the needed cyclotomic depths.

Killing the irreducible residual Selmer space. We next arrange (ii) when \(W\) is irreducible. Work first over \(\mathbb F_2\), initially and at new primes with all mutual bits zero, so \(f_j=0\). Write \(F,F^*\) for the resulting Selmer and exact-annihilator Selmer subspaces in global \(H^1(W)\), using the Weil pairing. At fixed odd places unramified is self-annihilator by isotropy, dimension \(h^0\), duality and the local Euler formula. At 2 unramified is isotropic of dimension \(h^0\) with annihilator two dimensions bigger. At the moving split primes the planes \(f,s\) pair by the mixed Weil pairing up to unit, separately isotropic (scalar Hilbert pairing alternating since \(r_j\equiv1\bmod4\)). At infinity the dual condition is strict. Thus by Wiles’s dimension formula (Poitou–Tate, subtracting ordinary \(h^0\) at each local place) \[\begin{gathered} h^*=h+\delta,\qquad h=\dim F,\qquad h^*=\dim F^*,\\[3pt] \delta=\begin{cases} 1&\text{if complex conjugation acts nontrivially on }W,\\ 0&\text{otherwise}. \end{cases} \end{gathered}\] Indeed in the first case real \(h^0=1,h^1=0\); in the second both equal 2. At a new prime with zero bits, switching there from unramified to \(f=0\) gives \[h_{\rm new}\le h-\operatorname{rank}{\rm loc}_r F +2-\operatorname{rank}{\rm loc}_r F^* .\] The strict kernel is unchanged by inflation and the new singular coordinate must annihilate old dual evaluations. This bound works also with changed fixed conditions and their exact duals.

At a new split prime evaluations can be prescribed arbitrarily on a joint \({\rm End}_{G_\mathbb Q} W\)-independent set of residual classes. Indeed restriction to the joint kernel of \(W\) and the pro-2 abelian data for bits, cyclotomic and fixed-twist splitting detects these classes: the quotient has a normal subgroup of order three with no invariants. Additive joint evaluations there form a submodule of a sum of the simple \(W\), hence surject by independence. The bits with old primes are freely imposed by disjoint quadratic ramification over the fixed split fields. Evaluation fields do not upset the trace-limit freedom by the same bounded-derived-length argument. Take Chebotarev representatives as above. Dimensions are uniformly bounded for a bounded tuple, so choices and a bounded iteration based initially on a bounded dimension can be fixed in pattern along the filter.

For cyclic cubic image the spaces with the stated zero-bit conditions are \(\mathbb F_4\)-spaces via that endomorphism field (annihilators also stable); if \(h>0\) arrange rank two over \(\mathbb F_2\) on both.

For \(S_3\) image arrange ranks \(\min(2,h),\min(2,h^*)\) simultaneously by choosing a basis including the intersection. This drops \(h\) until zero except possibly \(h=h^*=1\).

If the lines then coincide, the intersection problem (unramified at 2 and strict at infinity) has dual of dimension three (by the formula) containing that line. Switch at a new prime evaluating nontrivially on the line and surjectively on this larger dual. The same bound kills the intersection and keeps \(h=h^*\le1\).

Once the two lines are independent, add a zero-bit prime \(A'\) evaluating the generators on nonparallel vectors \(v,w\), respectively. If after switching \(h>0\) persists, both spaces still have dimension one and the singular coordinates at \(A'\) of their new respective generators are \(w,v\), by vanishing of strict kernels and orthogonality.

Add now a last prime \(B'\) evaluating trivially on both new lines but with bit 1 with \(A'\), others zero. With \(x=\lambda_{B'}/\lambda_{A'}\) the conditions are \(f_{A'}=x s_{A'},\ s_{B'}=x f_{B'}\), recovering the preceding problem at \(x=0\). Its generator cannot lift modulo \(x^2\): pair the linear coefficient globally with the preceding dual generator, giving \(\langle w,v\rangle\ne0\) at \(A'\) and zero at every other place. The modified local equations form a matrix over \(k'[[x]]\) on these fixed global and local spaces. A nonzero generic solution could be scaled to an integral solution with nonzero reduction. That reduction would be a nonzero multiple of the preceding one-dimensional Selmer generator, and rescaling would give the forbidden lift modulo \(x^2\). Thus we have vanishing on a nonempty open as required.

All avoidance polynomials can be fixed along the filter by bounded dimensions and finite base field; we work with \(\lambda_j\) in a sufficiently large common finite \(k'/\mathbb F_2\).

A one-parameter residual test. Set \(A=k'[[z_0]]\) and specialize \[2=t=0,\qquad u_j=\lambda_jz_0.\] In the irreducible case choose \((\lambda_j)\) in the nonempty open set furnished by (ii); in the reducible case retain the one- or two-prime configuration specified there. Choose the vector also so that each fixed finite place in (i) has nonconstant unramified scalar. The latter conditions are nonempty: the first nonzero term of the scalar is determined by the least valuation of its nonzero exponent vector, and cancellation excludes a proper closed condition on the \(\lambda_j\).

Over \(k'((z_0))\), Frobenius at a fixed finite place consequently has no invariants, on either coefficient. At a moving place the nontrivial inertia scalar gives the same conclusion. Local duality also gives \(H^2=0\) at every finite place. These are statements about the local limit models: at fixed places the coefficient actions stabilize at every precision; at moving places the inertia/residue models give the assertions. Evaluation on these local groups also shows that global \(H^0\) vanishes.

The remaining global obstruction. It remains to kill the kernel of finite localization on ordinary degree-one cohomology of the dual coefficient. Here is the precise Poitou–Tate reduction. At a finite stage and scalar precision \(A/(z_0^n)\), write \(K_n\) for that dual strict-at-finite kernel. Subscripts \(n\) below denote the corresponding finite-precision complexes. Finite Poitou–Tate gives \[\dim_{k'}H^2(C_n^+) \leq \sum_{v\in S_f\cup R_i}\dim_{k'}H^2(L_{v,n}) +\dim_{k'}K_n,\] and bounds \(\dim_{k'}H^3(C_n^+)\) by the dual global invariant dimension; higher positive cohomology vanishes. To see the first bound, the cokernel of ordinary \(H^1\)-restriction to the real place is dual to the real image of \(K_n\). The kernel of \(H^2\)-restriction to the real place maps to the finite local \(H^2\)’s, with kernel dual to the everywhere-locally-zero subspace of \(K_n\). These two contributions together have dimension at most \(\dim K_n\). Ordinary and Tate-modified real terms agree in the positive degrees used here. A perfect coefficient-ring functional identifies the finite duals in this calculation.

Apply this inequality stagewise and then to the limit matrices. For a finitely generated \(A\)-module, the dimension of its reduction modulo \(z_0^n\) is its free rank times \(n\), up to a bounded term. The local vanishings just proved therefore remove the local contributions after division by \(n\). The strict-finite kernel is likewise measured by its mapping-fiber model; finite local \(H^0\) has rank zero. Consequently a rank-zero dual strict-finite kernel implies that \(C^+\otimes_Ak'((z_0))\) is concentrated in degree one.

Reduction of the dual kernel. Let \(N\) be the full saturated preimage of this kernel in ordinary dual \(H^1\) over \(A\), including its torsion. Saturation and the cohomology base-change sequence inject \(N/z_0N\) into residual ordinary \(H^1\). At each fixed finite place its image is unramified: inertia acts constantly, so a restriction that becomes a coboundary after multiplication by a power of \(z_0\) has residual coefficient a coboundary on inertia.

At a moving place the tame cocycle relation on the limit is \[(\chi^{-1}(\sigma_j)-1)x({\rm Fr}_j) =(\chi^{-1}({\rm Fr}_j)-1)x(\sigma_j).\] Indeed, discard prime-to-two inertia and use \(r_j\to1\) in the Frobenius conjugation relation, including its cocycle sum. Dividing by \(z_0\) and reducing gives [eq:L2]. These conditions are retained in the finite diagrams: at a fixed place finitely many inertia evaluations detect a coboundary, and at a moving place the two displayed evaluations were marked. Thus the residual tests apply to the reduction of \(N\), not merely to unrelated stage classes. In the irreducible case (ii) gives \(N/z_0N=0\), hence \(N=0\).

The reducible case. For a trivial constituent, residual classes unramified away from the moving finite places are spanned by \([r_j]\), since \(r_j\equiv1\pmod8\). Their evaluations are the Legendre bits in [eq:L2], with own-prime Frobenius bit zero. With one prime, or with two primes of mutual bit one, the solution space has dimension one. The saturated preimage \(N\) already contains the torsion cocycle \((\chi^{-1}-1)/z_0\), with nonzero reduction; it can be defined compatibly at one higher precision. Since this torsion class occupies the sole allowed residual dimension, \(N\) has free rank zero. The preceding Poitou–Tate argument gives positive concentration for a trivial constituent. Exact triangles for extensions then give concentration for \(W\).

Concentration and twist comparison. Matrix cancellation now gives concentration over the full residual fraction field, and then over \(B_{(2)}\). The Euler characteristic of \(C^+\) is \(-2\): evaluate a stage at constant characteristic-zero coefficients, where the ordinary global Euler characteristic is \(-1\) and the real invariant line contributes one more subtraction. Thus the minimal model over \(B_{(2)}\) is a free rank-two module in degree one. The integral tensor \(a\wedge z\) has nonnegative valuation there; the fixed factors are units at \((2)\).

In the simultaneous twist construction the common diagram, determinant basis, and classes give identical raw reductions in the residual fraction field. The smoothing reductions agree too. After the remaining height-one tests prove \(U\in B\), the group-like residual factors \(D_q/P_q\) therefore show that the two reductions of \(U\) differ by a unit of \(B/2B\). ◻

Proof. Old local conditions and duality. We first check cohomology at the old places, over both the generic and residue fields when a DVR is being tested.

• In the first test dual local \(H^0\) vanishes at every finite place of the old support. At fixed odd \(q\) this follows from \(D_q\) (duality of inertia-invariants with coinvariants, including the Tate twist).

  At 2 choose an inertia element mapping to a generator in the real cyclotomic direction, and a Frobenius lift trivial in that direction. The invariance equations then require the simultaneous vanishing of a nonzero polynomial up to a unit in \(t\) alone (monic after writing the determinant condition with the inverse scalar cleared), and a nonzero series independent of \(t\). The latter uses property (i) of the tame exponents in the residual construction and constant Tate action of the chosen element. The first is regular in \(t\), so its distinguished polynomial has no factor independent of \(t\). It therefore has no height-one prime factor in common with the second series, by Weierstrass division.

  At a moving \(r_{j,i}\) invariance on inertia requires \(u_j=0\); on that divisor the Frobenius determinant condition is not identically satisfied, by \(\det(g_j)=1,\operatorname{tr}g_j\ne2\) on setting scalars centrally. These calculations test the actual local models (fixed groups by coefficient stabilization, moving odd primes by the inertia/residue complexes or Frobenius and inertia evaluations).

  In the second, generic \(t\)-test the same dual vanishing at fixed places uses inertia at 2 and nonconstant \(t\)-Frobenius scalar at every odd \(q\).

• In the third test we require the active characters such that already a fixed Frobenius lift at 2 trivial in the cyclotomic direction kills dual invariants, by its action with the constant part of the scalar.

  At each old odd \(q\) of the fixed-test support (including all now-fixed tame variable places), put the inflated residue cochains on the inertia invariant block for the unramified complex condition, similarly dually. This is well-defined here after inverting 2 with base change to both fields. Indeed inertia action of \(V\otimes\eta\) is constant; after exact prime-to-2 kernel invariants on inertia, the two-term procyclic \(2\)-inertia linear algebra is constant, split in cohomology over the rational coefficient space before imposing residue Frobenius. The quotient relative to the unramified condition computes residue cochains on the inertia coinvariants with negative Tate twist in degrees \(1,2\), by this calculation and inertia truncation (Frobenius conjugation on the generator evaluation modulo coinvariant relations inserts \(q^{-1}\)). The same formulas hold on the local models in subsequent switches. Its determinant of differential up to unit is \[D_{q,\eta}=\det(1-\chi_t({\rm Fr}_q)\rho_{V\otimes\eta}({\rm Fr}_q)/q \mid (V\otimes\eta)_{I_q}),\] where \(\chi_t\) denotes the \(t\)-scalar. This is nonzero by nonconstant cyclotomic action (and equals 1 at active tame variable conductors, by good reduction there). The quotient has zero \(H^1\) over the DVR itself by injectivity, though possibly not on its residue field.

  These local conditions include all \(H^0\) and inject in \(H^1\) there also. They give exact complementary dualities over the DVR: nullhomotope the cup via its unramified inflation factorization (residue cohomological dimension one with free models). Over a field the two unramified \(H^1\) dimensions equal the respective \(H^0\)’s; by local duality and the local Euler characteristic zero these are orthogonal complements. This and inclusion of all invariants, no \(H^2\), checks the quotient duality equivalence by base change.

In all tests global invariants vanish (also dually) by the open-image evaluation argument below. Global Euler characteristic in the problems is \(-1\), by the ordinary formula, or computing the positive-model Euler characteristic \(-2\) stagewise at constant characteristic-zero coefficients and restoring the real term. Unramified conditions at odd places of the third test have Euler correction zero. Consequently the cohomology on the fields has amplitude \(1,2\) by duality and exact orthogonality, with \[h=1+h^*,\] where \(h,h^*\) denote primal and exact-annihilator dual dimensions in degree one. Both degree-one spaces inject into ordinary unrestricted global \(H^1\) (dual strict uses the old dual local invariant vanishing where full old conditions are imposed). These assertions continue to hold with finitely many new line conditions in the argument.

Local lines at a switching prime. Switch at fresh good primes \(\ell=\ell_i\) tending to 1 in \(\mathbb Z_2^\times\), splitting on all base scalar data to increasing precision, with Tate Frobenius limiting to \[\tau=\begin{pmatrix}1&e\\0&1\end{pmatrix},\qquad e\ne0,\] deep in an open-image special linear subgroup. Here \(i\) runs over stages of the relevant limit (internal stages in the second and third tests). Inertia acts trivially. On the limit DVR the local complex is split free on cohomology in \(0,1,2\); degree one has axes the finite and singular lines \[f:\ M_{\mathscr O}/(\tau-1)M_{\mathscr O},\qquad s:\ \ker(\tau-1\mid M_{\mathscr O}),\] by evaluation on a Frobenius lift and a tame inertia generator respectively; \(M_{\mathscr O}\) denotes the coefficient there. Indeed \(e\) is a unit, and one uses the unramified and singular blocks from inertia/residue cochains (the conjugation exponent tends to 1). In particular the pure singular evaluation class sending inertia to the invariant basis and Frobenius to zero can already be computed to growing precisions with \(\ell\equiv1\), Frobenius action \(\equiv\tau\). The two pure lines, taking in addition all \(H^0\) but not \(H^2\), are self-annihilating against their respective opposite versions under the identification by the Weil form (inverse scalar there, also trivial locally in the limit). For the finite line use inflation isotropy; for the pure singular line the just-described cocycles take values along the same Weil-isotropic invariant line, giving the limiting cup zero also. Perfectness then follows by local duality (the mixed pairing is perfect); these are exact orthogonal complex conditions since local cohomology splits, using nullhomotopies as in Section 3. Use likewise lower taking only degree zero and upper taking both coordinates in addition, exact opposites. The finite condition identifies by inflation with proceeding without allowing this prime. Each intermediate line condition has Euler correction zero and takes in all \(H^0\).

Selection of the switching primes. For \(\mathcal G=\prod_i G_\mathbb Q\) take the product \(\mathcal H\) of the exact kernels of Tate action together with metabelian data \(\mathcal E_i\), containing \(\mathbb Q^{\rm ab}\) and the \(2\)-power roots of all old derivative primes (it suffices to take their composite). The scalar actions in use factor through those data. The image quotient contains as a normal subgroup arbitrary sequences of a fixed deep \({\rm SL}_2^b(\mathbb Z_2)\) in the Tate action alone, trivial on \(\mathcal E_i\), by open image for the non-CM curve and closed iterated commutators. Choose \(\tau\) as above in that subgroup (using its constant sequence). Thus by the product-group evaluation and diagonal-vanishing argument in Section 3, global \(H^0\) vanishes on all fields in question; restrictions to \(\mathcal H\) detect crossed global classes. On \(\mathcal H\) these evaluations are additive, and for a nonzero class the evaluations span the plane by equivariance and absolute irreducibility. These statements apply separately to primal and dual, whose Selmer degree-one classes inject as above.

If on the fiber \(h>1\) with a sole generic line already established, choose the nonzero fiber class from a primitive reduction of that line, and some nonzero degree-one dual class. In a generic-only surplus-rank test below instead take a specified nonzero primal class and some nonzero dual class. In a product-group coset of \(\mathcal H\) having action \(\tau\) and trivial metabelian action, we can simultaneously make both evaluations nonzero modulo \((\tau-1)\): each is a proper affine avoidance test on the additive evaluations, possible jointly by integer combinations in characteristic zero. Coboundary adjustments do not affect the tests. We require Frobenius agreement with the resulting sequence on the old diagram evaluations to growing precisions.

The depth margin for descent. Keep a full finite abelian base \(K_0=\mathbb Q(\mu_{M_i})\) per stage for the Kato constructions with \(2\)-part sufficiently deep for cofinal base scalar precisions, fixed before adding derivative primes. This contains any fixed scalar fields needed for the weighted trace and uses the chosen old omitted support as specified for each system (in square order include the moving primes just once).

Within the necessary termwise finite open constraints one can still keep \(\mathcal E_i\)-action exactly trivial while perturbing \(\tau\) by arbitrarily deep special linear elements so \(v_2\det(1-\rho_T)\) is large but finite. Indeed intersecting an open subgroup with that metabelian kernel still gives open image inside a deep special linear subgroup; one may perturb in the lower unipotent. Then choose \(A_i\) much larger still. Chebotarev gives \(\ell_i\) with agreements to the requisite finite precisions, splitting in \(K_0\) and in all previously chosen derivative fields, with \(\ell_i\equiv1\bmod 2^{A_i}\), and with \[v_2\det(1-\rho_T({\rm Fr}_{\ell_i}))\longrightarrow\infty,\qquad A_i-v_2\det(1-\rho_T({\rm Fr}_{\ell_i}))\longrightarrow\infty .\]

Impose also splitting on \(2^{A_i}\)-th roots of all previous derivative primes. Thus those primes split in the new degree \(s_\ell=2^{A_i}\) subfield of \(\mathbb Q(\mu_{\ell_i})\) (the new derivative field). Exclude old places, primes of smoothing integers and all level primes already present. This realizes the two nonzero tests by choosing the place representatives (cf. Chebotarev on evaluation matrices in Section 3). A Frobenius lift trivial also in the new derivative field can be used, by inertia adjustment invisible on the old data needed.

Further additions keep previous choices and diagram comparisons, reducing to slower cofinal precisions if needed. Splitting in \(K_0\) and the mutual splitting at all other derivative primes here are exact at each stage, not just limiting trace conditions.

Derivative classes and descent. For the square or rectangular Kato system being used, keep its base orders/mark/symbols/smoothings and add each derivative prime once to both orders and the determinant index. For \(I\) a subset of such primes at the current stage let \(K_I/K_0\) be the composite with their derivative fields. Trace first to \(K_I\), with \(T\)-coefficients denoted throughout by their identification there (so transport by the fixed quadratic twist is allowed, trivial as a character over \(K_0\)). Let \(c_I\) denote the integral resulting class, using [eq:K1]. Write \(\sigma_\ell\) for a tame inertia representative mapping to a generator on its own cyclic derivative factor and trivial on all the other factors. Consider \[D_I^{\rm der} c_I,\qquad D_I^{\rm der}=\prod_{\ell\in I} D^{\rm der}_\ell,\qquad D^{\rm der}_\ell=\sum_{j=1}^{s_\ell-1}j\sigma_\ell^j .\] The good Euler norm of Proposition 23, pushed by trace to these fields, gives norm down at \(\ell\) with scalar \(p_\ell=\ell^{-1}\det(1-\rho_T({\rm Fr}_\ell))\); the projection of the field translation is trivial there by the splitting. This is integral equality after a uniform fixed nonzero multiple \(L_0\). Indeed \(H^0((T\otimes\mathbb Q_2)/T)\) over all these abelian fields has bounded exponent by the commutator image, which bounds the torsion ambiguity in \(H^1(T)\); inflation into unrestricted cohomology is injective in degree one. All norm calculations can thus be made with the required enlarged supports also.

Work modulo \(2^n\), \(n\to\infty\) sufficiently slowly that \(s_\ell,\ p_\ell\equiv0\bmod 2^n\) and \(A_i>n\) for all current factors (using each one’s chosen depth). The identities \((\sigma_\ell-1)D_\ell^{\rm der}=s_\ell-N_\ell\), \(N_\ell\) the cyclic sum, then show \(L_0 D_I^{\rm der}c_I\) invariant modulo \(2^n\).

Multiply by a further fixed integer \(L_1\ne0\) for descent by inflation-restriction: the obstruction in degree two of the finite group with coefficients \((T/2^n)^{G_{K_I}}\), and similarly inflation ambiguity, are killed by a bounded power of 2 by the same open-image argument. This applies on unramified-outside-allowed-support groups (including the derivative ramification). Write \(L_*=L_0L_1\), the same for all subsets.

Descend \(L_*D_I^{\rm der}c_I\) modulo \(2^n\) to \(K_0\), then weighted corestrict from \(K_0\) to the global deformation coefficients (trivialize the scalar over \(K_0\) at the working precisions). Taking limits gives integral ordinary classes \(Z_I\) with \(Z_\varnothing=L_* z_{\rm sys}\), where \(z_{\rm sys}\) is the original system’s base class. Here for the empty set we simply use the original representative; compatibility with the true \(t\)-tower uses the \(2\)-power compatibilities of [eq:K1].

Choices need not give an Euler system of descents in \(t\): reduce as needed to cofinal Artin quotients, taking representatives to cycles on the models there and matrix limits. This preserves any previously chosen lower-subset classes through inflation. The weighted corestriction has no division by the group order.

The finite–singular comparison. We give the local comparison, allowing the moduli \(2^n\) to be decreased cofinally. At \(\ell\in I\) take integral cocycles \(b,b_\ell\) on \(G_{K_{I-\{\ell\}}},G_{K_I}\) respectively, for lower and upper classes after applying \(L_*\) and all other derivatives. Write \(\sigma=\sigma_\ell,\ m=s_\ell,\ \Phi={\rm Fr}_\ell\) a lift trivial on \(K_I\), and \(F=\rho_T(\Phi)\).

Both cocycles vanish on the respective local inertias: inertia acts trivially on \(T\), and even over the upper cyclic ramification \(H^1(I_{\rm loc},T)^\Phi=0\) by good reduction and Weil (no eigenvalue \(\ell\); the residue degree here is one). This applies at conjugate places as well. Put \(x=b(\Phi), x_\ell=b_\ell(\Phi)\). On \(G_{K_I}\) the exact norm equality gives, for some \(y\in T\), \[\sum_{j=0}^{m-1}(\sigma^j b_\ell)(u)=p_\ell b(u)+(u-1)y,\qquad (F-1)y=m x_\ell-p_\ell x.\] Here a translate conjugates the argument and acts on the coefficient. All the translates in the sum have the same value on \(\Phi\), by the inertia vanishing and trivial inertia action. Also \(\sigma^m b_\ell=b_\ell\) as cocycles by the same vanishing on \(\sigma^m\).

Let \(B_I\) be the mod-\(2^n\) descended cocycle before weighted corestriction. On the upper subgroup adjust by a coboundary (adjusting also on the whole group of \(K_0\)) to make its restriction equal to \(D_\ell^{\rm der}b_\ell\). Its value on \(\Phi\) after adjustment is zero by \(m(m-1)/2\equiv0\). Applying \(\sigma-1\) on these restrictions gives, for \(u\in G_{K_I}\), \[(u-1)B_I(\sigma)=-(u-1)y\bmod 2^n,\] using the crossed identity, the norm calculation and \(\sigma^m b_\ell=b_\ell\). Thus \(B_I(\sigma)\equiv-y\) losing at most bounded \(2\)-power precision by the invariant bound, independently of that coboundary adjustment. Now \[y=m(F-1)^{-1}x_\ell-\ell^{-1}{\rm adj}(F-1)x .\] The first term vanishes in the limit by the margin on the determinant valuation. And \(x\) agrees, up to a coboundary coordinate \((F-1)w\) at the common precision, with \(B_{I-\{\ell\}}(\Phi)\). Such changes disappear after applying \({\rm adj}(F-1)\) in the limit.

Upon restriction of weighted corestriction to \(G_{K_0}\), one obtains the sum of translates with weights. On each translate the same calculation applies with the same local representatives: the translated lower/upper classes still have the derivative and norm identities in cohomology (all factor actions commute there by abelianness), and inertia vanishing holds. Hence after summation and limit \[f(Z_I)=0,\qquad s(Z_I)={\rm adj}(\tau-1) f(Z_{I-\{\ell\}}) \quad\text{at }\ell . \tag{L3}\] All congruences here concern singular evaluations with trivial coefficient inertia and finite evaluations modulo Frobenius coboundaries. They therefore carry to the cohomology maps on the limiting models, then to the DVR or test field. The verification can likewise be made at each old \(\ell\in I\) separately to keep the pure singular conditions there.

The line map in [eq:L3] is a unit isomorphism over \(\mathscr O\) (off-diagonal \(\pm e\)). The \(Z_I\) lift in cohomology over the DVR with the prescribed pure conditions: degree zero is included and degree-one coordinates have the prescribed image. In the third test their old odd-place lifts to unramified conditions before residue-field specialization are automatic by zero \(H^1\) of the singular quotient over the DVR. The same facts apply using just generic fields.

Propagation of determinant divisibility. If in a generic test \(z_{\rm sys}\ne0\), there is no surplus \(h>1\) there: evaluate it and an old nonzero dual class as above at a first fresh prime; by [eq:L3] the derivative is then nonzero transverse, contradicting local cup reciprocity with that dual class (whose finite localization there is nonzero).

In a DVR test with the single generic line and nonzero class established, proceed successively if \(h>1\) on the fiber. At each new prime start with unramified inflation, arrange the two evaluations as above on the fiber (so also nonzero generic primal evaluation), and switch by the Line switch comparison. This drops \(h\) by one preserving generic nonvanishing of the class by [eq:L3]. The ordinary determinant valuations are exactly equal along the comparisons with the new derivatives; there is no bound lost at this DVR.

When \(h=1\) the terminal complex is a free line in degree one up to homotopy and the terminal class is integral. Since \(L_*\) is a unit here, backwards we obtain \[d(C_{\rm old},z_{\rm sys})\ \ge\ 0. \tag{L4}\] Here \(C_{\rm old}\) is the initial ordinary problem in the switch sequence, with the degree-one lift when unramified conditions are imposed. The clearing factor \(L_*\) is a fixed power of two up to a unit, and is a unit at each DVR used here. Consequently the equality of determinant valuations in the switches gives exactly [eq:L4]. ◻

Proof. Apply first the fixed-support \(t\)-test with square order and \(z_{\rm sys}=z\) there. Its ordinary class is generically nonzero: otherwise a nonzero bounded series annihilates it and it specializes to zero rationally off finitely many finite-order characters. This contradicts [eq:K4] and cyclotomic twist nonvanishing [50], in the form of [30], at high characters, including with the fixed quadratic twist when in use. Thus the test gives \(h=1\) generically.

Now specialize the \(B\)-diagram at \(u_j=0\) for all \(j\), retaining generic \(t\). The moving places have unramified action there and acyclic singular quotient (cyclotomic Frobenius scalar tends to 1, residue sizes tend to 1 and \(\det(1-g_j)\ne0\)). Inflation thus compares isomorphically with the fixed-support problem. Moreover the augmented ordinary square class is that from fixed support times nonzero limiting Euler factors: apply the fresh good norms removing the \(r_{j,i}\) sequentially at trivial tame weight, with bounded torsion clearing as above; \(\chi_h\) if used is split there, and the factors tend to \(2-\operatorname{tr}g_j\).

Hence there is a single ordinary line with nonzero class on this specialized generic line. By deformation, or using also the positive concentration at \(B_{(2)}\), we have generically on full \(B\) just the two-dimensional degree one in \(C^+\), with \(a\wedge z\ne0\) (the specialized generic \(t\)-test has these same dimensions). This holds also for analytically unknown center.

The first, \(B\)-DVR test now gives [eq:L4] for the ordinary full complex there, hence nonnegative order for [eq:L1] away from \((2)\) and the indicated fixed factors. ◻

Proof. Finite-character tests.

Represent \(U=g_0/f_0\) with integral series \(f_0,g_0\), \(f_0\ne0\), by determinant pivot formulas from the positive diagram, incorporating the factors of [eq:L1]. Retain their stage expressions \(g_{0,i},f_{0,i}\) before passage from the actual \(t\)-towers with finite tame orders. These use the integral smoothed lifts and basis choices; they compute the indicated rational coordinate when the pivots work with the same ranks.

Test at \(u_j=\eta_j-1\) for finite \(2\)-power roots \(\eta_j\ne1\), requiring \(f_0(\cdot,\boldsymbol\eta-1)\ne0\) and dual nonexceptionality on the Frobenius lift at 2 as in the third switching test. The latter is a nonidentical vanishing avoidance condition by the nonzero limiting tame-scalar exponent vector there. These tests still detect nonzero bounded series conditions on the tame variables by iterated one-variable Weierstrass. At each such test we have all the required finite characters and 2-local constant-scalar agreement at sufficiently precise original stages on the filter.

A rectangular system with the primitive odd factors. Fix one such stage for the third test. To retain the precise primitive odd factors use [eq:K1], [eq:K4] in the rectangular first-row-unfilled case. Take \(b=N_{\rm odd}\) and \(M\) supported only at 2 and the odd ramified places of the constant field weight (all active variable conductors and any active fixed quadratic ramification). These are coprime to \(b\); use sufficiently high \(2\)-part and a true \(2\)-power tower. Choose \(A=2^k,\xi\) with \(\delta_\xi(a)\ne0\) as in Lemma 25.

At any given height-one \(\mathfrak q\) of \(\Lambda_\eta\) not over 2, take smoothing \(c',d'\) whose factors \(\Delta'\) are units there. Indeed \(t\) at such a divisor can be realized at an algebraic point of the open unit disk by preparation. Take a CRT sequence tending dyadically to 1, congruent to 1 on required fixed symbol orders, respecting coprimalities and realizing a fixed nontrivial constant weight on an active variable conductor (values trivial on all others and any fixed quadratic factor). This uses the disjointness of the active variable conductors from those fixed levels. The \(t\)-scalar tends to 1; thus the vanishing equation in the smoothing is avoided.

Keep the choices fixed while switching. Write \(z'\) for this system. It has no extra odd imprimitive losses at the base finite cyclotomic characters: [eq:K4] gives the law with \(L^{(2)}\) since all omitted odd places are at ramified scalar characters of good-\(f\) primes. In particular its generic class is nonzero by the same cyclotomic nonvanishing theorem. The rectangular good Euler norm is exactly of the requisite kind. Thus generic single-line concentration for the unramified-old-odd problem, and [eq:L4] with \(z'\) at \(\mathfrak q\), follow by the third switching test. The same generic concentration holds on relaxing to full, since old odd singular quotient differentials have nonzero determinants.

Comparison of the two classes. In this common generic full cohomology line the original square system differs from \(z'\) by \[\frac{\Delta\,\prod_{q\in S_f,\ q\ne2}P_q}{\Delta'}\] times a nonzero constant, using the stage specializations for these factors. Indeed evaluate at infinitely many high \(t\)-characters with central nonvanishing, using [eq:K4] in the same representation with compatible weighted realizations; any geometric quadratic identification is common. The two exponential coordinates have precisely the displayed ratio times the fixed symbol ratio, up to signs. Odd omissions at the active variable conductors make no change. Generic ratios specialize in ordinary cohomology off at most finitely many such values, by clearing denominators, so the identity follows by one-variable Weierstrass (one can square to ignore signs). In particular the square raw coordinate uses a single generic ordinary line also on the tested stages.

On relaxation from the unramified-old-odd complex to full, determinant valuation changes for \(z'\) by the sum of \(d(Q_q,1)=-v_{\mathfrak q}(D_{q,\eta})\) for those singular quotients. These determinants are 1 at active variable conductors, and are precisely the specialized stage \(D_q\)’s at fixed odd places.

Consequently [eq:L4], the ratio comparison, and the ordinary real basis give \[\frac{g_{0,i}(\cdot,\boldsymbol\eta-1)} {f_{0,i}(\cdot,\boldsymbol\eta-1)} \ \in\ \mathbb Z_2[\eta][[t]][1/2].\] Here we use all the horizontal height-one primes, allowing \(z'\) or its smoothing choices to vary among them. On large stages the chosen pivots indeed have nonzero denominator generically on the test, by coefficientwise convergence and \(f_0(\cdot,\boldsymbol\eta-1)\ne0\), and ranks agree as just proved. No bound on a comparison constant or uniform \(2\)-denominator is being asserted, and this step uses no separate primitive-form characteristic-ideal divisibility theorem.

Integral Weierstrass division. The remaining possible prime divisors of poles divide \(\Delta,D_q,P_q\) at fixed odd corrections (or the smoothing alone); every such factor is regular in \(t\) over \(\mathbb Z_2[[\mathbf u]]\) as observed after [eq:L1].

For each irreducible such factor \(H_0\) use its distinguished representative, and \(H_0^a\mid f_0\) with \(a=v_{H_0}(f_0)\). At the above tests it stays distinguished. All its tested zeros to that multiplicity belong to \(g_0(\cdot,\boldsymbol\eta-1)\), by the integral numerator/denominator zero-persistence test in Section 3 (convergence on smaller disks and the stage containment).

Thus the integral division remainder of \(g_0\) by \(H_0^a\) in \(t\) vanishes at every such test, hence identically. This rules out the remaining horizontal poles. ◻

Proof of Proposition 27. Lemma 29 proves concentration and nonnegative order at \((2)\). Corollary 31 proves generic nonvanishing and the required divisibility away from the fixed factors, and Lemma 32 treats those factors. The ring \(B\) is normal, so these height-one tests give \(U\in B\). Generic nonvanishing gives \(U\ne0\). Finally, the common raw reduction of Lemma 29, the common smoothing reduction, and Lemma 28 show that the two integral reductions differ by a unit. They therefore have the same closed-point unit status, as asserted. ◻

Proof. Compact and discrete Selmer lattices.

For the central calculation first work at the actual trivial-parameter specialization at stages, for the curve of the branch, denoted temporarily by \(A_0\). Put \(S'=S\cup R_i\), \(O^j=H^j(G_{\mathbb Q,S'},T)\), \(R^j_\mathbb R=H^j(\mathbb R,T)\). Suppose \(s_2(A_0)=0\), which in particular holds for analytic rank zero.

Put \(s_0=\operatorname{length}\mathop{\mathrm{Sha}}(A_0/\mathbb Q)[2^\infty]<\infty\), and write \(\tau_g,\tau_v\) for lengths of global and finite local \(2\)-primary point torsion. Lengths here are over \(\mathbb Z_2\). The usual discrete Selmer has length \(s_0\). At compact coefficients the singular quotient by Kummer of local degree one is zero at odd and real places, and at 2 it is \(J_2\) free of rank one, dual to compact Kummer modulo torsion.

With \(j_2=\operatorname{length}(J_2/\operatorname{im}O^1)\), integral Poitou–Tate gives \[0\le j_2\le s_0,\qquad \operatorname{length}O^2 =s_0-j_2+\operatorname{length} R^2_\mathbb R +\sum_{v\in S'_f}\tau_v-\tau_g .\] Indeed the exact sequence goes from \(O^1\) to the local singular quotient, the dual of discrete Selmer, \(O^2\), the sum of local degree-two groups and the dual of global point torsion, ending surjectively. This uses modified real localization in Poitou–Tate (agreeing with ordinary real terms in those positive degrees), finite Kummer duality and passage to compact/discrete limits. At odd good places outside support unramified discrete \(H^1\) is already zero, giving the usual Selmer.

Also \(O^0=0,\ \operatorname{rank}O^1=1,\ \operatorname{length}O^1_{\rm tors}=\tau_g\), by the Euler and invariant formulas.

The determinant and local volume factors. Assume first that the analytic rank is zero. Then \(z\) has nonzero singular localization by [eq:K4]; write \(z\) also for its ordinary image. The positive complex with the integral real boundary basis and lift then gives raw valuation \[ \begin{aligned} d(C^+,a\wedge z) &=\operatorname{length}(O^1/\mathbb Z_2 z) -\operatorname{length}O^2-\operatorname{length}R^1_\mathbb R +\operatorname{length}R^2_\mathbb R\\ &=v_2(z,J_2)+2\tau_g-s_0-\sum_{v\in S'_f}\tau_v -\operatorname{length}R^1_\mathbb R , \end{aligned} \tag{1}\] where \(v_2(z,J_2)\) measures the singular image in that lattice. The first equality uses the long exact positive sequence through degree three: dividing degree one by the image of the real basis gives the kernel to \(R^1_\mathbb R\), and positive degree three is the cokernel to \(R^2_\mathbb R\). The second uses \(\operatorname{length}(O^1/\mathbb Z_2z)=\tau_g+v_2(z,J_2)-j_2\).

By the differential/log duality and Haar computation of the central lattices, for minimal \(\omega\), \(\exp^*z=\alpha\omega,\ \log_\omega A_0(\mathbb Q_2)=2^b\mathbb Z_2\), we have \(v_2(z,J_2)=v_2(\alpha)+b\), and \[\tau_2-b=v_2(c_2 L_2(A_0,1)^{-1}),\qquad \tau_q=v_2(c_q L_q(A_0,1)^{-1})\quad(q\ne2).\] The \(R^1_\mathbb R\) length inserts the real component factor. Thus the trivial-parameter law of [eq:K4] on the indicated curve (\(\Delta(0)\), omitted support \(S'_f\), connected period, evaluation 1) gives exactly \(X(A_0)+v_2\Delta(0)\) for the raw valuation.

Specialization of the limit determinant. These ranks and the determinant calculation pass to the limit at the center. Indeed the moving finite torsion lengths stay bounded by the trace-limit conditions. The formulas above bound \(O^2\), hence also positive degrees two and three; torsion in positive degree one injects into ordinary torsion. Pivots of the same ranks therefore work on the limiting specialization, allowing comparison of determinant and cycle entries there by bounded-denominator Gaussian formulas. Thus the raw valuation is unchanged in analytic rank zero, and the fixed \(D_q/P_q\) have central ratio 1, proving [eq:L5].

An initially unknown analytic center. In particular \(X(A_0)\ge0\) in analytic rank zero. If only \(s_2(A_0)=0\) is known but \(U\) is a unit, the same torsion bounds and ranks apply regardless of analytic nonvanishing. If \(L(A_0,1)=0\), then \(z\) has trivial rational singular image by [eq:K4], hence is ordinary torsion at the stages, so \(a\wedge z=0\) rationally there and also at the limiting center, a contradiction (all fixed corrections and smoothing have nonzero central values). Thus \(L(A_0,1)\ne0\); now [eq:L5] applies and gives \(X(A_0)=0\). ◻

Proof. After rescaling \(x,\beta\) by one common dyadic integer and \(k\) by its square, compactness makes all three cochains integral. Their fixed-support condition makes them unramified at the moving primes.

The Tate image and the additive evaluations. The quotient defining \(L/F\) has bounded derived length. The open-image theorem and iterated closed commutators therefore leave a deep subgroup of \(\mathrm{SL}_2(\mathbb Z_2)\) in the image of \(G_L\): the corresponding iterated Lie brackets contain \(\mathfrak{sl}_2\).

Restriction of \([x]\) to \(G_F\) is nonzero, since restriction followed by corestriction is multiplication by \([F:\mathbb Q]\). Its restriction to \(G_{F(T)}\) is still nonzero: otherwise it would inflate from the open Tate image, whose first cohomology on \(V\) vanishes by a central homothety. On \(G_{F(T)}\) its values are additive and equivariant, so irreducibility makes their span the full Tate plane.

The same holds on \(G_{L(T)}\). Indeed, if the restriction there were zero, those additive evaluations would factor through \(\operatorname{Gal}(L(T)/F(T))\). Since \(F(T)\) contains all dyadic roots of unity, this radical quotient has cyclotomic conjugation. A deep special-linear subgroup consequently acts trivially on that quotient, whereas it has no trivial quotient on the standard plane. This contradicts the full span just obtained. Irreducibility again makes the restricted span the full plane.

The scalar detected by the direction. Choose \(g\in G_L\) with the asserted Tate action, for example in a deep split torus. On the joint Tate kernel coboundaries vanish, so \(\beta\) agrees with \(\operatorname{pol}(x)\). The cochain identity gives, up to the fixed sign, \[k(h_1h_2)=k(h_1)+k(h_2)+\langle x(h_1),x(h_2)\rangle.\] A commutator there has \(x=\beta=0\) and \(k\)-value \(\pm2\langle x(h_1),x(h_2)\rangle\). The full span supplies a nonzero such value. Multiplying \(g\) by this commutator changes \(\eta\) without changing its Tate action or its \(x,\beta\) values. This gives (5). Only nonvanishing is used; the factor \(2\) is retained.

Frobenius and residue characters. Chebotarev gives primes \(r_i\) and Frobenius lifts \(g_i\) approaching \(g\) on all the fixed data, with roots and radicals fixed to depths sufficient for \(m_i\). Choose \(r_i\equiv1\pmod{2^{m_i+1}}\). In compatible primitive-root bases define \[ a^{(r_i-1)/2^{m_i}}\equiv \zeta_{2^{m_i}}^{\lambda_i(a)}\pmod{r_i} \qquad(r_i\nmid a). \tag{6}\] These residue exponents are surjective and define even characters. Fixing the prescribed radicals makes them vanish on \(\mathcal P\), and the Frobenius conditions give all the remaining horizontal data. ◻

Proof. Write \(y=\operatorname{pol}(P)\). Let \(G_y\) be the Barsotti–Weil extension of \(E\) by \(\mathbb G_m\) defined by the Poincaré biextension at \(y\), and choose \(B\in G_y(\mathbb Q)\) above \(P\). Such a lift exists because its fiber is a line torsor over \(\mathbb Q\). Choose a splitting of the underlying modules \(T_2G_y=\mathbb Z_2(1)\oplus T\); its Galois matrix has off-diagonal block \(\beta(g)g_T\), where \(\beta\in Z^1(T^\vee(1))\). This is, up to the fixed sign, the Kummer class of \(y\). To see the identification integrally, at order \(b\) the torsor of division points \(by'=y\) maps to the torsor of splittings by pushing out \(G_{y'}[b]\) along the \(b\)-power map on the kernel. Changing \(y'\) by \(b\)-torsion changes the splitting by its Weil pairing. These identifications commute with reduction.

The Kummer cocycle of \(B\) therefore has coordinates \((k,x)\), with \(x\) the Kummer class of \(P\), and \[ dk=-\beta\cup x. \tag{7}\] The data and representatives are unramified outside fixed support, by good integral models and the étaleness of division by \(2\) away from that support. The class of \(x\) is nonzero because \(P\) is nontorsion. Lemma 37 supplies horizontal data with \(\eta\ne0\). The height construction in the next lemma identifies \(h_\lambda^\mathbb Q(P,P)=\pm\eta\), proving the assertion. ◻

Proof. At either split place of \(r\), a local unit whose image dies in the ring class quotient must be a diagonal congruence modulo \(r^j\), up to the global units \(\{\pm1\}\). This follows directly from the idele description for the order \(\mathbb Z+r^jO\). Local Artin reciprocity and norm show that the even character becomes unramified over \(H_{r^j}\), and hence over \(F'\). It was already unramified elsewhere. Bad reduction and fixed model primes have character zero.

Choose any rational lift in the Poincaré biextension at the two arguments, with polarization on the second. Sum the local character valuations of this lift relative to the rigidified integral fibers at good places. Reciprocity makes the sum independent of the lift. The two integral biextension laws give bilinearity and adjoint compatibilities. On a relative Jacobian, degree-zero horizontal divisors with disjoint generic support compute this valuation, up to the common sign, by their intersection lengths. Indeed the Poincaré line under self-duality is the Deligne pairing, or its inverse according to orientation, and the change of its symbol lattice is exactly the intersection. Adding a whole vertical fiber has zero pairing with a degree-zero divisor. Only models outside the fixed killed support are needed.

For the downstairs heights, observe that \[\#E(\mathbb F_{r_i})=r_i+1-a_f(r_i)\longrightarrow2-a_*\ne0.\] Its two-adic valuation is bounded. Choose \(2^b\) which removes all these finite two-primary parts, so the scaled first argument \(X\) lies in the uniquely \(2\)-divisible factor of the local points. For a lift \(\mathcal B\) at \((X,Y)\), use at \(r_i\) the character of the scalar \[\mathcal B/([2^{m_i}]_1\mathcal B'),\] where \(\mathcal B'\) is any lift at \(([2^{m_i}]^{-1}X,Y)\), and division is in that factor. The biextension law makes this independent of the choice and bilinear modulo \(2^{m_i}\). If the normed character becomes unramified in an extension, compare with integral lifts to recover the valuation definition there. Division in the first argument can be performed downstairs; norming the lifts and scalar ratios, using the second biextension law, proves compatibility with trace in the second argument. At other places this is integral trace compatibility. Taking limits and dividing by \(2^b\) defines the rational heights. The field degree gives the stated factor \(2\) on rational arguments.

For the final assertion choose the fixed lift \(B\) above \(P\), scaled in the first direction. It is integral away from killed places, so only \(r_i\) contributes. Put \[w_i=(g_{i,T}-1)^{-1}x(g_i).\] Unramifiedness gives \(dw_i=x\) locally; enlarge \(b\) so \(2^bw_i\) is integral for every \(i\). Subtracting the coboundary of \((0,2^bw_i)\) from the Kummer cocycle of \(B\) produces invariant division points of the scaled first argument, in its uniquely divisible factor. Its remaining fiber coordinate is the unramified cocycle \(2^b(k-\beta\cup w_i)\). More concretely, a corrected division point differs from any rational lift above the divided base point by a root of the scalar ratio defining the height. Its Frobenius value in the root basis of (6) is precisely the residue character, up to the fixed sign; the valuation part of unramified Kummer is divisible by the root order. Passing to the limit and dividing by \(2^b\) gives \(h_\lambda^\mathbb Q(P,P)=\pm\eta\). ◻

Proof. In analytic rank one, the classical theorem gives rational rank one and finite \(\mathop{\mathrm{Sha}}\). The base dual exponential of \(z\) is zero by [eq:K4]. At odd finite places the rational \(H^1\) of the Tate coefficients is zero, by vanishing of invariants and dual invariants and the local Euler characteristic. Thus \(z\) lies in the rational Selmer line. Also \(\log_\omega P\ne0\), since the kernel of the local logarithm consists of torsion.

Work modulo \(2^{m_i}\) and \(u^2\), and write a cocycle for \(z_i\) as \(z_0+uz_1\). Then \[dz_1=-\lambda_i\cup z_0.\] Use the opposite sign throughout for the inverse scalar action. After one fixed common multiplier, the rational norm relation and Selmer line allow a coboundary adjustment with \[z_0=\alpha_i\beta,\qquad \alpha_i=\pm e_r\,\frac{\log_\omega z}{\log_\omega P}.\] The integral \(H^1\)-torsion in this adjustment has bounded exponent by global torsion finiteness. Suppressing that common multiplier, the root-coefficient two-cocycle \[z_1\cup x+\alpha_i\lambda_i\cup k\] is closed by (7). Its local invariant at \(2\) tends to \(Z'\log_\omega P\), by logarithm–dual-exponential local duality. Here localization has split ordinary group coefficients before differentiation, so all pairings are made on ordinary lifts with only fixed denominators. At the other fixed places the character is zero and the residual contribution has bounded torsion order; away from the support and \(r_i\), the cup is unramified.

At \(r_i\), the trivialization \(dw_i=x\) replaces this cocycle in cohomology by \(\alpha_i\lambda_i\cup(k-\beta\cup w_i)\). Local cup reciprocity identifies its invariant, up to sign, with \(\alpha_i\) times the fiber character computed in Lemma 39. Global reciprocity, followed by the limit and \(e_r\to2-a_*\), proves [eq:T2]. ◻

Proof. Begin with the row sum in its domain of absolute convergence. Row transformation gives the character opposite to that of \(\theta_\chi\). At every integral cusp transform, the row weights are periodic and odd under simultaneous change of sign. The \(c_1=0\) sum is regular at \(s=0\) by oddness and Dirichlet continuation. For \(c_1>0\), combine the two signs and apply Poisson summation to the second variable.

The zero frequency is a periodic Dirichlet sum at \(2s\), times \(y^{-s}\) and an integral factor regular at zero; the odd part of that integral cancels. At a nonzero frequency the relevant transform is that of \[(u_0+ic_1y)^{-1}|u_0+ic_1y|^{-2s}.\] After rescaling, shift to the lines \(\operatorname{Im}(u_0/(c_1y))=\pm1/2\) and integrate by parts. The differentiated tails are absolutely convergent and give exponential decay in \(c_1y\) times the absolute frequency, uniformly near \(s=0\). At zero only positive frequencies survive, by the pole in the lower half-plane. The resulting constants and positive modes are holomorphic. This reasoning also justifies continuation against cusp tests, as required in the unfolding below. ◻

Proof. The local theta transform. The dual of the lattice \((C_0,Q_C)\) is \((1/\sqrt{-D})C_0\). For \(p\mid D\), write \(d=D/p\). The \(p\)-primary discriminant subgroup is represented by \(x=dy/\sqrt{-D}\), \(y\in C_0\). Since \(C_0\) is prime to \(p\), conjugation is the identity on its residue field, and \(\mathrm Ny\equiv t^2\pmod p\) for the residue \(t\) of \(y\). Its quadratic form is therefore \[Q_C(x)=\frac{d\,\mathrm Ny}{pa_C},\qquad q_p t^2/p,\quad q_p\equiv d/a_C\pmod p.\] The square class of \(q_p\) is that of \((a_Cd)^{-1}\).

Let \(S_0=\left(\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\right)\) and \(T_0=\left(\begin{smallmatrix}1&1\\0&1\end{smallmatrix}\right)\). Choose \(a\equiv-1\pmod p\) and \(a\equiv0\pmod{(D/p)Nr^2}\). The word \(T_0^aS_0T_0^aS_0^{-1}T_0^a\) is \(S_0\) modulo \(p\) and identity on the other components. The finite theta transform has \[T_{x,y}=\delta_{x,y}\zeta^{q_px^2},\qquad F_s(x,y)=p^{-1/2}\zeta^{s2q_pxy},\qquad \zeta=e(1/p),\ s=\pm1.\] The scalar in the \(S_0\)-operator cancels against its inverse. Thus the word on this component is \(T^{-1}F_sT^{-1}F_{-s}T^{-1}\), whose \((x,z)\)-entry is \[\frac1p\sum_y \zeta^{-q_p(x^2+y^2+z^2)+s2q_py(x-z)} =\frac1p\sum_y\zeta^{-q_py^2}\zeta^{-2q_pxz}.\] Completing the square proves the equality. In particular its zero row and zero column are constant. Counting squares in \(\mathbb F_p\) gives the exact scalar \[ \kappa_p=\frac1p\sum_y e(-q_py^2/p) =p^{-1}\tau_p\epsilon_p(-a_CD/p). \tag{12}\] The other primary components are unchanged. Hence the partial \(S_0\)-branch replaces the local lattice by \(\mathfrak p^{-1}C_0\), with scalar \(\kappa_p\).

Trace representatives and character weights. For a subset \(J\) of the primes dividing \(D\), put \(D_J=\prod_{p\in J}p\), \(D_I=D/D_J\). The partial words just constructed give a representative \(\gamma_J\in\Gamma_0(Nr^2)\) which is \(S_0\) at the primes of \(J\) and identity at the others. Its branch of the trace has representatives \(\gamma_JT_0^j\), where \(j\) runs through the residues modulo \(D_J\) and is zero modulo \(D_I\). The product has Fourier denominators dividing \(D_J\). Summing these translations therefore kills nonintegral indices and multiplies an integral index \(m\) by \(D_J\). This is one factor \(p\) for each partial branch.

To track the Fourier twist, write \(\gamma=\left(\begin{smallmatrix}A&B\\C_\gamma&D_\gamma\end{smallmatrix}\right)\), where \(r^2\mid C_\gamma\), and \(n(x/r)=\left(\begin{smallmatrix}1&x/r\\0&1\end{smallmatrix}\right)\). Direct multiplication gives \[n(x/r)\gamma=\gamma'n(z'/r),\qquad z'\equiv xD_\gamma A^{-1}\equiv xA^{-2}\pmod r.\] Taking \(x,z'\) divisible by \(ND\) makes \(\gamma'\) integral and \(\gamma'\equiv\gamma\pmod D\). Thus \(\gamma'\gamma^{-1}\in\Gamma(D)\) acts trivially on the untwisted theta form. Expressing \(\theta_\chi\) by translates weighted by \(\bar\chi(x)/\tau(\bar\chi)\), the coefficient of index \(U\in D^{-1}\mathbb Z\) is consequently multiplied by \[\tau(\bar\chi)^{-1}\sum_{x\bmod r}\bar\chi(x)e(Uz'/r) =\chi(U)\chi(A^{-2}).\] The representatives divisible by \(D\) make this finite sum unambiguous for the fractional index. Characters are extended by zero.

For the Eisenstein rows, old and new coordinates satisfy \((c_0,d_0)=(c_1,d_1)\gamma^{-1}\), so \(c_0\equiv c_1A^{-1}\pmod r\). The factor \(\bar\chi^2(c_0)=\bar\chi^2(c_1)\chi^2(A)\) cancels the extra theta factor. We retain \(\chi(U)\bar\chi^2(c_1)\). At an identity prime \(p\mid D\), the row has \(p\mid c_1\) and weight \(\epsilon_p(d_1)\). At a partial \(S_0\)-prime it has \(p\mid d_1\) and weight \(\epsilon_p(c_1)\). Also \(N\mid c_1\).

Poisson summation and its scalar. For \(c_1>0\), a positive frequency has \(V=c_1l/D\), \(l>0\). Poisson summation at \(s=0\) gives \(-2\pi i/D\) times the positive finite Fourier transform. Indeed the transform of \((c_1z+u_0)^{-1}\), closed clockwise in the lower half-plane, is \(-2\pi i\,e(c_1lz/D)\). CRT evaluates the finite transform as \[\prod_{\substack{p\mid D\\p\notin J}}\tau_p\epsilon_p((D/p)l) \prod_{p\in J}\epsilon_p(c_1).\] Multiplying by \(D_J\prod_{p\in J}\kappa_p\), and by \(\sqrt D/(2\pi)\), leaves \[\frac{-i}{\sqrt D} \left(\prod_{p\mid D}\tau_p\epsilon_p(D/p)\right) \prod_{\substack{p\mid D\\p\notin J}}\epsilon_p(l) \prod_{p\in J}\epsilon_p(-a_Cc_1).\] Here the local cancellation is \[\frac{p\kappa_p}{\tau_p\epsilon_p(D/p)}\epsilon_p(c_1) =\epsilon_p(-a_Cc_1).\] The parenthesized product is the primitive Gauss sum \(i\sqrt D\) of \(\varepsilon\), since \(D\equiv3\pmod4\). Thus the common scalar is exactly one.

The lattice restrictions. Put \(y_0=\sqrt{-D}\,x\), \(a=DU=\mathrm Ny_0/a_C\), \(c_1=Nk\), and \(b=kl\). Then \(a+Nb=mD\). The zero frequency \(V=0\) contributes nothing: the index \(m\) cannot be a theta norm because its \(\ell\)-valuation is odd. Also \(a=0\) would imply \(N\mid mD\), impossible since \((m,ND)=1\) and \(N>1\). At an identity prime, \(p\mid k\); the relation forces \(p\mid a\), which is exactly the extra theta condition \(y_0\in\mathfrak pC_0\). The Fourier factor requires \(p\nmid l\). At a partial prime, \(p\nmid k\), while \(l\) is unrestricted. Thus \(J=\{p\mid D:p\nmid k\}\), the restriction is \(\gcd(k,l,D)=1\), and the remaining scalar is \(W(k)\). The sole factor \(1/2\) is the original theta normalization. This proves (11).

Since \(r\mid m\), the relation \(a+Nb=mD\) makes \(a\) a unit at \(r\) exactly when \(b\) is a unit. In that case \(k,l\) are both units. Thus \(\chi(a/D)\) gives precisely the stated mask. For a divisor pair, the ratio of the local factors of \(W(l)\) and \(W(k)\) is \(\epsilon_p(-a_CN)\). If neither divisor is divisible by \(p\), use the norm congruence \(\epsilon_p(-a_CNb)=1\) to obtain the same ratio. Hence \[W(l)=-W(k),\qquad \varepsilon(-a_CN)=-1.\] The latter uses \(\varepsilon(a_C)=1\), since \(a_C\) is an ideal norm, and \(\varepsilon(N)=1\), since the level primes split. The derivative of the character weight is \(\lambda_i(U)-2\lambda_i(N)-2\lambda_i(k)\). The first two terms are independent of the divisor \(k\) and cancel by antisymmetry. This leaves \(-2\lambda_i(k)\). ◻

Proof. Strip the common power \(r^j\) from \(a,b\), writing \[a=a^\circ r^j,\quad b=b^\circ r^j,\quad h=h_0+j,\quad \min(v_ra^\circ,v_rb^\circ)=0.\] The off-\(r\) weights do not change: \(r\equiv1\pmod D\), and the two primes over \(r\) are principal, so their parts can be inserted freely in \((y_0)/C_0\). With the off-\(r\) choices fixed, there are \(1+j+v_ra^\circ\) allocations between the two split primes in the theta norm, and \(1+j+v_rb^\circ\) allocations in the divisor pair \(kl=b\). Their product gives the multiplicity in \(B_h\): \[(1+j+v_ra^\circ)(1+j+v_rb^\circ).\] If \(h_0>0\), both primitive entries are units; the resulting \(C_h\)-multiplicities, beginning at \(h_0\), are \(1,4,8,12,\ldots\), whereas the masked derivative has multiplicity one only at \(h_0\). Their second difference has the coefficients \(1,2,1\) displayed in [eq:T4]. If \(h_0=0\), the \(C_h\)-multiplicities are affine for \(h\ge1\), and there are no masked terms there. These contributions have zero second difference, proving the formula. ◻

Proof. Isolation of the \(f\)-block. Apply \(\mathcal A\) by the usual Hecke formulas on coefficients, evaluate at \(n=\ell\), and difference by \(h\mapsto h+2\). On \(I_\chi\) this uses \(U_r^h(U_r^2-1)\). A cuspidal representation of conductor exponent one at \(r\) has the signed Steinberg eigenvalue; its complementary raises have zero eigenvalue. The displayed operator annihilates both for large \(h\). Conductor exponent two has \(U_r=0\). Eisenstein series with a ramified pair at \(r\) likewise have \(U_r=0\). All remaining unwanted old types, including the holomorphic special forms of the trivial pair, are killed by \(\mathcal A\). Thus only the \(f\)-block remains. Good Hecke adjoints and multiplicity one permit its coefficients to be computed by pairings against cusp tests.

Fricke transformation and unfolding. At level \(Nr^2\) use \[f_j=r^jf(r^jz),\qquad 0\le j\le2.\] Choose integers \(d_1,d_2,d_3\) with \[W_r=\begin{pmatrix}rd_1&d_2/r\\rd_3&r\end{pmatrix},\qquad d_2,d_3\equiv0\pmod{ND},\quad d_3\equiv1\pmod r,\quad r^2d_1-d_2d_3=1.\] Such choices follow from CRT. This determinant-one normalization of the exact-divisor matrix sends \(f_2\) to \(f\) and fixes \(f_1\), by level-\(N\) modularity. Use the same matrix before the trace at level \(NDr^2\). The identity \[n(x/r)W_r=\gamma'n(z'/r),\qquad z'\equiv-x^{-1}\pmod r\] for unit \(x\), with \(x,z'\equiv0\pmod{ND}\) and integral \(\gamma'\equiv1\pmod D\), gives \[\theta_\chi\big|W_r =\frac{\tau(\chi)}{\tau(\bar\chi)}\theta_{\bar\chi}.\] On the Eisenstein rows, write \((v,w)\) for the old coordinates. Clearing \(r\) gives \[v'=r^2(vd_1+wd_3),\qquad w'=vd_2+r^2w.\] These are a bijection onto the rows with \(NDr^2\mid v'\): the inverse formulas are \[v=v'-d_3w',\qquad w=d_1w'-d_2v'/r^2.\] The new weights are \(\varepsilon(w')\bar\chi^2(w')\), and the scalar is \(r^{1+2s}\).

Group the row greatest common divisors \(b_1\), which are prime to \(Dr\). The primitive row sum has level \(N_1Dr^2\), where \(N_1=N/\gcd(N,b_1)\). The terms with \(N_1<N\) pair to zero by newness of \(f\). This is also true against \(f_1\): tracing down at the \(N\)-factor in either degeneracy gives the corresponding zero trace. The extra levels are coprime to \(N\), so CRT separates the cosets; conjugation by \(\operatorname{diag}(r,1)\) for \(f_1\) is integral on those representatives and permutes the same \(N\)-coset data.

The remaining gcd factor is \(L^{(N)}(\varepsilon\bar\chi^2,1+2s)\). Unfold the primitive rows on \(\Gamma_0(NDr^2)\), counting the two signs once. The \(f_1\)-pairing is zero because its Fourier support is disjoint from that of \(\theta_{\bar\chi}\). The \(f\)-pairing has integral factor \(\Gamma(1+s)/(4\pi)^{1+s}\) and Dirichlet series \[\sum_{j\ge1}\frac{a_f(j)\rho(j)\bar\chi(j)}{j^{1+s}}.\] The Hecke recurrences identify this series with the product of the two twisted \(L\)-series divided by the row gcd factor. At \(p\mid N\), use \(\varepsilon(p)=1\) and the degree-at-most-one primitive local polynomial; at \(p\mid D\) the ramified twist factor is \(1\), and at \(r\) both factors are \(1\). Since \((f,f)_{Nr^2}=r(r+1)(f,f)_N\), evaluation at \(s=0\) gives \[ \begin{split} \frac{(I_\chi,f_2)_{Nr^2}}{(f,f)_{Nr^2}} &=\frac{C}{r+1}\frac{\tau(\chi)}{\tau(\bar\chi)} L(f,\bar\chi,1)L(f\otimes\varepsilon,\bar\chi,1),\\ (I_\chi,f_1)_{Nr^2}&=0. \end{split} \tag{13}\] In particular the factors \(\sqrt D/(2\pi)\), \(r\), \(1/(4\pi)\), and the level index \(r(r+1)\) give exactly \(C/(r+1)\).

The oldform Gram matrix. In the basis \(f_0,f_1,f_2\), the normalized Gram matrix is \((g_{|i-j|})\), where \[g_0=1,\qquad g_1=\frac{a^{\rm pr}}{r+1},\qquad rg_2=a^{\rm pr}g_1-1.\] These follow from norm under translation and cyclic-coset averaging, with eigenvalues for \(T_r\) and \(T_{r^2}-1\). At \(h\ge2\), the coefficient vector after removing the \(\ell\)-factor is \((a_h,ra_{h-1},r^2a_{h-2})\). The recurrence \(a_h=a^{\rm pr}a_{h-1}-ra_{h-2}\) shows that it lies in the span of the Gram rows for \(f_1,f_2\). The coefficient of the second of these rows is \[s_h=\frac{r^2a_{h-2}-g_1ra_{h-1}}{1-g_1^2}.\] The denominator has bounded dyadic valuation, since \(a_*\ne\pm2\). Taking the limit and using the determinant-one recurrence gives \[ \lim_i\frac{s_h}{r_i+1} =\frac{\gamma_h}{(a_*-2)(a_*+2)}. \tag{14}\]

Interpolation at the central character. All the factors in this old-block calculation admit polynomials with bounded denominators. The Gauss ratio is \(\tau(\chi)^2/r\). The Gauss polynomials have coefficients in \(\mathbb Z_2\): Frobenius on the \(r\)-th roots fixes them because \(\lambda_i(2)=0\). At the trivial character each Gauss polynomial has value \(-1\), so the limiting augmentation of the ratio is \(1\).

For \(L(f,\bar\chi,1)/\Omega_0\) use \(Z_i/d_0\). For \(L(f\otimes\varepsilon,\bar\chi,1)/\Omega_{\rm tw}\) use the even part of the \(\chi\)-weighted additive modular-symbol sum at the units \(x/r\), in positive Fourier Mellin convention, multiplied by \(\tau(\bar\chi)/r\). Before Gauss multiplication, its nontrivial value is \(\tau(\chi)L(f\otimes\varepsilon,\bar\chi,1)/\Omega_{\rm tw}\). At the trivial character its value is \[(a^{\rm pr}-2)L(f\otimes\varepsilon,1)/\Omega_{\rm tw}.\] Indeed summing the additive twists over the nonzero residues gives \[r\sum_{r\mid j}\frac{a_{f\otimes\varepsilon}(j)}j -L(f\otimes\varepsilon,1) =(a^{\rm pr}-2)L(f\otimes\varepsilon,1),\] by the Hecke recurrence and \(\varepsilon(r)=1\), with Mellin continuation understood. The fixed relative period lattices and the Manin–Drinfeld theorem bound the denominators [23]. Multiplying by the trivial Gauss value \(-1\) and letting \(r\to1\) therefore gives augmentation \[(2-a_*)L(f\otimes\varepsilon,1)/\Omega_{\rm tw}.\] Use (8) to compare all the products in the fixed dyadic embedding.

The character identities hold outside order at most two. In the characteristic-zero cyclic group ring, evaluation at all characters is injective. Multiplying the difference of the two sides by \(t^2-1\) also kills its evaluations at the two excluded characters, so gives a group-ring identity. The denominators of both sides are bounded, hence this identity passes to the bounded power-series limit. That ring is a domain, so its nonzero factor \(t^2-1\) can now be canceled. Differentiating the resulting identity uses \(Z_i(1)=0\), so only the central value of the second factor is needed. Equations (13) and (14), with the difference \(h\mapsto h+2\), now give exactly [eq:T5]. ◻

Proof. Since \(\gcd(k,l,D)=1\), a full prime power at \(p\mid D\) lies on one side of the divisor pair. Put all these powers initially on \(l\)’s side. Then \(W(k)=-\varepsilon(k)\). Moving \(p^{d_p}\) to the other side changes the sign by \[\epsilon_p(-bNa_C/p^{d_p}) \prod_{\substack{s\mid D\\s\ne p}}\epsilon_s(p^{d_p}),\] which is the local norm sign \(\varepsilon_{K,p}(-bNa_C)\), by the product formula. At the primes \(p\mid D\) not dividing \(b\), that sign is already positive by the norm congruence. Thus the allocations at the discriminant primes vanish unless all signs are positive, and otherwise contribute \(2^{\#\{p\mid D:p\mid b\}}\).

The remaining allocations are ordinary \(\varepsilon\)-weighted divisors. For \(d=d_q\), set \[A_q(d)=\sum_{j=0}^d\varepsilon(q)^j,\qquad M_q(d)=\sum_{j=0}^d(-2j)\varepsilon(q)^j.\] If \(q\) is split, or if \(q\) is inert and \(d\) is even, then \(M_q(d)=-dA_q(d)\). The weighted sum is therefore proportional to the undifferentiated sum, which is zero by \(W(l)=-W(k)\). If \(q\) is inert and \(d\) is odd, then \(M_q(d)=d+1\); all the other local factors give \(\rho(b/q)\), whose \(q\)-factor is one. The initial minus sign and the factor \(1/2\) give [eq:T6]. ◻

Proof. The generic supports are disjoint because \(v_\ell(m)\) is odd and \(\ell\) is inert. The cusps are disjoint at the good model places under consideration and retain their types under the prime-to-\(N\) Hecke correspondences; CM points have potentially good reduction, so there are no cusp intersections there.

At a split good prime \(q\), the endomorphism algebra of the ordinary CM reduction is \(K\). A matching homomorphism would be linear or conjugate-semilinear over \(K_\ell\), between Tate lattices free of rank one over \(O_\ell\). Its degree has even \(\ell\)-valuation, contradicting the choice of \(m\). The intersection is consequently zero in this case.

Suppose \(q\) is inert. Write \(\mathcal W=W(\overline{\mathbb F}_q)\). After omitting the killed bad definition and ramification primes of the fixed CM objects and level data, we have good lifts over \(\mathcal W\). Classical CM describes the Hilbert orbit, with its fixed type and orientation, by tensoring one object with ideal representatives \(I\); the level subgroup is cut out by \(\mathfrak n\).

Coarse intersections and lifting lengths. For each ordered pair of such objects, its intersection over \(\mathcal W\) is one half the sum, over degree-\(m\) homomorphisms of their reductions respecting level, of the individual lifting lengths. Here is the normalization. Since \(q\nmid m\), every subgroup of order \(m\) in the reduction lifts uniquely, and \(T_m\) uses all these subgroups with their multiplicities. After passage to a fine auxiliary level, each deformation disk is smooth with tame automorphism action, since \(q\ge5\). The coarse disk is the quotient by this action modulo its generic kernel \(\{\pm1\}\). Averaging an integral parameter with its tangent character linearizes the effective action. The tangent character is faithful: a tame kernel acting trivially on the parameter would act trivially on the disk, contrary to the generic automorphism group. The valuation of the coarse-parameter difference is therefore the sum of the framed congruence lengths over these translates. A framed congruence is exactly a lift of the specified special isomorphism. Thus counting all special isomorphisms gives division by \(2\). At an inert prime the residue degree over \(\mathbb Q\) is \(2\); this cancels that division in the weighted intersection of the proposition.

This is the proper-intersection formula of [19]; its hypotheses hold here because \((m,N)=1\) and the inert-\(\ell\) norm obstruction excludes generic intersections. We retain the calculation to identify its precise weight in the present divisor sum.

The quaternion lattice. Let \(\mathcal R\) be the supersingular maximal order of the base object. Its quaternion algebra has the description \[\mathcal B=K+Kj_0,\qquad j_0x=\bar xj_0,\qquad j_0^2=-c_{\rm B}<0.\] At a finite prime \(s\ne q\), choose a generator of the base Tate lattice over \(O_s\). Maximality identifies \(\mathcal R_s\) with its full integral endomorphisms, and \(j_0\) acts as \(b_s\) times conjugation, with \(\mathrm Nb_s=-c_{\rm B}\). At \(q\) use the unique division maximal order; \(v_q(c_{\rm B})\) is odd. The fractional ideal \(\mathfrak c\) defined by \[vb_s\in O_s\ (s\ne q),\qquad c_{\rm B}\mathrm Nv\in q\mathbb Z_q\] has norm \(q/c_{\rm B}\).

For tensor objects indexed by \(I,J_1\), the integral Hom lattice is \(J_1\mathcal RI^{-1}\), with degree \((\mathrm NI/\mathrm NJ_1)\) times reduced norm. The degree factor follows from the tensor inclusion isogenies of ideals of given norms, also after reduction. Count pairs by writing \(J_1=C_0I\). For \(u_0+vj_0\) in rational Hom, put \(y_0=\sqrt{-D}\,u_0\), \(z_0=\sqrt{-D}\,v\). The lattice and level conditions become \[ \begin{gathered} y_0\in C_0,\qquad z_0\in\bar{\mathfrak n}\mathfrak cC_0I/\bar I,\\ y_0+z_0b_p\,\bar i_p/i_p\equiv0\pmod{\mathfrak p} \qquad(p\mid D), \end{gathered} \tag{15}\] where \(i_p\) generates \(I\) locally. To check this, away from \(q\) test \[u_0t+vb_s(\bar i_s/i_s)\bar t\in C_{0,s} \qquad(t\in O_s).\] At étale places the linear and antilinear conditions separate. The split level condition on \(\mathfrak n^{-1}O_s/O_s\) adds the factor \(\bar{\mathfrak n}\) on the antilinear side. At an odd ramified prime, set \(v_s^*=vb_s\bar i_s/i_s\). Testing on \(1,\sqrt{-D}\) requires \(u_0+v_s^*\) and \(\sqrt{-D}(u_0-v_s^*)\) to be integral; these are the two inverse-different conditions and the extra congruence in (15). Here \(C_{0,s}=O_s\). At \(q\), integrality separates because the linear and antilinear norm valuations have opposite parities.

The degree identity is \[Dma_C=\mathrm Ny_0+c_{\rm B}\mathrm Nz_0.\] Set \[Z=(z_0)/(\bar{\mathfrak n}\mathfrak cC_0I/\bar I), \qquad b=q\,\mathrm NZ.\] Because \(\mathrm N\mathfrak c=q/c_{\rm B}\), this gives \(a+Nb=mD\), with \(a=\mathrm Ny_0/a_C\), exactly as in the coefficient formula. Neither component vanishes: a purely linear map violates the inert-\(\ell\) degree condition, whereas a purely antilinear map would force \(N\mid mD\).

Genus conditions and the number of maps. Fix \(y_0,C_0\). For an ideal \(Z\) of norm \(b/q\), the condition that a generator \(z_0\) exist is that \([Z\bar{\mathfrak n}\mathfrak cC_0]\) be a square. Its norm is \(bNa_C/c_{\rm B}\). At \(p\mid D\), the quaternion algebra splits, so \(-c_{\rm B}\) is a local norm. After moving the ideal away from \(D\) by principal scaling, Gauss genus theory identifies the square condition with \[\varepsilon_{K,p}(-bNa_C)=1\qquad(p\mid D).\] If soluble, there are \(2^{\#\{p\mid D\}-1}\) classes \(I\), and two generators \(z_0\) for each before the congruences. At \(p\mid b\) the congruence in (15) is automatic. At the other discriminant primes the nonzero residues match up to a sign.

These signs are jointly equidistributed. Replacing \(I\) by \(\mathfrak pI\) leaves \(I/\bar I\) unchanged and flips the generator-ratio sign only at \(p\). Principal changes of representative transport \(z_0\) by the corresponding quotient with its conjugate and preserve the test. The relation obtained by multiplying all ramified ideals is the principal ideal \((\sqrt{-D})\); its transport changes the generator by \(\sqrt{-D}/\overline{\sqrt{-D}}=-1\). Thus that relation is absorbed by the two generator choices. Of the \(2^{\#\{p\mid D\}}\) pairs, the \(\#\{p\mid D:p\nmid b\}\) nonautomatic signs leave \(2^{\#\{p\mid D:p\mid b\}}\) possibilities. Summing over ideals \(Z\) gives \[2^{\#\{p\mid D:p\mid b\}}\rho(b/q)\] homomorphisms per summand. Moreover \(d_q\) is odd, since \(b=q\,\mathrm NZ\) and ideal norms have even valuation at an inert prime.

The length of a lift. Each counted homomorphism has lifting length \((1+d_q)/2\). Use the Grothendieck–Messing filtration criterion and Serre–Tate theory over \(\mathcal W/q^j\), with the canonical nilpotent divided powers on \(q\); here \(q\ge5\). The integral first crystalline cohomology splits into two unramified CM lines. Up to units, Frobenius has \[F(e_1)=e_2,\qquad F(e_2)=qe_1;\] the Hodge summand is \(e_2\). Source and target have the same CM type. The antilinear component interchanges the lines, say \(e_1\mapsto Ae_2\), \(e_2\mapsto Be_1\). Commutation with Frobenius gives \(v_q(B)=v_q(A)+1\). Their sum is \(d_q\), the determinant valuation of this component, so preservation of the Hodge line modulo \(q^j\) is exactly \[j\le v_q(B)=\frac{1+d_q}{2}.\] Prime-to-\(q\) level adds no lifting obstruction. This is also the inert lifting exponent in [19]. Combining the map count, this length, and the residue-degree cancellation gives the negative of [eq:T6], with its stated normalization. ◻

Proof of Proposition 36. By Proposition 46, \(B_h(n)\) computes, up to the common sign, the partial character-weighted intersection away from \(r\). We explain why the combinations in [eq:T4] compute the full heights. In particular, we do not impose an unramified-character description downstairs at \(r\).

The difference \(C_h(n)=B_h(n)-B_{h-2}(n)\) corresponds to \(T_n(T_{r^h}-T_{r^{h-2}})\) on the second divisor. Only cyclic \(r^h\)-paths remain: a noncyclic kernel contains full \(r\)-torsion and factors through multiplication by \(r\). Exactly two cyclic kernels at each full CM object remain at maximal order, one on each split factor. Their Hilbert sums are both the original sum, independently of \(h\ge1\), including the corresponding cusp degrees. The second difference on the left of [eq:T4] removes these terms.

Every remaining Heegner orbit before \(T_n\) has conductor a positive power \(r^j\). Indeed a cyclic over-lattice that remained maximal locally would be \(O_r\)-stable and lie on one split factor. Ring class theory therefore places its field of definition over \(K\) over \(H_{r^j}\). Write the orbit with its multiplicity as a trace from such a field \(F'\), including its own cusp subtraction, and pull the first divisor up by the projection formula. The \(r\)-intersections of these pulled-up terms are zero. Their reductions are ordinary, since \(r\) is split, and both objects before \(T_n\) have full order at \(\ell\). A matching homomorphism of degree \(n\) would violate the same linear or conjugate-semilinear norm obstruction at the inert prime \(\ell\). The cusps also contribute nothing. We may consequently add the \(r\)-place intersections as zero. The character over \(F'\) is now unramified, so Lemma 39 identifies the sum with \(h_i^{F'}\), using the normalized norm valuations in the projection formula.

Applying \(\mathcal A\) to [eq:T4] at \(n=\ell\) inserts \(\mathcal A T_\ell\) between the height arguments. The prime-to-\(Nr\) Hecke recurrences in \(n\) commute with the cyclic-path expansion already made. On the Jacobian, \[\mathcal A T_\ell =\frac{\mathcal A_fa_f(\ell)}{\delta}\phi^*\phi_*\] rationally: \(\phi^*\phi_*/\delta\) is the orthogonal projector onto the elliptic factor. Biextension functoriality over each \(F'\) projects the pairing to \(E\), with this scalar; only fixed denominators are cleared. Multiply the first argument by the fixed \(2^b\) of Lemma 39, and retrace the second to \(K\).

The resulting second arguments on \(E\) are the indicated Hecke second differences of \(Q_X\), up to bounded-order cusp torsion. Each cyclic operator has eigenvalue \(a_f(r^h)-a_f(r^{h-2})\). After passage to the limit and the additional difference \(h\mapsto h+2\), the geometric side is \[ \pm\mathcal A_fa_f(\ell)(a_*-2) (\gamma_{h+2}-\gamma_h)\, \frac{h_\lambda^K(Q_X,Q_X)}{\delta}. \tag{16}\] Here the second difference uses \(\gamma_{h+1}-2\gamma_h+\gamma_{h-1}=(a_*-2)\gamma_h\). Cusp torsion disappears in rational heights; the fixed multipliers cancel in these rational equalities.

On the other side, apply [eq:T5] at \(h+1,h,h-1\), with coefficients \(1,2,1\), as required by [eq:T4]. The determinant-one recurrence now inserts the factor \(a_*+2\). If \(\alpha,\alpha^{-1}\) are the limiting eigenvalues, then \(\gamma_h=\alpha^h+\alpha^{-h}\). Their distinct non-root-of-unity property ensures that \(\gamma_{h+2}-\gamma_h\ne0\) for a sufficiently large \(h\). Cancel this quantity and \(\mathcal A_fa_f(\ell)\ne0\). The remaining scalar \[\frac{(a_*+2)(2-a_*)}{(a_*-2)(a_*+2)}=-1\] on the holomorphic side gives exactly (4). If \(Q_X\) is torsion its rational height is zero, while \(C\Omega_0L(f\otimes\varepsilon,1)\ne0\); hence \(Z'=0\). No rational spanning assertion has entered this proof. ◻

Proof of Theorem 34. Now assume \(\mathop{\mathrm{an}}(E)=1\). The companion has analytic rank zero, so the classical low-rank theorem gives the rational rank decomposition over \(K\). In particular \(Q_X\) is a rational multiple of \(P\) modulo torsion; write \(Q_X=t_P P\) in \(E(K)\otimes\mathbb Q\). The exact split Gross–Zagier normalization [eq:GZ]–[eq:GZ-E] gives \[ C L'(f,1)L(f\otimes\varepsilon,1) =\frac{2H_{\rm abs}(Q_X)}{\delta} =\frac{2t_P^2H(P)}{\delta}. \tag{17}\] This is the single Hilbert-class trace and the absolute Poincaré-height diagonal, with the restricted polarization; see [27] and [15]. Together with (8), it also proves the rationality of \(L'(f,1)/(\Omega_0H(P))\).

Choose the direction of Lemma 38, so that \(h_\lambda^\mathbb Q(P,P)\ne0\). By Lemma 39, \[h_\lambda^K(Q_X,Q_X)=2t_P^2h_\lambda^\mathbb Q(P,P).\] Substitution of (17) into (4) therefore gives \[Z'=\pm d_0(2-a_*)\, \frac{L'(f,1)}{\Omega_0H(P)} h_\lambda^\mathbb Q(P,P). \tag{T3}\] The factor \(2\) from field extension is the same factor as in (17); neither is discarded. Finally combine [eq:T3] with the cup reciprocity formula [eq:T2]. Since \(2-a_*\), \(h_\lambda^\mathbb Q(P,P)\), and \(\log_\omega P\) are nonzero, cancellation yields [eq:T1]. ◻

Proof. The compact Selmer line. Let \(A_0\) be \(E'\) or the indicated positive twist. Work with the notation \(O^j,R^j_\mathbb R,J_2,\tau_g,\tau_q,b\) of Lemma 33, at the actual stages (so \(S'=S\cup R_i\), \(\omega\) minimal on \(A_0\)). Write \(P\) for a rational point generating the free Mordell–Weil lattice, and \(s_1=v_2(\#\mathop{\mathrm{Sha}}(A_0/\mathbb Q))\), finite by the low analytic rank theorem.

The Poitou–Tate map from \(J_2\) to the dual of the discrete Selmer group is injective. Indeed its composite with the dual of \(A_0(\mathbb Q)\otimes\mathbb Q_2/\mathbb Z_2\) is evaluation by compact point localization. Log duality and \(\log_\omega P\ne0\) make this composite injective, with cokernel of length \(v_2(\log_\omega P)-b\). The Kummer exact sequence adds the finite quotient of length \(s_1\), so the cokernel into the dual Selmer has length \(s_1+v_2(\log_\omega P)-b\).

The preceding term in Poitou–Tate is the singular image of ordinary \(O^1\), which must therefore be zero. At the other places compact \(H^1\) is already Kummer. Global Kummer and finiteness of \(\mathop{\mathrm{Sha}}\) now identify \(O^1\) with the compact Mordell–Weil group, and the remaining terms of Poitou–Tate give \[\begin{split} O^1&= A_0(\mathbb Q)^\wedge_2,\\ \operatorname{length} O^2&=s_1+v_2(\log_\omega P)-b+ \operatorname{length}R^2_\mathbb R+ \sum_{q\in S'_f}\tau_q-\tau_g . \end{split}\]

Removing the moving Euler factors. Write \(z^{\circ}_i\) for the ordinary image after specializing all \(B\)-parameters centrally on the given branch. The moving omitted factors there are \(e_r=1-a_{A_0}(r)/r+1/r\); the fixed twist in the comparison is split at these primes. Removing them all gives the fixed-support class \(z\) of the tame calculation, with \(z^{\circ}_i=(\prod_{r\in R_i}e_r)z\) rationally after inflation, by the norm and tower relations. In particular by [eq:T1] \[v_2\!\left(\frac{\log_\omega z^{\circ}_i}{(\log_\omega P)^2}\right) =v_2\!\left(\Delta(0)\,\prod_{q\in S'_f}L_q(A_0,1)^{-1}\cdot \frac{L'(A_0,1)}{\Omega_0 H(P)}\right). \tag{P1}\] Here \(\Delta(0)\) is the evaluated smoothing including any fixed quadratic scalar. To use [eq:T1] at the stripped base one makes its own choice of horizontal sequence if necessary, killing the fixed support and fixed data required there and adapting to the point \(P\) on this actual curve; it need not be any current variable of \(B\).

The determinant valuation. The first equality in (1) still applies, since \(O^1\) has a line, with nonzero \(z^\circ_i\) rationally by [eq:P1], and \(O^2\) is torsion. Now \(\operatorname{length}(O^1/\mathbb Z_2 z^\circ_i)=v_2(\log_\omega z^\circ_i/\log_\omega P)+\tau_g\). Thus the raw valuation is \[v_2\!\left(\frac{\log_\omega z^\circ_i}{(\log_\omega P)^2}\right) +b+2\tau_g-s_1-\sum_{q\in S'_f}\tau_q -\operatorname{length}R^1_\mathbb R = X(A_0)+v_2(\Delta(0)),\] by the Haar and real-component computations. Here \({\rm Reg}_{A_0}=H(P)\) on the full free lattice. Just as at rank zero these are bounded-torsion-length central specializations: the moving \(\tau_r\) are bounded, since \(r\to1\) and the Tate trace limits are not 2 there. The displayed cohomology formulas and positive triangle bound positive torsion and keep the rational degree-one dimension two with other degrees zero. Thus pivots for those dimensions work at the center of the limit diagram also; the class determinant specializes there and has the same finite valuation. The fixed corrections have central ratio 1. This proves [eq:L5] in analytic rank one. ◻

Proof. Integrality in Proposition 27, the central equality [eq:L5] in both ranks, and isogeny invariance prove the lower bound. Under the positive twist hypothesis, the central value of \(U\) on that branch has valuation zero. Since \(B\) is local, \(U\) is a unit on that branch. By the simultaneous construction and reduction-unit transfer for [eq:L1], \(U\) at \(E'\) is then a unit as well. If the given twist is trivial we simply have the unit directly. This conclusion works for any of the completed choices in that construction, including with the prescribed-sequence option. Thus for \(s_2(E')=0\), Lemma 33 gives analytic nonvanishing and \(X(E')=0\). ◻

Proof. The inverse limit of the finite Selmer groups fits into the Kummer sequence with the rational-point completion and the Tate module of \(\mathop{\mathrm{Sha}}\). Its rational dimension is therefore the corank \(s_2=1\) of the usual divisible-coefficient Selmer group. These local conditions may be imposed on \(G_{\mathbb Q,S}\), since at good odd places outside \(S\) they are unramified.

The class of \(\beta\cup x\) vanishes locally by Kummer isotropy. Moreover, \(H^2(G_{\mathbb Q,S},\mathbb Q_2(1))\) injects into the sum of local cohomology groups by Poitou–Tate: the opposite everywhere-locally-zero kernel for \(\mathbb Q_2\) is unramified everywhere and vanishes by class field theory. Thus the global cup is a coboundary, giving \(k\). Compactness bounds the denominators of these continuous cochains. Rescale \(x,\beta\) together and \(k\) quadratically to make them integral. No rational-point representative of \(x\) is required. ◻

Proof. Apply Lemma 37 to the nonzero compact Selmer class and its self-cup primitive from Lemma 50, with the displayed field \(L\). Its hypotheses require only a nonzero cohomology class and these fixed-support cochains. The prescribed list includes every spectral-height condition and the fixed data of both branches.

The lemma gives \(g\) satisfying [eq:P2] and horizontal Frobenius/residue-character data approaching it. Splitting on \(E[2]\) and in the twisting field, together with the determinant-one trace condition, also gives the prescribed-prime hypotheses of Proposition 27. Write \(v=u_1\), so the Galois scalar is \((1+v)^{\lambda_i}\), up to the consistent inverse orientation. ◻

Proof. Suppose that \(\delta(v)\) has order at least two, including the possibility that it vanishes identically. Then the central class \(\beta\) lifts to the Selmer problem over \(\mathcal R/v^2\). We first realize this hypothetical lift on finite-stage cochains, then test it by cup reciprocity.

Realizing the first-order lift. After a fixed nonzero dyadic multiplier \(m\) and passage to a slower cofinal precision, the lift gives cocycles on \(G_{\mathbb Q,S\cup R_i}\) with the deformed coefficients, \[b_i(v)=m\beta+v z_{1,i}\pmod{(2^{n_i},v^2)}, \qquad n_i\le m_i,\qquad n_i\longrightarrow\infty,\] with the following two properties: \[ d z_{1,i}=-m\lambda_i\cup\beta\pmod{2^{n_i}}, \qquad \operatorname{inv}_2(z_{1,i}\cup x)\longrightarrow0. \tag{18}\] Use \(-\lambda_i\) instead throughout for the inverse scalar action. These derivatives are consequences of the hypothetical lift; they are not Kato classes.

Here are the finite-model details. After the one-variable specialization, transfer restriction at \(2\) to constant local models; the coefficients there are already split at each stage. At dual-number precision retain the cup with the fixed \(x\), the local invariant map on root coefficients, and the global model image of the inflated central \(\beta\). A rational lift is a model cocycle modulo \(v^2\) whose constant term represents \(\beta\). Adjust it by a deformed coboundary to match that model image, then clear one fixed common denominator.

The cycle and constant-term equations pass to stage representatives modulo \((2^{n_i},v^2)\) at a slower cofinal precision, by the marked finite-model construction. Include these representatives back into actual cochains. The central contraction homotopy expresses the difference of their constant terms from \(m\beta\) as a coboundary. Lift the adjusting degree-zero cochain and subtract its deformed coboundary. The constant term is now exactly \(m\beta\), so the coefficient of \(v\) in the cocycle equation gives the first identity in (18).

On the constant local models at \(2\), the derivative of the hypothetical Selmer lift is rationally finite. Its cup invariant against \(x\) is therefore zero by Kummer isotropy. The marked cup and invariant maps make the stage invariants converge to this value. The last coboundary adjustment does not change them because \(\lambda_i=0\) locally at \(2\). This proves the second assertion in (18) using actual finite-stage cochains.

The closed scalar cocycle. Suppress the index on \(z_{1,i}\). The cochain \[z_1\cup x+m\lambda_i\cup k \tag{P3}\] is closed: its two differentials cancel by \(dz_1=-m\lambda_i\cup\beta\) and \(dk=-\beta\cup x\). At \(2\), the second term is zero and the first has invariant tending to zero by (18). At the other fixed places, \(\lambda_i\) vanishes and the rational \(x\) is a coboundary, so one common multiplier kills their local contributions. Outside the allowed support the cup is unramified.

The moving local invariants. At every moving prime there is a bounded-denominator local cochain \(w_{j,i}\) with \(dw_{j,i}=x\): since \(x\) is unramified, take \[w_{j,i}=(\rho_T(\operatorname{Fr}_{j,i})-1)^{-1} x(\operatorname{Fr}_{j,i}).\] The trace bounds give the common denominator. After clearing it, [eq:P3] is locally cohomologous to \(m\lambda_i\cup(k-\beta\cup w_{j,i})\). The second factor is an unramified root-coefficient cocycle. At a supplementary prime the character \(\lambda_i\) is also unramified, hence \[\operatorname{inv}_{r_{j,i}}\bigl(m\lambda_i\cup (k-\beta\cup w_{j,i})\bigr)=0 \qquad(j\ne1).\]

At the distinguished prime \(r_i\), local cup reciprocity and the residue-exponent choice instead give \[\operatorname{inv}_{r_i}\bigl(m\lambda_i\cup (k-\beta\cup w_{1,i})\bigr) \longrightarrow \pm m\eta\ne0,\] with the same fixed denominator clearing understood. To see the normalization, at the working precision the local field contains the relevant roots of unity. An unramified root cocycle is the Kummer class of a unit, and its Frobenius value records raising the residue unit to \((r_i-1)/2^{n_i}\). This is exactly the additive residue exponent defining \(\lambda_i\), in the compatible root basis. Frobenius convergence gives the scalar \(\eta\) of [eq:P2].

All other local invariants tend to zero. The nonzero limit at \(r_i\) contradicts the global sum-of-invariants law for the closed cocycle [eq:P3]. Thus \(\delta(v)\) has order one. ◻

Completion of the proof of Proposition 3. The inequality and the corank-zero implication were proved in Corollary 49. Suppose \(s_2(E)=1\), and use the direction and the unit constructed above. By Lemma 52, the local-condition triangle gives a single generic degree-one line for the ordinary global problem on this germ. Restoring the ordinary real triangle gives just degree one of dimension two for the positive complex. These are the same dimensions as on the full ring \(B\), so the generic determinant identity defining \(U\) specializes to \(\operatorname{Frac}\mathcal R\), changing pivots if necessary.

The real basis generates the constant real contribution, and every correction factor in [eq:L1] is a unit in the germ. Write \(z^{\rm all}\) for the Kato class with all supplementary moving primes still present. The local-condition triangle and the inverse determinant convention now give \[ \begin{split} 0=\operatorname{ord}_v U &=d_{\mathcal R}(C_f^{\rm Sel},1) +d_{\mathcal R}(J,\operatorname{loc}_{\rm sing}z^{\rm all})\\ &=-1+\operatorname{ord}_v(\exp^*z^{\rm all}). \end{split} \tag{19}\] Thus the dual exponential has a simple zero, in any fixed nonzero rational differential coordinate.

Remove the supplementary primes at each stage by the good Euler norm, with the true cyclotomic variable trivial and the conductor-\(r_i\) variable retained. The Euler polynomials restoring those primes may have nontrivial unramified scalar values, but their limiting central values are \(2-\operatorname{tr}g_j\ne0\). They are therefore units in \(\mathcal R\). The rational norm identities pass to the local scalar polynomials and their limits: the Euler factors are integral, and the split local dual-exponential coordinates have uniformly bounded denominators. Consequently the one-prime horizontal comparison also has a simple zero, and its derivative \(Z'\) is nonzero, up to a possible variable inversion.

If \(L'(f,1)=0\), the nonzero companion value and [eq:GZ] make \(\phi_*P_X\) torsion. Proposition 36 would then give \(Z'=0\), a contradiction. Its torsion implication requires no rational spanning point, and the prescribed direction satisfies all its field, eigenvalue, and residue-character conditions. Since the functional sign is odd, we conclude that \(L'(E,1)\ne0\) and \(\mathrm{an}(E)=1\).

We may now apply Lemma 48. The unit \(U\) gives \(X(E)=0\); the classical low-rank theorem gives the corresponding rational rank and finiteness of \(\mathop{\mathrm{Sha}}\). Isogeny invariance transports these conclusions from \(E'\) to the original curve. ◻

Proof. Let \(w^h=(\alpha)\) as ideals, \(\alpha=(a+b\sqrt{-d})/2\), \(a^2+db^2=2^{h+2}\). Both \(a,b\) are odd: otherwise \(\alpha\) would have the same reduction at both primes over 2, impossible. Each odd prime of \(b\) has \((2/q)=1\) (here \(q\) denotes that prime), by the norm equation and \(h\) odd; hence \(b\equiv\pm1\bmod8\). For \(h\ge3\), \(a^2\equiv9\bmod16\), and the unit \(\bar\alpha\) at \(w\) is congruent to \(a\) modulo 8. For \(h=1\) we have \(d=7\), and the unit root of \(X^2\pm X+2\) is again \(\pm3\bmod8\).

Thus \(-1,2,\bar\alpha\) give a basis of local squareclasses there. They are also a basis of global squareclasses with only dyadic support (odd \(h\)). With trivial scalar \(\mathbb F_2\) and no odd places yet allowed, the strict problem is acyclic: restriction to \(w\) is an isomorphism in degree one by that basis and in degree zero on constants, so use the above amplitude and Euler characteristic.

On either individual residual constituent with scalar twist \(\overline\Psi_t\), now adding the two places over each odd \(q\mid N\) gives singular blocks by the unramified inflation comparison. Each has determinant over \(\mathbb F_2[[t]]\) of order \(g_q=2^{v_2(b_q)}\) in \(t\), \(b_q\in\mathbb Z_2\) the exponent at one place over \(q\). Indeed Frobenius-minus-one on the scalar inertia \(H^1\) coefficient is \((1+t)^{\pm b_q}-1\) (the cardinality twist reduces to one). Extension of the two constituents is only needed at full support. There are two places over each \(q\) and two residual constituents, so summing these orders gives [eq:A1]. Determinant multiplicativity for the residual filtration allows the two constituents to form a nonsplit extension. ◻

Proof. For existence, prescribe a compatible ray class action over \(K\) of moduli \(2^m\sqrt{-d}\), using principal ideals with generators having residues 1 at the 2’s and a fixed nonsquare at the ramified prime. This fixes 2-power roots of unity by norm. \(K(i,\sqrt[4]{2})/K\) is dihedral of order 8 (disjoint quadratic data); its maximal abelian subfield is 2-cyclotomic in the sense of being contained in \(K(\mu_{2^\infty})\). Thus its nontrivial commutator element can be prescribed compatibly with the ray condition. The \(\sqrt q\)’s are independently switchable by their new disjoint ramification.

The full constraint field has finite commutator image over \(K\), so the subgroup fixing it has Tate image containing a finite-index part of the closed commutator of the image over \(K\). This includes an open subgroup in determinant one by the non-CM open-image theorem [54]. The coset therefore permits trace different from 2. After fixing such an element, Chebotarev to increasing precisions (degree-one places with chosen lifts) gives the sequence; in particular \(r_i\to1\) 2-adically.

The Sylow-2 quotient of \(\operatorname{Pic}(\mathcal O_{r_i})\) is cyclic of order \(2^{v_2(r_i-1)}\) by the unit ratio at \(r_i\) and odd class number. Use its group-ring Galois character \(\Psi_{u,i}\) with generator value \(1+u\). Its quadratic quotient is \(K(\sqrt{r_i})\) (pull back the discriminant character by norm, trivial on the defining principal order relations).

At the places over odd \(q\mid N\), Frobenius exponents in this variable are odd: for \(\mathfrak q^h=(x)\), the ratio \(x/\bar x\bmod R\) has symbol \((q/r)=-1\).

At \(w,\bar w\) the exponents have valuation exactly one. Indeed \(\alpha\bmod R\) is square by Hilbert reciprocity on \((\alpha,\pi_i)\), primary \(\pi_i\) at both 2’s and units at other odd places. Thus \(\alpha/\bar\alpha\) is a square with nonsquare root \(\alpha/(\sqrt2)^h\bmod R\); both roots have the same symbol. Inertia at either place over \(r\) surjects onto the cyclic quotient, while \(\Psi_t(\mathrm{Fr}_R)\to1\) since \(\pi_i\) is increasingly primary. ◻

Proof. At \(u=0\), unramified inflation supplies two singular local differentials with Tate matrices \(g-1\), up to conjugation and determinant units. The cardinality twist tends to one, and \(\det(g-1)=2-a_*\). This proves [eq:A2].

For [eq:A3], set \(2=t=0\) and consider each trivial residual constituent with its \(u\)-scalar twist over \(D_0=\mathbb F_2[[u]]\). Discard the odd \(q\mid N\) for generic acyclicity on the constituent, by the singular inflation comparison using the nontrivial unramified scalar there.

In the remaining strict problem, at \(u=0\) degree one injects into unrestricted cohomology and has basis \(\pi_i,\bar\pi_i\) (radical classes, primary at \(w\), using the preceding dyadic basis). This notation identifies such classes by the ultrafilter.

Suppose the two-term differential over \(D_0\) has a nontrivial free kernel; its saturation gives \(c\in H^1\) reducing nontrivially. Lifting modulo \(u^2\) forces the global cup of that reduction with \(\chi=[r_i]\), the linear scalar character coefficient, to vanish.

At \(R\) both basis elements pair nontrivially with \(\chi\). Indeed \(r\equiv1\bmod8\), and \(\bar\pi_i\bmod R\) has symbol \((2B/r)=-1\), \(2B=\operatorname{Tr}(\pi_i)\). Here \(B\) is an integer, \(B\equiv1\bmod4\) by primarity, \(4r=(2B)^2+d\gamma^2\); at every odd prime \(q\mid B\) we have \((r/q)=(d/q)\) with both units (norm and the ramified-prime condition). Quadratic reciprocity also for possible \(B<0\) gives \((B/r)=(B/d)=-1\). Since the local \((r,r)\) Hilbert symbol is trivial and \(r=\pi_i\bar\pi_i\), this proves the assertion. Hence \(c\) reduces in global cohomology to \(\chi\).

In unrestricted global cohomology put \[D_\chi=\frac{\Psi_u-1}{u}.\] This is a twisted cocycle: at a finite stage it is defined by retaining one additional coefficient precision. It reduces to \(\chi\). The coefficient exact sequence for multiplication by \(u\) therefore gives an integral class \(z\) such that \[[c]-[D_\chi]=u[z].\] Only existence of this quotient is needed; unrestricted \(H^1\) need not be \(u\)-torsion-free.

At \(w\), \(D_\chi+uz\) is a boundary, and \(\Psi_u=1+u^2{\rm ur}\bmod u^3\) with \({\rm ur}\) the nontrivial unramified quadratic additive character. Thus \(\bar z_w={\rm ur}\) (boundaries are divisible by \(u^2\)).

At \(\bar w\), lifting \(\bar z\) modulo \(u^3\) with the same leading coefficient forces \(\bar z_{\bar w}\cup {\rm ur}=0\), hence even valuation as a radical.

These linear-term obstruction and boundary calculations hold on actual cochains to fixed Artin precisions by the finite models; fixed dyadic local actions here stabilize modulo each precision in the ultrafilter sense. The available global radical classes are generated by \[-1,\quad 2,\quad\bar\alpha,\quad\pi_i,\quad\bar\pi_i.\] The class \(\bar z\) has even valuations at both dyadic places: at \(w\) this follows because its image is unramified, and at \(\bar w\) from the cup obstruction. These two valuation conditions remove the generators \(2,\bar\alpha\), since \(h\) is odd. Thus only \(-1,\pi_i,\bar\pi_i\) remain. The unramified class at \(w\) is the radical class of \(5\), whereas \(\pi_i,\bar\pi_i\) are local squares and the remaining image is contained in the line generated by \(-1\). This contradiction rules out a free kernel on either scalar constituent, proving [eq:A3]. ◻

Proof. Work on the smooth ordinary formal curve of a fine full odd auxiliary level sufficiently large for the tame data, over \(\mathcal V=W(\overline{\mathbb F}_2)\) (and extend constants as needed). Use the canonical finite flat \(2\)-level from the multiplicative part, and the étale Igusa tower trivializing the multiplicative formal group and its differential. We use the ordinary deformation and tame elliptic moduli theory of [20, 31]: the multiplicative part lifts over the ordinary deformations (its dual is étale); these trivialization covers extend étale on the ordinary compactified charts, with the multiplicative trivialization on Tate charts.

The log Kodaira–Spencer differential given by the square of the trivialized differential gives \(d_{\rm mod}\) by its dual derivation on scalar functions; derivations lift along the étale levels and the completion. This uses the elliptic Kodaira–Spencer isomorphism on the fine prime-to-2 curve, with \(dq_{\rm Tate}/q_{\rm Tate}\) normalization and unit cusp widths there. It raises equivariant weights by two. \(V_2\) uses the quotient by the first connected subgroup with canonical choices on the result (for weight computations take target differential with pullback twice the source differential); \(V_\ell\) similarly uses the first subgroup in raised cyclic \(\ell\)-level with the stated Tate normalization.

Pullbacks of the algebraic characteristic-zero forms along the indicated ordinary maps have bounded denominator: these are rigid sections on the quasi-compact ordinary formal region, holomorphic also on cusp charts (the connected level and quotient on Tate curves give the usual Tate-level choices). One can first bound on a finite formal-affine covering in the Hodge lattice, before trivializing.

Integrality, once some bound exists, can be checked by expansion at all trivialization lifts of a \(\mu\)-cyclic test cusp in each tame determinant component. Indeed the prime-to-2 full level special curve is smooth with geometrically connected determinant components. Each special Igusa component at any fixed finite height surjects onto its ordinary component, and formal expansions there detect vanishing of the special-fiber section. An integral section on the completed infinite tower uses just finite height modulo each precision. This gives the integrality test by division one uniformizer factor at a time. Choose the full-level cusps so that all the nested cyclic tame choices involved are of \(\mu\)-type.

The weights of the inverse-derivative sequence tend 2-adically to zero, so expansion weight factors on all differential trivializations tend uniformly to 1. Its odd-exponent expansion converges uniformly modulo each scalar precision and the even terms vanish. The same expansion test on differences gives integral convergence on the tower; invariance of the limit descends it to weight zero (étale torsor descent levelwise modulo finite precisions). Also \(d_{\rm mod}F_0=f^{[2]}\). The Hecke recursions give the displayed depleted expansions. The same argument checks lattice congruences on a common raised tame level. For a tuple of coefficient components, first allow a common initial denominator and then apply the argument in coordinates for the prescribed lattice. ◻

Proof. The integral expansion in \(z-1\) defines the measure by Mahler duality. We claim that \[\mathcal F(z)+\mathcal F(-z)=0.\] The points \(z,-z\) are the two lifts over the same canonical degree-two quotient \(B\). The Serre–Tate pairing on étale and dual étale bases gives the relation \(z_{\rm target}^{e}=z_{\rm source}^c\) with the étale and connected factors \(e,c\) of the isogeny.

Trace along the connected quotient morphism vanishes by expansion at Tate cusps (all exponents odd). More explicitly one does the trace on the tame ordinary curve, transporting tame data by the quotient. This is a finite-flat square correspondence: the quotient map reduces to Frobenius on the identified special tame curves, is locally a square map in deformation parameters, and the Tate map on cusp charts. Finiteness likewise lifts from the special Frobenius on the adic formal curves. Its trace identity on weight-zero functions follows by the same \(\mu\)-cusp detection (the two Tate expansions with opposite root arguments, and source cyclic choices still \(\mu\)-type). The left side is the transform of the measure whose multiplier is \(1+(-1)^x\). Uniqueness of the transform and torsion-freeness of the coefficients show that the restriction of \(\mu\) to \(2\mathbb Z_2\) is zero. This proves the support assertion. ◻

Proof. The log \(G\) used on such interior disks about CM reductions is analytic on the disk, with the prescribed group-log values at algebraic points. One justification, also for abelian projections, is by Néron extension on smooth charts. Near an interior ordinary tame point use the algebraic relative subgroup scheme of the fine elliptic moduli, i.e. the finite-flat-subgroup Hilbert moduli in the universal \(2^b\)-torsion, for subgroups of order \(2^b\) if the level requires it. It is of finite presentation and smooth over the trait at the canonical-subgroup special point: in Artin deformations there the subgroup must map trivially to the étale quotient (by étaleness and its special connectedness), so the unique lift over any ordinary elliptic deformation is the multiplicative subgroup. Thus the chart has the same smooth deformation germ as the tame chart.

On its smooth open neighbourhood our characteristic-zero map to the abelian variety extends by the Néron property (in the generic fiber use the map on the cyclic locus; other subgroup types, if present, are open and closed and can use a constant). One may enlarge the local base first. Thus the disk maps into a single Néron tube, where log differences from the center are analytic by the formal group. On smaller closed disks one may first multiply into the small log range, giving the same analytic group log by division. This suffices for all level choices and their quotient disks here.

To prove [eq:M1], compare differentials. They agree by the parametrization differential, quotient pullbacks and the expansion normalization above (\(V_2 G\) differentiates with the extra Tate factor 2, \(V_2^2G\) with factor 4; these differential identities also follow by expansion on the ordinary curve). The possible constant is zero by summing over \(z,-z\). Indeed the two sources can be seen as the two quotients of \(B\) by noncanonical order-two subgroups, with inherited cyclic tame data. The \(T_2\) or \(U_2\) action gives \[G(z)+G(-z)=a_2 G(B)-\epsilon_2 G(V_2 B).\] For level at 2 these two noncanonical kernels are exactly the \(U_2\) quotients, retaining canonical \(2\)-level on the result. This Hecke normalization on divisors pulls back the usual differential operator on forms; cuspidal terms are torsion. Thus \(P_2(V_2)G\) also has trace zero in the pair.

For the Eisenstein assertion, roots of unity are killed by the logarithm and \(\log 2=0\). The displayed weight-zero discriminant ratio is nonvanishing on the interior disks, with analytic log (the normalized variation of an analytic unit on an open disk from its center has values in principal units, e.g. by its Newton polygon). The differentials agree by the log derivative of \(\Delta\). The norm of the ratio in the source pair is one by the Tate identity \(\Delta(q)\Delta(-q)=-\Delta(q^2)^3/\Delta(q^4)\); use the same correspondence expansion (the Tate orders of the ratio cancel at canonical cusps). This fixes the remaining disk constant and proves the Eisenstein identity. ◻

Proof. The conductor-\(s\) points are canonical lifts because reduction is ordinary at the split prime, the endomorphisms split the \(2\)-divisible directions, and the Serre–Tate parameter must be one by the endomorphism lifting relation. Consequently the geometric special end ring is exactly \(\mathcal O_s\) (all such endomorphisms lift at the trivial parameter). Make the disk evaluations over the completed unramified base, enlarging constants if needed for the form coefficients. Choose Serre–Tate generators in both directions by polarization. For each \(\mathfrak a\) use a prime-to-\(2s\) and prime-to-level ideal action as representative, viewed also at higher 2-conductors. Write \(\delta_{\mathfrak a}\in\mathbb Z_2^\times\) for its Serre–Tate exponent from the base (connected divided by étale factor); take the identity at the trivial label.

Points at \(z=\zeta_{2^m}^{\,j}\), \(j\) odd for a primitive \(\zeta_{2^m}\), have endomorphisms \(\mathcal O_{2^m s}\): among endomorphisms of the reduction the two local actions must agree modulo \(2^m\). Canonical cyclic \(2\)-level is preserved by the order, as are the oriented tame groups. Thus by the main theorem of CM these \(X_0\)-points are defined over \(H_{2^m s}\): the units of the completed endomorphism order stabilize the cyclic level.

Prime-to-level ideal actions transport the groups (including the canonical group at 2) and the disk parameters by the isogeny exponent. In particular the kernel of lowering the order class map to conductor \(s\), the unit ratios \((\mathbb Z/2^m)^\times\), acts simply transitively on the primitive roots in a disk.

For an ideal principal below, extending to \((\gamma)\), the exponent is the local ratio \(\gamma_w/\gamma_{\bar w}\) up to reversing conventions (action by the corresponding endomorphism on the underlying conductor-\(s\) reduction). Thus in the inverse limit the exponent on the order-class kernel maps to \(\Gamma\) by one fixed \(\eta:\mathbb Z_2^\times\to\Gamma\), independent of \(s\) by the order class sequence. Here isogeny actions in all labels are identified by their actual Artin actions on the CM orbit, inverting ideals consistently if required by reciprocity. The cyclic \(2\)-level throughout is the canonical connected subgroup; there is no independent variation of a \(2\)-level structure along the orbit. ◻

Proof. The class-group exact sequence makes \(\theta\) factor through conductor \(2^m\). Fourier summation over odd \(j\) gives [eq:M2]: by Lemma 58, its nonzero Gauss sum multiplies the inverse-character moment of each disk measure. The corrected weight within disk \(\mathfrak a\) at \(\zeta_{2^m}^{j}\) is \(\theta(t_{\mathfrak a})\theta_l(j/\delta_{\mathfrak a})\psi_s(\mathfrak a)\), precisely the full character value by the parameter description. Each tame \(V_\ell\) translates the orbit at raised level with the requisite orientations retained after lowering. In (M1), \(V_2,V_2^2\) disappear by primitivity (same quotient in the pair).

The two primitive sums of \(G\) at opposite odd orientations differ only by a translation character unit and a sign: the Atkin–Lehner operation at the full odd part of the primitive level has this eigenlaw up to killed cusps, reverses all original odd choices and translates the orbit by the indicated odd ideals, preserving canonical level at 2.

At \(t=0\) there is instead the disk-center version: sum once over \(\operatorname{Pic}(\mathcal O_s)\) with weights \(\psi_s\); the same Euler translations occur at the odd primes, and an additional \(P_2(\psi_s(\sigma_2))\) at the inverse connected-isogeny translation by the canonical centers. This follows by augmentation and (M1). Taking \(s=1\), the two orientation sums differ by a sign and a translation of the same Hilbert-class trace, giving [eq:M3]. The factor \(c_E^{-2}\) is retained exactly; no assertion that \(c_E\) is a 2-adic unit is used. ◻

Proof. Return to the scalar residual test setting. For conductor \(r_i\) use a base disk chosen by an order-\(r_i\) descending isogeny from the conductor-one base (non-eigenline), transporting generators so the Serre–Tate exponent is \(r_i^{\pm1}\) by polarization. Thus for fixed \(m\) above, the evaluation basepoint is eventually also the corresponding \(r_i\)-descendant (the exponent is one modulo \(2^m\)). Trace down that conductor at \(u=0\) in (M2) gives multiplier \(a_{r_i}-2\) eventually, by the good-prime Hecke relation subtracting the two split prime-ideal translations, whose \(\theta\)-values become exactly 1. The two orientations give its square. Applying the high-character identity test to infinitely many such \(\theta\) proves [eq:M4].

Finally compute modulo 2 at \(s=1\) by the Eisenstein congruence on the disks. The odd depletions on each measure contribute \(1-(q+1)[\pm b_q]/q+[\pm2b_q]/q\) at \(q\), by translation as above (test at high characters by the same Fourier summation), of residual order \(2g_q\).

The measure using just \(F_{{\rm Eis},0}\) has unit value at the center, which is its unweighted CM trace. Indeed the product of \(V_2\Delta/\Delta\) over the conductor-one class orbit is \((\alpha/2^h)^{12}=\bar\alpha^{-12}\), viewing \(\alpha\) at the chosen embedding with \(w\) connected. The connected isogeny permutes the classes, normalizing differential pullback by 2 in \(V_2\); the ordinary isogeny pullback factors for independently chosen differentials have product the principal-generator factor around the cycles, altogether \(\pm\alpha\). Similarly the product of \(V_2^2\)-ratios is \(\bar\alpha^{-24}\). Thus the log-ratio trace with factor \(-1/48\) is \(-\log(\bar\alpha)/4\), odd by the dyadic basis congruence above. Thus the Eisenstein measure before the odd depletions is a unit at the center. Each odd depletion contributes order \(2g_q\) to each of the two orientations, which proves [eq:M5]. ◻

Proof. At the places of \(r\), use the inertia/residue calculation on limits: a pro-2 tame generator acts with difference \(u\) (up to unit); where this vanishes on a residue field, the residue Frobenius and its singular twist have action-minus-one invertible by \(r_i\to1,\ \operatorname{tr}g\ne2\) and \(\theta(\mathrm{Fr}_R)\to1\).

At each fixed odd place \(v\), replace the full condition by \(U_v\). Its quotient in the full local complex is represented, for cohomology and determinant purposes, by \[[\,M_{I_v}(-1)\longrightarrow M_{I_v}(-1)\,] \qquad (1,2)\] with Frobenius differential. Indeed the scalar is unramified and the Tate inertia differential (after discarding the pro-(prime-to-2) kernel) has constant free invariant and coinvariant spaces after inverting 2. Relative to its degree-zero invariants the inertia cochains have just the singular degree-one space with the cardinality twist. These calculations can be taken first over the fixed local groups at compact precisions before inverting 2, with inflation using lattice invariants, and the descriptions hold also on characteristic-zero field tests over \(A\). Thus \(U_v\) includes local \(H^0\) and injects in \(H^1\).

For the exact complementary conditions on conjugate duals, cup on unramified restrictions factors through unramified cochains (null for the invariant map), and the orthogonality quasi-isomorphism can be checked on residue fields using the displayed dimensions, local duality and the degree-zero invariants. For example the unramified \(H^1\) dimension is the full invariant \(H^0\) dimension; local Euler characteristic is zero. The product of the two singular determinants over \(q\in S^\circ\) is up to units \[D_q=P_q(\chi(\mathrm{Fr}_v))P_q(\chi(\mathrm{Fr}_v)^{-1}),\] nonzero here by the Frobenius-exponent choice. For multiplicative reduction the coinvariant line has arithmetic Frobenius \(a_q=\pm1\); for additive reduction there is no coinvariant line in characteristic zero. ◻

Proof. We construct the lines and prove exact orthogonality before defining their logarithmic coordinates.

The local Shapiro tower.

Put \(F_v=K_v=\mathbb Q_2\), and let \(D_j/F_v\) here denote the unramified extension of degree \(2^j\) (field notation at \(v\)). In the present case the local \(\theta\)-field \(P_v/F_v\) is cyclic totally ramified. The \(u\)-action is unramified with Frobenius value \((1+u)^b\) for \(v_2(b)=1\). Use first the Shapiro scalar \[\Theta[\operatorname{Gal}(P_v/F_v)] [[\operatorname{Gal}(D_\infty/F_v)]]\] with tautological character, \(1+s_0\) the Frobenius generator in the unramified factor. Local cohomology is inverse cohomology over \(H_j^{\rm loc}=P_vD_j\) with corestriction. Project the finite factor by \([\sigma]\mapsto\theta(\sigma)\) (idempotent splitting after inverting 2) and substitute \(1+s_0=(1+u)^b\), a finite flat change by Weierstrass or a cyclic-generator change. This computes exactly the desired diagram after these changes: fixed local groups/actions are available by stabilization at each finite precision, and Shapiro, free coefficient models and the fixed projector denominator apply there and in compact inverse limits.

More generally for several conductor variables unramified at \(v\), split \(\theta\) if needed locally as a totally ramified cyclic-field character \(\theta_P\) (on \(P_v\)) times a finite unramified \(\theta_U\), by the abelian local description. Use the finite factor for \(\theta_P\), and substitute for \(1+s_0\) the product of \(\theta_U(\mathrm{Fr}_v)\) and the conductor-variable Frobenius powers. Provided some exponent is nonzero, the change is again flat, finite flat after adjoining the other variables.

Uniform bounds for logarithmic lattices. At local finite levels the integral Kummer sequence on \(H^1\) with Tate coefficients has ends \(\mathcal A(H_j^{\rm loc})^\wedge_2\) and \(\operatorname{Hom}_{\mathbb Z_2}(\mathcal A^\vee(H_j^{\rm loc})^\wedge_2,\mathbb Z_2)\) for the abelian variety \(\mathcal A=E\), or on full modules before a newform projection; identify via polarization allowing fixed denominators for other varieties. Transitions are norm and dual restriction. Compactness permits exact limits. After finite-character projection the log and dual lattices compare to corresponding additive lattices in the Lie spaces and their trace duals up to bounded 2-powers uniformly in \(j\), and projected log torsion kernel has bounded exponent (one can use the unnormalized projector). Both \(\theta_P^{\pm1}\) have this property. Here and below, “bounded” means independent of the unramified degree \(2^j\). The constants may depend on the fixed character \(\theta\), its ramified field \(P_v\), the fixed extension used for semiabelian reduction, the abelian variety, and its coefficient projector. Uniformity as \(\theta\) varies is not needed for these characteristic-zero tests.

Indeed a uniform small lattice exponentiates by fixed ramification, using a model first over \(P_v\), then its unramified extensions.

For the upper and torsion bounds extend to the compositum with a fixed Galois \(L/F_v\) of semiabelian reduction. Component exponents there are bounded since after \(LP_v\) the extensions are unramified. An inertia element \(\nu\) over \(L\), chosen with \(\theta_P(\nu)\ne1\), acts trivially on geometric points of the special identity component there.

To see this last fact one can use the special-fiber maps from the Néron model over \(L\). On identity components they are surjective on geometric points: on auxiliary odd-adic Tate modules rationally they identify inertia invariants (Néron reduction over maximal unramified fields is surjective with odd-divisible formal kernel without odd torsion, and components are finite), unchanged under finite base extension by semistable unipotent inertia. Both components are semiabelian by the semistable reduction criterion, so these Tate images suffice for surjectivity. Thus \(\nu\) indeed fixes the special points by functoriality from \(L\).

After a common component multiple, \((\nu-1)\) of any point therefore enters the formal group, and a uniform further multiple enters the small log range. This proves both upper and torsion bounds on applying the finite projectors (their eigenvalue difference from 1 costs only a fixed denominator). The dual log comparison uses the dual of the torsion-free log lattice, and the different denominator is bounded.

Normal bases and free local cohomology. Now \(\mathcal O_{H_j^{\rm loc}}=\mathcal O_{P_v}\otimes_{\mathbb Z_2}\mathcal O_{D_j}\). Use trace-compatible integral normal bases on the unramified tower. More explicitly, choose \(\beta_0=1\) and choose successively \(\beta_j\in\mathcal O_{D_j}\) with \[\operatorname{Tr}_{D_j/D_{j-1}}(\beta_j)=\beta_{j-1}.\] The unramified trace is surjective, so this is possible. The total trace of \(\beta_j\) is a unit. Modulo \(2\), the trace operator for the cyclic group of order \(2^j\) is the top power of its generator minus one. Nonzero total trace therefore makes \(\overline\beta_j\) a generator of the residue-field module over the cyclic group algebra. Lifting gives an integral normal basis, and the chosen bases are trace-compatible.

Use also a rational normal basis on the fixed finite factor. The projected norm-limit Kummer terms on both ends therefore become free via log and its dual after inverting \(2\). On the dual end, use the trace-dual lattice; a self-dual normal basis is unnecessary. The transition dual to inclusion is again trace in additive coordinates, and the different of the fixed ramified extension contributes only a fixed denominator. Sandwiching the lattices between fixed multiples is compatible with these limits by compactness.

Degree zero of the full cohomology upstairs vanishes, and the projected degree-two inverse cohomology vanishes after inverting 2 by local duality and the uniform projected torsion bounds. For an elliptic plane the two free ends have rank one each. The same holds on each Hecke-field plane and differential embedding for its modular abelian variety, by characteristic-zero projection (all projector, isogeny and polarization losses there are fixed). Thus \(H^1\) is itself split free with a direct Kummer line: its two free rank-one ends form an exact sequence, and the quotient is projective.

Exact orthogonality. Cup on conjugately paired Kummer lines vanishes, by Shapiro on finite group quotients using ordinary Kummer cups of translates, then on the inverse limits and under the scalar changes. Hence by ranks and perfect local duality this gives exact orthogonality, also on field tests. The restrictions from the split degree-one lines admit isotropy nullhomotopies, so this is an assertion about the local complexes and their comparison triangles, rather than just the dimensions of their generic cohomology. ◻

Proof. For \(E\) use \(\log=\log_{\omega_E}\); more generally on a Hecke plane take a nonzero corresponding differential defined over the coefficient extension. With evaluations in the fixed embedding, \(\lambda_w\) is the line functional which, before the Frobenius-variable substitution, takes a Shapiro Kummer system at \(w\) to the group-ring sums \[y_j\longmapsto \sum_{\sigma\in\operatorname{Gal}(P_wD_j/F_w)} \theta_P(\sigma)(1+s_0)^{k(\sigma)}\log(\sigma y_j) \pmod{(1+s_0)^{2^j}-1},\] where \(k(\sigma)\) is the unramified exponent. Extend linearly on coefficients. The convention comes from restricting the character corestriction back up; group multiplication on the Shapiro scalar sends the point-system by the inverse Galois action. These sums respect trace and kill the wrong finite-character components. They are bounded after fixed denominators by the lattice comparison; unramified values may use our completed unramified constants.

This gives the line isomorphism claimed: for example a small-log generator before projection uses a fixed small factor times the product of a rational normal generator for \(P_w\) and the above unramified normal generators. Its evaluation is a nonzero constant times a power-series unit. Indeed the finite cyclic normal resolvent at the embedding is nonzero: after scalar extension, the circulant matrix of a rational normal basis is invertible, and its determinant is the product of the character resolvents. Conjugate coordinates are permuted up to units. The unramified group sum has augmentation \(\operatorname{Tr}_{D_j/F_w}(\beta_j)=1\) at every finite level, so its inverse limit is a power-series unit.

Specialization and membership. We explain how this comparison detects membership and logarithms on transferred arithmetic classes. At simultaneous cyclic group quotients of order \(2^k\) in the conductor variables with at least one Frobenius exponent nonzero at 2, the unramified substitution uses cyclic order \(2^{k-\min v_2(b_j)}\) for large \(k\), \(b_j\) the exponents. It is from the corresponding group ring by free scalar extension there (coset bases, and rescaling by the fixed \(\theta_U\) factor if present). The Kummer line specializes, after inverting 2, exactly to the corresponding projected Kummer conditions at those finite levels: use the trace-compatible small-log generators above and ordinary Kummer dimensions under Shapiro.

Suppose as here the stage base ring class fields for \(2^m\) and the tame conductor together locally contain \(P_vD_n\) for every fixed \(n\) on large stage sets. This follows from containing the \(\theta\)-field and the increasing unramified layers by the Frobenius condition. A corestricted twisted class locally Kummer at the stages up to growing precision then lies in \(U_v\) in the test. Indeed at each fixed cyclic quotient its localization is obtained by first norming to these fixed fields by transitivity (and summing the several local terms, with group translates/weights). Thus it specializes into finite-level projected Kummer by the fixed-group comparisons, closedness and congruences at growing coefficient precision there.

Vanishing in the complementary quotient of \(H^1\) is detected on all such finite group quotients (bounded power series, clearing fixed denominators).

Compatibility with point logarithms. For an un-differenced class from global points this also computes its \(\lambda_w\)-value by the unnormalized full orbit sum of logs with character weights. Grouping as above uses actual trace logs in a fixed local extension at each fixed quotient; Kummer congruences at growing precision imply log congruences there, so even the limit into \(\mathcal V^*\) with fixed extensions agrees with the indicated evaluation. Every projection used here has a fixed denominator. The orbit sums are unnormalized throughout, including as the cyclic quotients increase. ◻

Proof. Use just the unramified Shapiro tower. A uniform polynomial in its Frobenius action with value nonzero at 1, times a fixed scalar, now bounds the log comparison in both directions and kills the possibly unbounded reduction torsion. Indeed first use the component order, which is bounded under unramified Néron base change. On the special identity component, the abelian Frobenius polynomial annihilates the abelian part; its roots have complex absolute value \(\sqrt2\). For a torus split over a residue extension of degree \(e\), one may use \(F^e-2^e\); the unipotent part is killed by a fixed power of \(2\). For an elliptic curve these are respectively the good-reduction Frobenius polynomial, the toric polynomial, and the scalar \(2\) in the additive case. Multiplying the relevant factors and, if necessary, another fixed scalar gives a polynomial \(Q(F)\) with \(Q(1)\ne0\). None of the Frobenius polynomial factors has a root-of-unity zero. This enters the formal group, and a further fixed scalar enters the small formal log lattice at all levels.

Thus the map from a fixed small additive lattice by exponentiation, compatible with norm and inclusion, has cokernel killed by such a bound, as does the dual comparison with the opposite action; degree-two projected cohomology is controlled by the same torsion bound. For general varieties likewise use the abelian Frobenius polynomial, the torus Frobenius relation and a characteristic power on the unipotent part.

After localizing (Frobenius corresponds to the unramified group generator or its inverse) we again have exact Kummer/quotient lines on the plane, specializing to ordinary Kummer and quotient at the center by small-log normal generators. Weighted log is a line isomorphism there, and finite-level transfer and membership work as above after first multiplying by the Frobenius polynomial bound (which is invertible at the center). All the bounds canceled in this argument are units at the characteristic-zero center. The assertion is consequently local near that center; it does not assert an unramified logarithmic isomorphism over the entire power-series ring. ◻

Proof. Write \(f_{\rm loc}(z)=z(\gamma^2)\), \(s_{\rm loc}(z)=z(\sigma_\ell)\) in limiting degree one at this new place of \(K\), for an inertia element \(\sigma_\ell\) generating tame 2-inertia (in its pro-2 quotient) and the whole relative ring-class inertia of order \(\ell+1\) to be used at \(\ell\). The prime \((\ell)\) splits completely in the prime-to-\(\ell\) lower conductor fields by the principal ideal/order criterion, so the base scalar character is trivial there.

Also \(\gamma^2\) acts trivially on all the ring class fields of \(K\), even at upper conductor, by generalized dihedral action.

In the limit the full local complex is split with free cohomology \(M,M\oplus M,M\) in degrees 0,1,2, with the indicated degree-one coordinates (here \(M\) in the local coordinates means the underlying coefficient plane). Indeed discard prime-to-2 inertia kernel as in the local calculations; action is trivial locally modulo increasing precision and the tame conjugating power \(\ell^2\) tends to 1. Thus the inertia/residue cohomology or procyclic resolutions give the free coordinates, with zero differentials in the limit. Unramified inflation is the finite plane (\(s_{\rm loc}=0\)); the transverse plane uses \(f_{\rm loc}=0\), including degree zero and omitting degree two as for the finite plane. They fit the lower/upper comparisons in the square switch.

In particular under conjugate transport the pure planes are self-annihilating, and the cross pairing is, up to a unit, \(\langle x,J y\rangle\) by the Tate alternating plane form. Indeed the pure scalar cups are zero as in the local calculation following the square switch, and the mixed scalar cup is a unit. Transport at the place can use \(\gamma\) instead of global conjugation by the inner identification on the cochains; it fixes the Frobenius argument, and changes the tame argument by the \(\ell^{\pm1}\)-power; on coefficients there is the Tate transport with possible scalar character unit. This yields exact complementary dualities also for lower/upper and their triangles (with split local comparisons); for untwisted coefficients one also has the ordinary nonconjugated version with pairing \(\langle x,y\rangle\). For \(T_2 E\) use the unimodular integral plane form, and after inverting 2 any fixed constant plane-form normalization on a characteristic-zero polarized Hecke component costs only units in horizontal valuations. The local identity itself below does not require inverting \(2\) in the Tate limit. ◻

Proof. The conductor-product system. For a subset \(I\) of switched primes at a stage, use CM points with conductor \(b_i\prod_{\ell\in I}\ell\), obtained by simultaneous product quotients of the base point by one descending isogeny at each \(\ell\), transporting the level. These choices can extend the previous ones when adding a switch. They are defined over the corresponding upper ring class fields by CM reciprocity: the order changes as stated at the new inert primes, transported level still has stabilizer containing the completed order units. The relative groups over \(H_{b_i}\) are products of independent cyclic inertia groups of orders \(\ell+1\); this is the order class exact sequence and Chinese remaindering, using the unit hypothesis. In particular choose the above generators independently from inertia, acting trivially on the fields with \(\ell\) omitted.

At an upper and omit-\(\ell\) lower pair of points \(Y,X\) respectively in \(\mathcal A\), one has \[{\rm Tr}_\ell Y=a_\ell X,\qquad \widetilde Y=F_\ell\widetilde X ,\] where the first formula if needed is after killing fixed cuspidal torsion, tilde is good reduction at the chosen prime, and \(F_\ell\) is arithmetic residue Frobenius.

These identities work for all matching translates. The trace traverses all the cyclic lines in the good Hecke correspondence, since \(\ell\) is inert and prime to lower conductor. Reduction below is supersingular; all the degree-\(\ell\) quotients there reduce via Frobenius with the transported prime-to-\(\ell\) data. This gives the second relation at the modular point and hence on the abelian image (avoid any fixed primes of bad models/maps). Use maps based at a rational cusp. Also \(F_\ell^2-a_\ell F_\ell+\ell=0\) on the abelian factor of good reduction by the good Eichler–Shimura relation (it can be checked on the rational Tate realization).

Derivative operators and global descent. Use \[D_I=\prod_{\ell\in I}D_\ell,\qquad D_\ell=\sum_{j=1}^{\ell} j\sigma_\ell^j .\] The Kummer of the upper point acted on by \(D_I\), modulo \(2^a\) after a common cusp/point multiplier, is invariant relatively by \((\sigma_\ell-1)D_\ell=\ell+1-{\rm Tr}_\ell\) and the divisibilities.

A further uniform multiple descends it to \(H_{b_i}\) in cohomology. Indeed \(\mathcal A[2^\infty]\) invariants over these upper fields have bounded exponent. Take an element \(h\in G_K\) acting by a near-one scalar with square not one on \(T_2\mathcal A\) by Bogomolov’s homothety theorem for an abelian variety over a number field [6]. If the scalar is \(c\), then \(h\tau h\tau\) acts by \(c^2\) on the full Tate module. It fixes every ring class field: the image of \(\tau h\tau\) is the inverse of the image of \(h\) in every ring class quotient. Consequently \(c^2-1\) kills all the \(2\)-power torsion defined over any of the upper fields. Its valuation is fixed, giving the required uniform exponent bound.

In inflation–restriction, both the obstruction to descending a relatively invariant Kummer class and the ambiguity of its descent have coefficients in these torsion invariants. They are therefore killed by a fixed power of \(2\). Multiplying by that power gives descent; reducing the coefficient precision by a further fixed amount removes its ambiguity. The resulting retained precision still tends to infinity. These losses are independent of the moving primes and are absorbed into the common point multiplier.

Descent at the fixed places. At \(2N\) the upper extension for these derivatives is unramified relative to the base, with base component exponents uniformly bounded. In fact the ring class fields for the conductor products are composita over the Hilbert field by the same order class description; at 2 the base is thus unramified above a fixed ramification field (fixed \(m\)), and at fixed prime-to-conductor odd support the base itself is unramified over \(K\). Use unramified Néron base change.

The kernel on local Weil–Châtelet in unramified descent is killed by the component group order: for connected points the cyclic unramified \(H^1\) vanishes by Lang on the reduction and the induced additive successive quotients on the formal kernel (unramified residue normal bases, then passage up the complete filtration). Thus a uniform additional multiple ensures the local Kummer condition modulo \(2^a\) on the base there. At fixed odd places as above another component multiple gives unramified Kummer by 2-divisibility of connected points over the maximal unramified field and lifting.

At good places outside allowed support and all derivative primes, the descended class is already unramified since inertia doesn’t change going upstairs and prime-to-residue Kummer there is unramified. Use a common multiplier \(L\ne0\) incorporating the fixed losses. Across further switches the bounds allow common scales. If a scale is enlarged, it is enlarged on all classes in the same comparison. Thus each matched derivative relation uses the same \(L\) on its two sides. For a fixed finite sequence of switches, all losses can be incorporated in one such nonzero \(L\). ◻

Proof. The singular coordinate. Check the identities before corestriction. Apply the other derivatives to \(Y,X\) first and keep the notation; the matching trace and reduction relations still hold. Choose divisions \[2^a Q=L X,\qquad 2^a Z=L D_\ell Y\] so the respective descended cocycles restrict to their Kummer coboundary expressions upstairs (absorb changes by a torsion coboundary). Thus the finite evaluation below at \(\gamma^2\) reduces to \((F_\ell^2-1)\widetilde Q\). Write \(b\) for the upper descended cocycle on \(G_{H_{b_i}}\). Then \[U=(\sigma_\ell-1)Z-b(\sigma_\ell)\] is fixed by the upper-field subgroup. Indeed, for an element \(g\) of that subgroup, normality gives \(\sigma_\ell^{-1}g\sigma_\ell\) in the same subgroup, and the cocycle identities together with \(b(g)=(g-1)Z\) yield \(gU=U\).

Multiplication by \(2^a\) gives \[2^a U=L\big((\ell+1)Y-a_\ell X\big).\] The endomorphism divisibilities ensure that \[L\left(\frac{\ell+1}{2^a}Y-\frac{a_\ell}{2^a}X\right)\] is itself an upper-field point. Its difference from \(U\) is \(2^a\)-torsion defined over that field, whose exponent is bounded by Lemma 68. Inertia acts trivially in reduction. Hence \(b(\sigma_\ell)\) reduces, ignoring bounded-exponent torsion, to \[(a_\ell-(\ell+1)F_\ell)\widetilde Q =(F_\ell-a_\ell)(F_\ell^2-1)\widetilde Q.\] The equality follows directly from the Eichler–Shimura polynomial: \[(F_\ell-a_\ell)(F_\ell^2-1) =F_\ell(F_\ell^2-a_\ell F_\ell)-F_\ell+a_\ell =a_\ell-(\ell+1)F_\ell.\] Projection to the retained precision kills the bounded-exponent errors. Prime-to-\(\ell\) torsion injects under good reduction, so the reduction computation determines the Tate coordinate. Since \(F_\ell-a_\ell\) tends to \(J\), it gives the first identity in [eq:K].

The transverse condition. For the other identity \(b(\gamma^2)\) reduces to \((F_\ell^2-1)\widetilde Z\), and \(\widetilde{L D_\ell Y}=L\ell(\ell+1)F_\ell\widetilde X/2\), with \(F_\ell^2\widetilde X=\widetilde X\). Because \(2^{a+1}\mid\ell+1\), an \(F_\ell^2\)-fixed \(2^a\)-division of this reduced point is explicitly \[L\frac{\ell(\ell+1)}{2^{a+1}}F_\ell\widetilde X.\] Any other division differs from it by \(2^a\)-torsion. The imposed Frobenius-square condition makes \(F_\ell^2-1\) kill that ambiguity to the retained precision. Hence the finite coordinate of the derived class vanishes.

These calculations work on every translate from the base group over \(K\) (commuting on the points with the relative action), before summing with character weights. Local actions make the coboundary choices in the coordinates immaterial to retained precision. This proves [eq:K]. Repeating the calculation at an earlier switched prime, with all the other derivative operators already applied, gives the same vanishing there. A fixed equivariant change to a projected coefficient commutes with the calculation; its denominators are cleared in the common \(L\).

Old cochains/classes in successive comparisons can always be used by unramified inflation and reduction to slower precisions; [eq:K] only uses the matched restrictions of the descended classes. ◻

Proof. Corestriction preserves the unramification just checked away from dyadic and moving support. At odd bad places the inertia restriction class vanishes at every compact precision, hence in the local model (inertia cochains here can be taken on the fixed group; no inverse-limit obstruction from the finite invariants); use inflation.

At 2 use the finite-level Kummer membership check in the preceding local comparisons. Thus the global classes lift, uniquely in degree one since the maps of the conditions include \(H^0\) and inject in degree one.

For \(Z=Z_\varnothing\), the logarithm comparison of Lemma 65, together with [eq:M2] in both orientations, gives [eq:B1].

The Shapiro weights evaluate the full \(L\)-scaled class sums, the opposite primitive-level orientation differs by Atkin–Lehner and a translation, and the Gauss sums, \(c_E,L\) are fixed nonzero constants with 2 inverted. The two Euler translations at each odd prime multiply exactly with the conjugate arguments. The bounded logarithmic lattice comparisons justify all these limits while retaining the unnormalized sums over the varying class groups. ◻

Proof. The strict comparison. Take the high-\(\theta\) test over \(A\) as above; \(L_{\rm alg,r}(\theta,u)\ne0\) by [eq:A3]. First replace the odd bad full conditions by \(U_v\) there, still using strict at \(w\), full at \(\bar w\); this strict problem is generically acyclic since the removed singular blocks are. Put \[L_{\rm prim}= L_{\rm alg,r}(\theta,u)\Big/\prod_{q\in S^\circ}D_q .\] Thus at each height-one DVR of \(A\) the strict problem has \(d(-,1)=-v(L_{\rm prim})\). Now write \(C_F\) for the problem using \(U_v\) also at both dyadic places. If \(B_r(\theta,u)\ne0\), [eq:B1] gives nonzero localization of \(Z\); writing \(\operatorname{loc}_w Z=p_0 e\) for an \(A\)-basis \(e\) on that Kummer line, [eq:D] applies by the local comparisons. Thus \(C_F\) has generic ranks one and one, and [eq:D] gives the equality in [eq:D2].

Detecting two independent fiber classes. We prove nonnegativity at every such DVR by the square-switch test of Section 3, using [eq:K]. At the generic and DVR-fiber fields all Selmer degree-one spaces used in the square switch inject into unrestricted degree one and thus into abstract crossed classes on \(\mathcal G_K=\prod_i G_K\) by the evaluation lemma. Global invariants are zero and so are the Selmer degree zeros. Exact duality using conjugate transport gives amplitude in degrees 1 and 2; this works also on the paired lower and upper conditions at the extra inert places described above (old diagrams identified via unramified inflation).

In detail let \(k_0\) be the fiber field at this DVR. At a step with old finite condition, degree-one generic rank one and nonzero generic derived point class \(Y=Z_I\), suppose \(H^1\) on the fiber has dimension \(>1\). Take two independent classes there including a primitive reduction from the generic line in the two-term model. Use Shapiro also abstractly into the induction on \(\mathcal G=\mathcal G_K\rtimes\langle\tau\rangle\), with diagonal conjugation. The two \(M\)-planes of the induction over \(\mathcal G_K\) are absolutely simple: already constant \(G_K\) sequences act on Tate via the absolutely irreducible representation (non-CM open image), up to the scalar twists. Their determinants are unequal: the two summands have scalar twists \(\chi\) and \(\chi^{-1}\), and equality of their determinants would force \(\chi^4=1\), contrary to the high dyadic inertia condition. The involution \(\tau\) exchanges these nonisomorphic absolutely simple summands, so the induction is itself absolutely simple. This check works just the same given Tate absolute simplicity over \(K\) on a Hecke plane and the stated high-character condition. For a non-CM elliptic curve the required characteristic-zero absolute simplicity follows from the open image theorem [54].

Take the product kernel \(\mathcal H\) of all termwise full Tate and ring class actions within \(\mathcal G_K\), normal also under \(\tau\). Restriction detects the crossed classes since the constant element \(h\tau h\tau\) used in the descent is central modulo \(\mathcal H\) and acts by a scalar whose difference from 1 is invertible here. Thus joint evaluations of the two classes on \(\mathcal H\) span two copies of the induction by the simple-module evaluation test. On \((\tau\eta)^2\) for \(\eta\in\mathcal H\) these evaluations are additive in \(\eta\), given by \(1+\tau\) on the evaluations at \(\eta\). Projection back to the original summand over \(\mathcal G_K\) consequently still spans its two copies. Some \(\eta\) gives rank two there, by the quadratic determinant test.

A switch preserving the determinant valuation. By Chebotarev choose new good inert primes with \(\gamma_i\) agreeing with \(\tau\eta_i\) to the required increasing precisions on the previous evaluation data and full Tate actions. Then \(J=\rho(\tau)\), and in the old problem inflated to the new finite plane the \(f_{\rm loc}\) evaluation is exactly the above test on the fiber. Hence its rank there is two; it is nonzero also generically (use the primitive class) and so tests \(Y\) generically nontrivially.

The new derivative class belongs to the transverse problem and gives exactly [eq:K], with previous plane conditions retained. The compatible square switch comparison now preserves the class-functional valuation at this DVR and cuts fiber dimension by two, retaining generic rank one and a nonzero class there. Continue on the same ultrafilter. At each step the fiber dimension drops by two, whereas the nonzero generic class persists, so the dimension remains at least one. After finitely many steps there is no surplus and both degree-one and degree-two fiber dimensions are one.

For completeness, at this terminal DVR the minimal model has one free term in each of degrees one and two. The generic cohomology is also one-dimensional in each degree, so its differential is zero. Exact self-duality identifies these two free cohomology lines as integral duals. The terminal derived class is integral, hence its class-functional tensor has nonnegative valuation. The square-switch equalities carry that valuation back to \((Z,Z^\vee)\). All common fixed-scale multipliers are units in the horizontal DVR, since \(2\) has been inverted. This proves [eq:D2]. ◻

Proof. For each sufficiently high \(\theta\) with \(B_r(\theta,u)\ne0\), Proposition 71 gives \(p_0^2/L_{\rm prim}\in A\) by the height-one valuation criterion in the regular UFD \(A\). By [eq:B1] and the logarithmic line isomorphism, \[B_r(\theta,u)\in L_{\rm alg,r}(\theta,u) A'.\] This is automatic if \(B_r(\theta,u)=0\). Weierstrass division in \(u\) using [eq:A3] and the high-character remainder test therefore prove \[B_r\in L_{\rm alg,r}\,\mathcal V^*[[t,u]]\] integrally. By [eq:M4] and [eq:A2] at \(u=0\), cancelling the same nonzero \((a_*-2)^2\), \(B\) is likewise divisible by \(L_{\rm alg}(C(t))\). By [eq:M5] and [eq:A1], the reductions modulo \(2\) of its numerator and denominator have the same finite order in \(t\). Their integral quotient therefore has nonzero constant coefficient modulo \(2\), and is a unit. More explicitly, specializing the two-variable quotient at \(u=0\), relations [eq:M4] and [eq:A2] identify it with \(B/L_{\rm alg}(C(t))\) up to an integral unit, after cancellation of the same nonzero factor \((a_*-2)^2\). Thus its specialization at \(u=0\) is a unit. Its constant coefficient at \((t,u)=(0,0)\) is consequently a unit, proving that \(B_r/L_{\rm alg,r}\) is already a unit in the two-variable ring. ◻

Proof. We work at \(t=0\), retaining the tame variable \(u\). The nonvanishing of the specialization of a Selmer generator is part of the argument, so no central localization hypothesis is imposed. Localize on that tame line at the \((u)\)-DVR after inverting 2 (and similarly for extended coefficients on the measure side, still testing order in \(u\)). Both series there are nonzero by [eq:A3] and the unit equality. The odd fixed-place complexes are contractible near this test center: at \(u=0\) they vanish by local duality, the odd-place Euler formula and absence of rational Tate invariants. The added \(r\)-places likewise vanish by construction (\(2-a_*\ne0\)). At \(2\) use Lemma 66, which supplies the Kummer lines and logarithmic isomorphism at the center DVR. Let \(C_F\) now use these two Kummer conditions, and \(Z(u)\) be the global Kummer class by weighted traces of the conductor-\(r_i\) CM points (disk centers, original level); it belongs to those conditions. The \(t=0\) measure formula following [eq:M2] gives there \[B_r(0,u)=\text{unit}\cdot\lambda_w(\operatorname{loc}_w Z(u))^2 .\] Indeed this is the disk-center logarithm with both orientations as before, retaining additional connected-2 Euler translations, all Euler factors units at this DVR by their values at \(u=0\). For taking the limit of the weighted logs one may first multiply by the polynomial bound of the unramified comparison, evaluated on the group power translating by Frobenius (or its inverse as appropriate). This moves the actual log points into uniformly bounded lattices and specializes to a nonzero scalar at the center. At each fixed cyclic quotient the resulting sums agree in the limit with the line-functional values by corestricting first to the fixed unramified levels, as in the local membership and log checks. Thus one cancels only units of this DVR in asserting the identity.

The two free central cohomology lines. Strict generic acyclicity, the nonzero logarithm, and [eq:D] again apply. They now give generic ranks one and one for \(C_F\) and \(d(C_F,(Z(u),Z(u)^\vee))=0\), by the unit quotient and the log isomorphism.

The special fiber is also of ranks one and one only: the two lines specialize to usual Kummer; the new allowed \(r\)-places at \(u=t=0\) cause no characteristic-zero change by unramified inflation and local acyclicity. We therefore have the actual ordinary rational Selmer in degree one by fixed-diagram comparison, so use \(s_2(E/K)=1\) and duality. Let \(R\) denote this characteristic-zero \((u)\)-DVR. Its special fiber has just one cohomology line in each of degrees one and two. A minimal free model for \(C_F\) over \(R\) therefore has the form \[[\,R\xrightarrow{\delta}R\,]\qquad\text{in degrees }1,2.\] The generic cohomology already has rank one in both degrees, which forces \(\delta=0\). Hence both cohomology groups are free lines over \(R\), paired perfectly by the integral duality over \(R\).

Choose a generator \(y\) of the degree-one line and write \(Z(u)=a(u)y\), with \(a(u)\in R\). The class-functional determinant tensor then has valuation \(2v_u(a(u))\): the two free lines are integral duals and \(y\) has unit determinant volume. Its valuation is zero by the unit comparison. Thus \(a(u)\) is a unit, and \[Z(0)\ne0.\] This establishes nonvanishing in global cohomology at the specialization from the Selmer-corank hypothesis.

The specialized class \(Z(0)\) is \((a_*-2)\) times the Kummer of the Hilbert trace \(P\) of [eq:M3], up to any common point multiplier used: at the stages trace over the order-\(r_i\) layer gives the good Hecke trace less the two horizontal translates before the Hilbert sum; cuspidal torsion is harmless. These are equalities via inflation of the true old classes at constant coefficients, so pass by the finite diagrams. The factor \(a_*-2\) and the common point multiplier are nonzero in characteristic zero. It follows that \(P\) is nontorsion. Formula [eq:GZ-E], with the split twist sign, now shows that the product has a simple zero [27, 15]. ◻

Proof. Compute at the one-variable center \(C(t)|_{t=0}\). Here we have rank one and finite Tate–Shafarevich over \(K\) by classical Gross–Zagier–Kolyvagin and the quadratic isogeny [27, 33], with \(P\) spanning and having nonzero local log.

Impose integral Kummer \(U_v\) at odd places of the support first, keeping strict/full at \(w,\bar w\); denote the resulting complex by \(C_*\). This is rationally acyclic by the converse line comparison of [eq:D] with all-Kummer. Thus \(C(0)\) is rationally acyclic too by local duality at the added full places.

Use \(v=v_2\), \(n_P,\tau_g,s_K,l,\tau_\ell\) as for [eq:Kum] and the Haar comparisons (in particular \(s_K=v_2(\#\mathop{\mathrm{Sha}}(E/K))\)). At \(w\) a degree-one free basis with log \(2^l\) has \(d(U_w,e)=\tau_2\). At each odd place over \(q\mid N\), the full-place quotient is the shifted dual of integral Kummer with only degree-two torsion of length \(\tau_q\). Thus \[\begin{split} d(C_*,1) &=2n_P+2\tau_g-s_K-2\big(v(\log_{\omega_E}P)-l+\tau_2\big),\\ -v(L_{\rm alg}(C)(0))&=d(C(0),1)=d(C_*,1)-2\sum_{q\in S^\circ}\tau_q, \end{split}\] by [eq:D], [eq:Kum] and the comparison triangles. The integral unit quotient in Corollary 72 specializes at \(t=0\), so \[v(L_{\rm alg}(C)(0))=v(B(0)).\] Formula [eq:M3] evaluates the right-hand side as \[v(B(0))=-2v(c_E)+2v(\log_{\omega_E}P) +2\sum_{\ell\mid2N}v(P_\ell(1)).\] Combining the displayed determinant comparisons with this equality cancels the two logarithm valuations and yields \[2(n_P+\tau_g) =s_K+2v(c_E)+2(\tau_2-l) +2\sum_{q\in S^\circ}\tau_q -2\sum_{\ell\mid2N}v(P_\ell(1)).\] The Haar identities \[\tau_2-l=v(c_2)+v(P_2(1)),\qquad \tau_q=v(c_q)+v(P_q(1))\quad(q\ne2)\] cancel every Euler factor and give exactly [eq:E]. The two places of \(K\) above each odd support prime account for the factor \(2\) in the singular determinant contribution; the dyadic Kummer volume retains \(\tau_2-l\).

Finally, \(n_P\) is the index valuation in the full free Mordell–Weil lattice and \(\tau_g\) records its torsion. Therefore \[n_P+\tau_g=v([E(K):\mathbb ZP]),\] and [eq:G] identifies [eq:E] with \(X(E)+X(E^{-d})=0\). Together with Proposition 73, this proves Proposition 53. ◻

Proof. The Heegner class and its sign.

Formula [eq:GZ] gives a nonzero projected Hilbert point up to torsion [27, 15]; its Kummer class \(y\) projects nontrivially on \(V_f\). The rational Mordell–Weil space for the newform orbit is a vector space over its Hecke field. A nonzero vector stays nonzero after scalar extension through each embedding of that field; scalar extension to a field is faithful. Kummer injectivity is Hecke-equivariant, so the projected Kummer class is nonzero at the specified \(2\)-adic coefficient embedding.

Conjugation reverses all level orientations in the Hilbert sum. This agrees up to ideal translation with the full Fricke action (divide by the oriented level subgroup and use the dual subgroup); base cusp differences are torsion. The Fricke eigenvalue is \(-w(f)\), giving the asserted sign \(s_y\) for \(y\).

Local signs and simultaneous evaluations. Apply [eq:K] at base conductor one, with no scalar twists, working with the usual untwisted Tate pairings and variable inert primes. Use the alternating plane pairing on \(V_f\); up to coefficients this comes from the polarized ordinary pairing and Hecke projection (Rosati fixes the field). The initial problems therefore use exact orthogonal conditions. At fixed places descent gives ordinary Kummer by the proof above (it only needs fixed base local ramification here).

Switched places use the two pure planes; all these conditions are conjugation stable in the limit. At a new place with \(\gamma_i\) tending to complex conjugation on full Tate, the action of global conjugation on localizations of classes is \(J=\rho(\tau)\) in \(f_{\rm loc}\) and \(-J\) in \(s_{\rm loc}\), by the same comparison with \(\gamma_i\) and tame transport. In this proof a “new prime” means a fresh auxiliary prime sequence with its limiting arithmetic diagram, as in Proposition 69. All finite and singular coordinates, local pairings, and reciprocity comparisons below are taken in those compatible limiting diagrams.

Because complex conjugation is odd on a weight-two Tate plane, each eigenspace of \(J\) is one line. Write \(M_+\) and \(M_-\) for those lines. The alternating pairing vanishes on either line with itself and is nondegenerate between the two lines.

For any nonzero conjugation eigenclasses of sign \(s_x\) in a current Selmer problem, finite evaluation at a new place can be made nonzero simultaneously for finitely many such classes. Use again the product kernel \(\mathcal H\) and cochain evaluations; restriction is injective by the same central element (now even on the plane alone), and the evaluations of each class span the simple plane. On \((\tau\eta)^2,\ \eta\in\mathcal H\), its values are \((1+s_x J)\) of those evaluations, since conjugation comparison is on classes and coboundaries restrict trivially there. Thus they span the corresponding sign line \(M_{s_x}\). Explicitly, on the joint kernel, where the coefficient action is trivial, \[x\big((\tau\eta)^2\big)=(1+s_xJ)x(\eta).\] The same formula is valid on the cohomology classes because conjugation comparison holds on classes and coboundaries restrict trivially to that kernel. By additivity in characteristic zero, simultaneous nonzero tests for finitely many classes exist; use Chebotarev as above, retaining all the data including closeness on the full \(\mathcal A\). This works also in the extended diagrams with earlier switches.

Excluding the opposite sign. Suppose an initial nonzero Selmer class \(x\) has sign \(-s_y\). Choose a first prime at which both \(x\) and \(y\) have nonzero finite evaluations. Then \[f_{\rm loc}(y)\in M_{s_y}\setminus\{0\},\qquad f_{\rm loc}(x)\in M_{-s_y}\setminus\{0\}.\] The derivative of \(y\) (using common nonzero scaling by \(L\) here) has singular evaluation \(J\) times its nonzero finite evaluation. Pair the derivative with \(x\). At the new prime the mixed local cup is nonzero, because its singular vector \(Jf_{\rm loc}(y)\) lies in \(M_{s_y}\setminus\{0\}\), opposite to the nonzero finite vector of \(x\). Every other local cup vanishes by the self-annihilating conditions. Global cup reciprocity says that the sum of these invariants is zero, a contradiction.

Excluding a second line of the same sign. The initial Selmer space is therefore wholly in sign \(s_y\). Suppose its dimension exceeds one, and choose a first prime at which \(y\) evaluates nontrivially. Some nonzero old class \(x\) is in the finite evaluation kernel since the evaluation is in just one \(J\)-line. Its finite and singular coordinates at this prime are both zero, so it lies in the common lower condition and persists in the transverse problem, still in sign \(s_y\). The new derivative \(y'\) has nonzero singular coordinate \(Jf_{\rm loc}(y)\in M_{s_y}\). Conjugation acts as \(-J\) on this coordinate, so the \(-s_y\)-eigenprojection of \(y'\) is nonzero. At a second prime, make the finite evaluations of \(x\) and that eigenprojection nonzero simultaneously. Differentiate the full class \(y'\), using its place in the derivative system. By [eq:K], its twofold derivative has singular coordinate \(Jf_{\rm loc}(y')\). The component of this vector in \(M_{-s_y}\) is nonzero and pairs nontrivially with \(f_{\rm loc}(x)\in M_{s_y}\setminus\{0\}\). Any component in \(M_{s_y}\) pairs to zero with \(f_{\rm loc}(x)\), so it cannot cancel the nonzero contribution.

At the first prime, both classes remain in the transverse condition, which is self-annihilating. At all fixed places the Kummer conditions are retained. Thus the second prime gives the only nonzero local cup invariant, again contradicting reciprocity. The Selmer space is exactly the line generated by \(y\), with sign \(s_y=-w(f)\), proving [eq:K0]. ◻

Proof. Quadratic reciprocity determines the localization of \(p_i^*\) at each odd place of \(S\). At \(2\), its squareclass is determined by the symbol of \(2\), since \(p_i^*\equiv1\pmod4\); its real sign is determined by the symbol of \(-1\). These descriptions are linear in \(l_i\), giving \(\xi\). Global Hilbert reciprocity gives \[\langle\operatorname{loc}_S(z),\xi(y)\rangle=y(z) \qquad(z\in H,\ y\in H^*).\] The subspace \(\operatorname{loc}_S H\) is isotropic: outside \(S\), both arguments are odd local units and their Hilbert symbols are trivial. If \(\operatorname{loc}_S(z)+\xi(y)=0\), pairing with this subspace first gives \(y=0\), and valuations and sign then give \(z=0\). Notice also that \(\dim V=2m\).

The displayed edge relation is quadratic reciprocity in signed-prime notation. Choose the primes successively. Their labels prescribe congruence classes at the fixed support, while the edges to earlier primes prescribe additional quadratic-residue classes. The Chinese remainder theorem and Dirichlet’s theorem realize these prescriptions; reciprocity then determines precisely the reverse edges. This also realizes every label list. ◻

Proof. The quotients of the old and new conditions by their intersections pair perfectly. Their global localization images annihilate one another by reciprocity. The common strict kernel has dimension \(\dim\operatorname{Sel}_{\rm old}-j_0\), while the new localization image has dimension at most \(b_0-j_0\). This proves the first inequality. The same calculation for orthogonal scalar problems gives \(n-j-j_*\).

For the difference assertion, compare each configuration with the one in which the first local conditions are enlarged to the full local spaces. Finite Poitou–Tate duality identifies the image of the enlarged global Selmer group in the local quotient with the annihilator of the old dual image. The difference of the two Selmer dimensions therefore increases by the sum of the local dimension increases. That sum is the same for either of the transverse-line configurations, proving the assertion. These are the strict/relaxed localization calculations of finite duality; see also [40] and [41]. ◻

Proof. The radicals supported at \(S\) and the new signed primes give independent coordinates on the global classes unramified outside their union. This gives the coordinates in the statement and its boundary condition at \(S\).

At a new prime \(p_i\), the untwisted two-power torsion is unramified because the curve has good reduction. Twisting makes a ramified inertia generator act by negation. Its invariant two-power torsion is therefore exactly \(W\). The two-adic completion of the local points is this torsion group: the remaining open subgroup is pro-\(p_i\). On a half of \(t\in W\), the twisted Kummer cocycle has inertia value \(t\) and Frobenius value \[B(l_i)t+\chi_h(\operatorname{Fr}_{p_i})t.\] Comparing with the global coordinates sets \(t=u_i\). The contribution of \(p_i^*\) itself occurs on both sides and cancels; the other radical contributions give exactly the site equation in [eq:graph]. The nonempty ramification set likewise forces the global two-primary torsion to be exactly \(W\). Its Kummer coordinates are \(u_i=t,w=Bt\), as asserted. ◻

Proof. The image in \(E/A\) of a half of \(a\) is an additional two-torsion point. It is rational exactly when \(B(-)a\) takes values in \(A\). Thus \(f_A=0\) exactly when \(E/A\) has full rational two-torsion.

The map \(H^1(A)\to H^1(W)\) is injective. Projecting a two-Kummer class to \(H^1(W/A)\) tests divisibility of the point by the dual isogeny; inside the kernel, the classes are precisely the images of the \(\phi\)-Kummer condition. Consequently \(L\cap(V\otimes A)\) is the sum of the local isogeny Kummer spaces. Since the rational kernel has order two at each place, \(k_A-m\) is the two-valuation of the product over \(S\) of local cokernel orders divided by local kernel orders for \(\phi:E^h\to(E/A)^h\).

When \(f_A=0\), this product equals the target/source ratio of \(\Omega\prod_q c_q\). Indeed, the product over all places has that value by Haar change of variables: the finite Néron volumes are \(c_q/L_q(E^h,1)\), the Euler factors are isogeny-invariant, and the pullback scalar on differentials cancels by the product formula. At unused good odd primes the local ratio is one, by the unramified isogeny Kummer condition. At used primes \(p_i\), both two-primary point completions have order four and the map has kernel of order two, so the ratio is again one. The product at \(S\) is fixed by the prescribed local squareclasses. This argument uses the total real period.

Consider the connected graph of full-rational curves joined to the starting curve by full-to-full two-isogenies. Every vertex is good outside \(S\); Shafarevich finiteness makes this graph finite up to rational isomorphism. After one allowed twist, choose a vertex maximizing the relative two-valuation of \(\Omega\prod_q c_q\); its ratios to the other vertices are rational by the preceding calculation. At every outgoing full-to-full isogeny the target/source valuation is nonpositive, which is [eq:orient]. Isogenies preserve the conductor, and the same inequality holds throughout the prescribed family. ◻

Proof. Choose a label list containing at least \(2m+2\) copies of every element of \(H^*\). Adjust multiplicities to obtain the prescribed total label. Among all admissible edge assignments, minimize the pure dimension \(d\) in [eq:graph]. Lemma 78 realizes each assignment by primes, so the minimum is even. Suppose for a contradiction that \(d\ge2\), and fix a minimizing graph.

We will vary only the edges. Minimality first forces all differences \(u_i+u_j\) of pure solutions into one line \(A\subset W\). Edge switches preserving that line then give more independent site errors than the boundary condition at \(S\) permits. The orientation inequality [eq:orient] closes the argument when \(f_A=0\).

Proof. If a full-rational curve is available by a rational two-isogeny, use Lemma 82 on that curve. Otherwise first take an even-functional-sign twist away from \(2N_E\), possibly the trivial twist, using the same local prescriptions. Denote the current curve by \(E\) and enlarge \(S\) to include all current and original exclusions. Choose a model \[E:y^2=x(x^2+ax+b),\qquad D=a^2-4b,\] with kernel \(A=\langle T\rangle\), \(T=(0,0)\), and two-isogeny \(\phi:E\to E/A\). The standard quotient equation is \[E/A:y^2=x(x^2-2ax+D).\] It follows from this equation, or the duplication law, that the remaining two-torsion fields of the source and target are respectively \(\mathbb Q(\sqrt D)\) and \(\mathbb Q(\sqrt b)\). Thus \(D,b\) are both nonsquare. Write \(W=E[2]\) and \(\widehat\phi\) for the dual isogeny. The module \(W\) is an extension of the scalar module \(W/A\) by \(A\) with extension character \(D\). Quadratic twisting preserves both two-torsion modules and these squareclasses: a twist by \(h'\) replaces \((a,b)\) by \((h'a,(h')^2b)\).

Proof of Proposition 77. Apply Lemma 82 or Lemma 83, according to the rational two-torsion of the initial curve. All twisting primes are fresh and avoid the original \(2N_E\), so their product has the required discriminant and coprimality properties. ◻

Proof. Resume notation \(E\) for the original curve and put \(W=E_0[2]\). Choose \(S\) for \(E_0\) containing \(2,\infty\) and all primes of \(hN_E\); in the one-line case also include the support of its final model, whose coefficients we again denote by \(a,b,D\). Choose a new prime \(d\equiv7\pmod{16}\) whose Frobenius acts as complex conjugation on \(\mathbb Q(\sqrt H)\). These prescriptions are compatible: the elementary subfield of \(\mathbb Q(\mu_{16})\) is contained in \(\mathbb Q(\mu_8)\), on which the exponent \(7\) acts as complex conjugation. We will add one further compatible condition in a case below. Signed-prime reciprocity makes \(-d\) a square at every odd finite place of \(S\), and \(-d\equiv1\pmod8\) gives the same conclusion at \(2\). The two-Kummer conditions of \(E_0\) and \(E_0^{-d}\) thus change only at \(\infty,d\), and Frobenius at \(d\) acts on \(W\) as conjugation. Quadratic twisting preserves the global rational two-torsion dimension, so every change of Selmer dimension gives the same change of pure dimension.

Proof. Apply the reducible split-product comparison of Sections 8 and 9 to \(E_0,K\) from Proposition 84. The curve remains non-CM, its level and \(2\) split in \(K\), and its ordinary Selmer corank over \(K\) is one. That comparison gives a simple zero of the product and, by [eq:E] and [eq:G], \[\operatorname{an}(E_0)+\operatorname{an}(E_0^{-d})=1, \qquad X(E_0)+X(E_0^{-d})=0.\] At this point both factors have analytic rank at most one, so the classical theorem gives finiteness of their Tate–Shafarevich groups. We may therefore use the isogeny arithmetic-volume comparison of Section 2 to transfer these conclusions from \(E_0\) and \(E_0^{-d}\) to \(E^h\) and \(E^{-hd}\).

Every signed-prime product used to form \(h\) was squarefree, odd, and coprime to \(2N_E\), with the empty product allowed. The fresh prime \(d\) avoids its support, and \(K\) splits all primes of \(hN_E\) as well as \(2\). Thus \(k=-d\) and \(h\) satisfy all discriminant, coprimality, and splitting requirements of Proposition 4. ◻

Proof. By classical CM and theta-series theory [52], \(N\) is divisible by the primes ramified in \(L\); the Hecke character \(\varphi\) has type \((1,0)\) in the convention \(\varphi((x))=x\) near 1 modulo the conductor (exchange orientations if necessary). Its conjugate-source character equals its complex conjugate, by purity and the trivial determinant character of the form: on rational arguments \(n\) away from the modulus, \(\varphi((n))=n\chi_L(n)\) for positive \(n\). Thus \(F\) is real.

The field of values of \(\varphi\) on good ideals is \(F_c\). Indeed an automorphism fixing \(F,L\) changes it at most by a finite character, and leaves split-prime traces and products unchanged. Interchanging the two values at a split prime cannot give a root-of-unity ratio (power up to ray principal ideals). Thus the finite character is trivial by Chebotarev, and the reverse generation follows by traces and ray-principal values.

The Tate realizations are inductions of the character lines by the CM theorem [49], or by traces at good primes and semisimplicity. On each Hecke embedding the representation over \(\mathbb Q\) is absolutely irreducible and the two lines over \(L\) differ to infinite order. For distinct embeddings the planes are inequivalent, so also no lines between them can coincide over \(L\).

Faltings’ theorem [24] implies that \(A\) is \(\mathbb Q\)-simple with endomorphism algebra \(F\), and over \(L\) its endomorphisms have algebra \(F_c\). Indeed this latter algebra contains \(F\); locally at all finite coefficient places its base changes are the commutants of the separate character pairs, the quadratic étale algebras of \(F_c/F\) by the character-value description. This determines the quadratic étale algebra globally. The \(F\)-Lie algebra over \(\mathbb Q\) has rank one, so the action of the full field on it over \(L\) induces CM type from \(L\). ◻

Proof. Start with a large rational conjugation-invariant modulus containing the discriminant; prescribe a multiplicative residue character \(\eta(n)=\chi_L(n)\) on rational units and \(\eta(e)=e^{-1}\) on global units. For sufficiently divisible modulus the images intersect only at \(\pm1\), compatibly. Extend to all residues that are units. Prescribe \(x\eta(x)\) on the group of prime-to-modulus principal ideals \((x)\); extend to prime-to-modulus fractional ideals by extracting roots. This gives type \((1,0)\) with trivial determinant character on the theta form (at its primitive level), cuspidal by non-invariance.

The residual inducing character of the plane has odd order and is inverted by conjugation. Multiply the Grössencharacter by a finite odd Teichmüller character to arrange that the new residual inducing character equals the desired order-three one. This uses abelian class field theory to twist by the quotient (or inverse according to realizations). The finite character is anticyclotomic and is trivial on rational positive arguments off the modulus (split pairs by inversion, inert primes by inversion and odd order). Thus it does not change the determinant character. The two residual orientations give the same induction. ◻

Proof. Split the ray-tower group into its torsion and free factors. Choose \(\Lambda(\chi)\) from the torsion character of \(T_\alpha\psi\) before the Tate twist, and evaluate the free factor at the remaining character. The conductor of \(\chi\) away from \(2\) is accounted for by finite inertia. Increase the dyadic part of \(\mathfrak f\), by cofinality, until the full divisibility \(\mathfrak f_\chi\mid\mathfrak f\) holds. Use the vector \(\gamma\) as basis, enlarging \(\mathcal O\) if necessary.

Write \[C_\Lambda=R\Gamma\bigl(\mathcal O_L[1/2\mathfrak f], \Lambda(\chi)(1)\bigr).\] Let \(\mathfrak q\) be the characteristic-zero specialization prime of \(\Lambda_{\mathcal O}\). The residue-field complex at \(\mathfrak q\) is concentrated in one line in degree one by hypothesis. Cancellation in a minimal perfect model therefore identifies the localized complex itself with one free term in that degree. Consequently the generic determinant generator of Theorem 88 specializes to its actual cohomological push-forward. This comparison may be made first with a smoothing multiplier nonzero at \(\mathfrak q\), and then with the rational unsmoothed class.

The integral determinant line specializes by derived base change: \[\mathcal D(C_{\Lambda})\otimes_{\Lambda_{\mathcal O}}\mathcal O \simeq \mathcal D\bigl(R\Gamma(\mathcal O_L[1/2\mathfrak f], T_\alpha\psi)\bigr).\] The specialization of the chosen elliptic-unit determinant generator has just been identified, after inverting \(2\), with the distinguished class \(z(T_\alpha\psi)\). Hence equality of these lattices is precisely [eq:C2]. The arithmetic/étale comparison and finite models on the imaginary base justify the displayed cohomological base change. The argument uses concentration rather than the nonzero-\(L\)-value hypothesis of [11]. ◻

Proof. The notation for omissions here and later means omit the indicated prime ideals over \(L\) from its character \(L\)-function (below whole sets over rational primes, giving omitted factors of the primitive newform twist).

Apply the reciprocity equation of Theorem 88 to the type \((1,0)\) character modified by the finite twist, using \(\gamma\otimes e_\psi\) with \(e_\psi\) an Artin basis normalized on the identity label, in compatible Betti and étale comparisons. Indeed this motive over \(L\) occurs with coefficients on \(H^1(\operatorname{Res}_{H/L} A_H)\), for \(H\) cutting out the finite character, projecting the CM and finite Artin factors.

To compare to the CM realization in the reciprocity law one needs only rational isomorphisms: the type \((1,0)\) character motive is realized on isogeny factors via Weil restriction of CM elliptic curves (Hecke-character construction), and the projected Tate character lines agree. Faltings’ theorem [24], projected over \(L\), then supplies an algebraic map with extended number-field coefficients, nonzero on this component (tensor at the chosen prime then choose a nonzero map before completion), hence an isomorphism for the period comparisons.

The cohomology push of units twists exactly by the indicated vector, as tensor twisting and taking images compose. Conjugating the Hecke character in the law gives the same modular \(L\)-series, with the consistent finite norm-character twist. Under Shapiro, restriction and projection to \(V_\alpha\) preserve \(\exp^*\); restriction compatibility uses the trace in local duality, without a doubling. Only one base embedding over 2 contributes to the \(H^0(\Omega)\) of this component, the one matching the holomorphic period, also when \(2\) is nonsplit, after extending scalars.

An Artin de Rham basis, evaluated geometrically over the splitting field on this label, uses a nonzero algebraic factor \(b\) times \(e_\psi\); its complex and dyadic comparisons there use the two embeddings of the very same \(b\). It thus cancels when taking the scalar relative to \(\omega\) by the stated local trivialization. This proves [eq:C3]; the coefficient push-forward is unnormalized.

Similarly for a fixed quadratic character split at 2 one may think of \(B=A^\varepsilon\) with transported polarization, basis, and \(\omega\); with these geometric identifications (extend base scalars for \(\omega\) if needed) the denominator is again \(p_\gamma\). ◻

Proof. The Tate characters and Faltings’ theorem [24] identify \(B\), up to \(F\)-equivariant isogeny, with the orbit factor of \(f_B\). Good-prime traces show that its Hecke field is still \(F\). In particular \(B\) is \(\mathbb Q\)-simple.

Multiply \(P\) into the domain of the \(2\)-adic logarithm. Its full logarithm is nonzero: the logarithm is injective on a sufficiently small formal neighborhood, so a point in its kernel is torsion. If \(\log_\omega P=0\), this nonzero full logarithm lies in the proper algebraic subspace \(\ker\omega\) of the Lie algebra. The \(p\)-adic analytic subgroup theorem [26] gives an algebraic subgroup \(H\) containing this multiple of \(P\), with \(\operatorname{Lie}H\subseteq\ker\omega\). The Zariski closure of the subgroup generated by a rational nontorsion point is defined over \(\mathbb Q\); by \(\mathbb Q\)-simplicity it is all of \(B\). Thus \(H=B\), contradicting the properness of \(\ker\omega\). ◻

Proof. Choose a split Heegner imaginary companion \(K'\) with nonzero central value, as in [eq:T1], at level \(N_B\). It differs from \(L\): a prime ramified in \(L\) divides \(N_B\) and must split in \(K'\). The induced plane remains absolutely irreducible over \(K'\), because its two characters are still distinct on the compositum. The Heegner argument [eq:K0], applied to the known simple product zero, gives the rational Selmer line. The nonzero rational-sign Hilbert trace and Kummer injectivity give \(\dim_F B(\mathbb Q)_{\mathbb Q}=1\). By [eq:C3], the class \(z_B\) belongs to that Selmer line. Indeed, its dual exponential vanishes, and odd local \(H^1\) vanishes by local Tate-torsion finiteness and duality. Lemma 91 gives \(\log_\omega P\ne0\).

Choice of primes and the exponential coordinate.

Take the auxiliary index prime \(\ell\) there inert in \(K'\), split in \(L\), away from the operators and bad primes. Such \(\ell\) has \(a_{f_B}(\ell)\ne0\) by the ray principal value argument. Use primes \(r_i\) and even tame characters from \(\lambda_i:(\mathbb Z/r_i)^\times\to\mathbb Z/2^{m_i}\), \(m_i\to\infty\), with the prescriptions of the horizontal comparison, splitting also in a fixed finite normal \(F_0\) containing \(L\), the twisting data, Hilbert class field of \(K'\), and fields imposing fixed congruences. Kill fixed support, smoothing norms and the additional finite list of primes as there. Take smoothed classes from [eq:C3] in the corresponding group coefficients (include the split pair of ideals at \(r_i\)), with polynomial exponential coordinate \(Z_i\) orienting the variable to interpolate \(\bar\chi\) for characters \(\chi=t^{\lambda_i}\). These classes and the exponential polynomials have bounded denominators, by split group coefficients at 2. For the primitive characters used in [eq:T4]–[eq:T5] they evaluate on the exponential side as \(\Delta e_S L(f_B,\bar\chi,1)/p_\gamma\), by [eq:C3]; at \(t=1\) the ordinary class is \(\Delta e_r z_B\), \(e_r=1-a_{f_B}(r_i)/r_i+1/r_i\). In particular \(Z_i(1)=0\). Write \(Z(v)\) for the resulting bounded-denominator coefficientwise limit of \(Z_i(1+v)\), and \(Z'\) for its linear coefficient.

Frobenius and height transversality.We can prescribe \(r_i\to1\) and limiting Frobenius \(g\) with no root-of-unity Tate eigenvalue on any dyadic component, determinant one on each plane. In fact add to \(F_0\) all dyadic roots of unity and dyadic radicals of the primes to be killed, obtaining \(D_0\). Its maximal abelian part over \(F_0\) has bounded exponent over the cyclotomic tower, by conjugation on the translations by one fixed cyclotomic scalar differing from one. Each CM character has infinite image over \(F_0\) on the cyclotomic kernel (otherwise compare conjugation transports using normality and inversion on the infinite anti-ratio). The image is therefore still infinite on \(G_{D_0}\), by abelianness over \(F_0\).

Avoid the finitely many characters’ torsion preimages simultaneously (closed with empty interior). Frobenii approximating such \(g\in G_{D_0}\) are realized by Chebotarev, with the roots and radicals split to sufficient simultaneous depth; use the same primitive root/residue exponent convention as before. Write \(a_*\) for the limiting trace on the chosen plane.

We can additionally arrange a nonzero \(F\)-height \(h_{\lambda}^{\mathbb Q}(P,P)\) for this tame limit character, with polarization \(\lambda_B\) and the specified coefficient projection in this notation. Indeed form the biextension lifts used before with first argument \(P\) (Kummer \(x\) in full Tate), taking linear combinations of dual arguments by rational \(F\)-endomorphisms and scalar extension so as to project to our embedding (clear fixed denominators).

The resulting block and fiber cochains \(\beta,k\) have \(dk=-\beta\cup x\), with \(\beta\) representing the projected \(\lambda_{B,*}(P)\) up to sign; ordinary evaluations with \(x\) compute the projected pairing. The projected \(x\) remains nonzero on \(G_{F_0(T_2B)}\), by finite-extension restriction in characteristic zero and central homothety vanishing on the image. A homothety acts simultaneously by a nonidentity scalar \(m\): multiply an element over \(F_0\) of cyclotomic value \(m\ne1\) by its complex-conjugation transport. Adding the radical tower cannot kill the restriction: its translation quotient has conjugation by this element of homothety action according to the cyclotomic scalar \(m^2\), precluding such a nonzero continuous equivariant map. The joint kernel is normal over \(\mathbb Q\); the projected \(x\)-values there span the plane. The commutator of two elements there has zero \(x,\beta\), and its \(k\)-value is up to sign twice the projected symplectic pairing of the \(x\)-values, hence can be nonzero. This adjusts \(g\), preserving the preceding requirements, to give \[k(g)-\beta(g) g_T (g_T-1)^{-1}x(g)\ne0\] where \(g_T\) is the action on Tate. The division calculation of [eq:T2], valid on each of the biextensions in the combination and with \((g_T-1)^{-1}\) bounded on all components at the approximating primes, gives \(h_{\lambda}^{\mathbb Q}(P,P)\) equal to this value up to sign. All retrace rules for these heights use just the two Poincaré laws; norm is unramified over the same ring class fields of \(K'\), and pre-division downstairs uses the uniform 2-part bound for the residue-group size of \(B\).

The cohomological derivative.The cup computation modulo the squared parameter, as in [eq:T2], gives \[ Z'\log_\omega P =\pm\Delta(2-a_*)\alpha h_{\lambda}^{\mathbb Q}(P,P). \tag{20}\] Indeed after a fixed multiplier the degree-one derivative term of the cocycle satisfies \(d z_1=-\lambda_i\cup z_0\) up to character orientation with \(z_0=\pm\Delta e_r\alpha\beta\); make the adjustment in full dual Tate with coefficients if needed, the cohomological equality being by the norm and Selmer line and torsion bounded. Cup with \(x\) and add the corresponding \(\lambda_i\cup k\). At 2 the cup uses ordinary differential evaluation (projected class against \(x\)), and at \(r_i\) one trivializes \(x\) by the bounded-denominator Frobenius divisions, giving exactly the fiber calculation; remaining local terms have limit zero as before.

The spectral and intersection derivative.Use the holomorphic and intersection calculation of [eq:T3]–[eq:T6] at level \(N_B\), on the block of \(f_B\). It yields \[ \begin{split} C_Bp_\gamma L(f_B\otimes\varepsilon_{K'},1) \frac{Z'}{\Delta e_S} &=\pm(2-a_*)h_{\rm pr}^{K'}(P_0,P_0),\\ C_B&=\frac{\sqrt{|\operatorname{disc}K'|}} {8\pi^2(f_B,f_B)_{N_B}}. \end{split} \tag{21}\] Here \(P_0\) is the projected Hilbert trace on the Jacobian orbit factor with its restricted polarization, and \(h_{\rm pr}\) is the corresponding \(F\)-valued tame height at the chosen embedding. The two comparison steps are as follows.

1. Use the isolating Hecke polynomial before [eq:T5] with algebraic coefficients, clearing fixed denominators. The Gram and difference calculation is identical at the indicated embedding. For the second \(L\)-factor use a Betti period \(p'\) for the primitive companion form as in [eq:C1] and the same additive modular-symbol polynomials. It differs from \(p_\gamma\) by a nonzero algebraic multiple by quadratic twisting, isogeny and the CM eigen-differentials. Likewise \(p_\gamma\) is algebraically proportional to the Betti period in [eq:C1] at level \(N_B\). Thus \(C_B p_\gamma p'\) is algebraic; the period-normalized companion modular symbols have bounded denominators by the fixed relative period lattices (Manin–Drinfeld and the CM Betti line). These facts justify the bounded-coefficient interpolation and differentiation giving [eq:T5], now with \(p_\gamma,\Delta e_S\) in place of \(\Omega_0,d_0\).

2. The finite intersection matching is on the same modular curve and oriented Hilbert orbit of \(K'\), before projection. Inserting the Hecke projector into the pairing applies the restricted orbit polarization split by \(F\) (no divisor by a map degree in this notation). One can project by linear combinations of the correspondences after the unramified pull-ups in that proof. Dividing the scaled first argument downstairs on the orbit factor and retracing over \(K'\) then works by the same biextension rules and the uniform residue torsion bound from the Frobenius choices. This gives (21) after the cancellation in [eq:T4].

This spectral calculation uses the Hilbert trace itself and the nonzero companion value. It does not use \(P\), the Selmer-line identification, or the nonzero-height prescription.

Comparison with the absolute height.In the simple-zero case, [eq:GZ] gives \[C_B L'(f_B,1)L(f_B\otimes\varepsilon_{K'},1) =2H_{\rm pr}(P_0,P_0),\] with the absolute projected pairing. An \(F\)-equivariant isogeny to \(B\) sends \(P_0\) into \(F P\) rationally (also by the sign and rank in [eq:K0]); its pullback of \(\lambda_B\) differs from the restricted polarization by an element of \(F^\times\). Thus \(H_{\rm pr}(P_0,P_0)/H_{\lambda_B,F}(P,P)\) is nonzero algebraic, and \(h_{\rm pr}^{K'}(P_0,P_0)\) is twice that multiple of \(h_\lambda^\mathbb Q(P,P)\), by bilinearity, adjunction and extension from \(\mathbb Q\) to \(K'\). Consequently the complex height-normalized \(L'/p_\gamma\) in [eq:C4] is interpreted dyadically by an algebraic scalar, and the two derivative computations imply [eq:C4], cancelling the nonzero factors. This uses [eq:GZ] with its absolute-height normalization. ◻

Proof. The spectral calculation (21) remains valid: its proof used neither a simple zero of \(L(f_B,s)\) nor a point generating the Selmer line. For torsion \(P_0\), its tame height on the right is zero. All factors multiplying \(Z'\) on the left are nonzero, including \(L(f_B\otimes\varepsilon_{K'},1)\). Hence \(Z'=0\). ◻

Proof. Ordinary global cohomology of \(M\) (or \(M^*\), commensurable with it) rationally has a line in degree one only. Indeed Kummer Selmer has the rank given by classical GZK; strict dual localization has zero kernel inside it by nonzero local log in rank one, and local \(H^2\) at finite places vanishes by dual invariant vanishing. Thus Poitou–Tate kills rational \(H^2\) globally and the ranks follow by Euler characteristic. By [eq:C2] and Shapiro, a determinant generator for the positive complex of \(M^*\), written in generic cohomology, is \((\gamma-c\gamma)(1)\wedge z\), the first vector denoting real boundary and the second lifted rationally. Indeed the real action on \(M^*\) is regular with a free invariant basis as indicated, taking the twist sign into account.

We normalize \(v_2(2)=1\). To calculate the index, the positive Euler length of a finite quotient in comparing these lattices is minus its module length. Filter it residually (use a common sublattice or scalar multiple); constituents by CM semisimplification are trivial or irreducible induced from one nontrivial residual line on \(L\), the conjugate line then being its inverse. For a trivial constituent this Euler calculation is global squareclasses and Brauer reciprocity: positive \(h^1,h^2\) are \(|S_f|, |S_f|-1\), respectively, including the ordinary real restriction in defining positive, and there are no other terms. The arithmetic-cohomology comparison for squareclasses/Brauer follows as in the diagram conventions (the universal cover contains imaginary layers). For the induced line use Shapiro and the ordinary Euler characteristic \(-1\) on \(L\), with one real invariant line further removed. Additivity proves the length claim. The determinant change of module bases from \(M\) to \(M^*\) has valuation \(v_2(\mu)\) by actual polarization unimodularity; thus the change of \(d\) on fixed positive tensors from \(M^*\) to \(M\) is \(+v_2(\mu)\), by the length calculation. And \(a^*=\pm(\gamma(a)/\mu)(\gamma-c\gamma)\) before twist, by evaluation duality and anti-invariance. This proves [eq:C5]. ◻

Proof. In rank zero [eq:C3] gives for \(\exp^*z\) the scalar \(e_S L(E,1)/p_\gamma\) relative to \(\omega\). In rank one [eq:C4] gives, using any free Mordell–Weil basis \(P\), \[v_2\left(\frac{\log_\omega(\lambda_{E,*}^{-1}z)}{(\log_\omega P)^2}\right) =v_2\left(e_S\frac{L'(E,1)}{p_\gamma H(P)}\right)\] in the period-normalized sense there. Since \(p_\gamma\gamma(a)=\pm\Omega_0\) (connected period), these are exactly the log/Haar inputs of the central positive calculations giving [eq:L5] and [eq:P1], with a common multiplier of valuation \(v_2(\gamma(a))\) and no smoothing. Explicitly the determinant raw valuation formula there uses the integral real boundary basis \(a^*(1)\), subtracts \(\sum_{v\in S_f}\tau_v+\operatorname{length}H^1(\mathbb R,T_2 E)\) and the dyadic \(\mathop{\mathrm{Sha}}\)-length, adds \(2\tau_g\), and inserts \(v_2\) of the above scalar (exponential scalar at rank zero) plus the local log-image exponent \(b\). Haar absorbs \(v_2(e_S)\) and \(b\), giving the Tamagawa factors and the real component correction to the whole period. Thus \(d(C^+(M),a^*(1)\wedge z)=X(E)+v_2(\gamma(a))\). This calculation extends scalars by flatness, and if the generic elements require denominators to lift integrally the length formula applies first to a multiple, then by homogeneity. Comparing to [eq:C5] proves \(X(E)=0\). The basis \(P\) used here is a basis of the full Mordell–Weil group modulo torsion; the calculation therefore uses the regulator stipulated in Theorem 1. ◻

Proof. Poitou–Tate gives rational concentration of ordinary global cohomology in one degree-one line. The zero Selmer hypothesis makes its map to the dyadic singular quotient injective, hence nonzero. By [eq:C2], \(z\) generates that line; by [eq:C3], its nonzero singular localization implies \(L(E,1)\ne0\). Apply Corollary 95. ◻

Proof. Only the assumed simple zero of \(L(f/K,s)\), the Heegner rank argument [eq:K0], and the known-simple-zero comparison of Proposition 92 are used here. In particular, this proof does not use the CM corank-one converse proved below.

Concentration and the integral induction sequence.

Here \(K\ne L\) and the plane remains absolutely irreducible over \(K\) as in [eq:C4]. [eq:K0] gives ranks \(0,1\) on the even, odd functional-sign constituents over \(\mathbb Q\), the latter generated on primal Tate by \(P\) in Kummer, viewed as a rational point up to denominator of the sign-appropriate curve/variety by untwisting (\(-w(f)\) conjugation sign). Logarithm on that generator is nonzero as before. Thus ordinary global cohomology for each constituent over \(\mathbb Q\), on \(M^*\) or its quadratic twist by \(\varepsilon_K\), has just a line in degree one rationally by the same Poitou–Tate test. Initially include also the primes of \(D\), writing \(e,e'\) for the resulting omission multipliers of \(f,f\otimes\varepsilon_K\). [eq:C2] gives generators \(z_0,z_1\) for these inverse determinants by Shapiro from \(L\). The integral induction exact sequence with summands (as sub and quotient) \(M^*,M^*\varepsilon_K\) on diagonal and difference, and Shapiro from \(K\), then give \(\operatorname{res} z_0\wedge\operatorname{res} z_1/2\) as ordinary inverse determinant generator over \(K\). Indeed anti-diagonal insertion, corresponding to twisted restriction, composes to twice the identity on the quotient.

The localized determinant.By [eq:C3] and [eq:C4], localization of this pair at \(w\) is rationally an isomorphism on degree-one spaces, by nonzero exponential on the rank-zero factor and finite nonzero Kummer log on the other; local cohomology likewise has no other degrees rationally. Its inverse determinant volume is self-dual via the cup and invariant using the alternating Tate form divided by \(\mu\), by integral local duality. Hence the \(d\)-value of this localized wedge is the valuation of the cross pairing of the two localized vectors (including division of the wedge by 2). Indeed the discriminant of the determinant pairing on that basis is up to sign the square of the cross value since the finite vector is isotropic. The triangle for strict thus gives before removing \(D\) \[-d(C_{*,{\rm big}},1) =v_2\left(\frac{e e'\,(\log_{\omega,w}P)^2 L'(f/K,1)} {2\mu p_\gamma^2 H_{\lambda_A,F}(P,P)}\right).\] The mixed cup before division by \(\mu\) evaluates the dual exponential against primal Kummer log by \(\lambda_A^{-1}\) in the \(F\)-pairing. The sign-appropriate twist in [eq:C3] or [eq:C4] is identified geometrically over \(K\) and at \(w\), with the indicated \(\omega,p_\gamma\), and transports the absolute pairing unchanged. The exact period cancellation is \[\frac{L'(f/K,1)} {2\mu p_\gamma^2H_{\lambda_A,F}(P,P)} =\frac{\pm i}{\sqrt D},\] by [eq:C1] and [eq:GZ]. It is a dyadic unit because \(D\) is odd.

Removing the discriminant primes.Removing the primes of \(D\) by localization subtracts the valuations of just their \(f\)-Euler multipliers (good unramified representation over each ramified place, singular Frobenius block on \(M^*(-1)\); the twist Euler factors there are 1). The remaining primes contribute \(e_S^2\) since split, proving [eq:C-pair]. This proves the claimed determinant formula. ◻

Proof. We work over the DVR \(R_{\mathfrak p}\), for a height-one prime \(\mathfrak p\) not containing \(2\), and prove nonnegativity by successive line switches over \(L\). On the fiber of \(R_{\mathfrak p}\), all local dual invariants at the original allowed finite places vanish: at fixed places by constant local Tate torsion finiteness, at old moving places by inertia if \(v\ne0\) and the Frobenius conditions if \(v=0\). Global invariants also vanish by the kernel calculation below. Thus for the current primal problem (initially full), with unramified or transverse conditions as below at auxiliary new places and complementary conditions on the Tate-dual problem, we have \(h=\dim H^1=1+h^*\) on the fiber by Euler and Poitou–Tate, where \(h^*\) is the dual Selmer dimension in degree one, injecting into global cohomology also (full local \(H^0\) included on the switch planes). The Euler assertion persists under switches by the equal local quotient ranks; unramified inflation recovers the old complexes.

If \(h>1\), we arrange a fresh place \(Q=Q_i\) outside the current support, splitting in stage scalar fields and in preceding derivative fields, with \(\mathrm N Q\to1\), Tate Frobenius tending to 1, and with both a primitive reduction of the generic primal cohomology line and some nonzero dual fiber class evaluating nontrivially on its Frobenius. Over \(R_{\mathfrak p}\) the local evaluations are a split finite axis (value on Frobenius) and a singular axis (on a tame generator), of rank one each; conditions include degree zero and omit degree two. Each pure axis pairs trivially with the same opposite-side axis, with perfect mixed pairing: the limit action is trivial, one uses increasing roots-of-unity precision and the tame/invariant computation of the diagram conventions. Thus these are the local comparisons of Lemma 13, with the unramified plane the pure finite one.

Class-group corrections in the auxiliary ray fields. Denote by \(B_i/L\) the base cyclic extension from \(\lambda_i\), totally ramified at both places over \(r_i\). Preceding switches will use cyclic \(F_Q/L\) of order \(s_Q=2^{A_i}\) (parameters possibly different per \(Q\)), totally ramified just at \(Q\), unramified elsewhere, with each auxiliary place split in the base and all the other derivative fields. Choose fixed prime ideals \(I_j\) giving cyclic generators of the class group (cyclic factors with orders \(h_j\)), \(I_j^{h_j}=(x_j)\). For old auxiliary \(Q\) choose \((y_Q)=Q\prod I_j^{-n_j}\). Avoid these \(I_j\) in choosing new places. Include in \(D_i/L\) all \(2\)-power roots of unity, dyadic radicals of \(x_j,y_Q\), and \(B_i\) and all old \(F_Q\). Splitting at a new place to sufficiently high finite depth on this data permits \(F_{Q_{\rm new}}\) as indicated of any required finite \(2^A\) order for that depth: take a primitive residue exponent killing units and all \(x_j,y_Q\), prescribe it on principal ideals, and extend to ideals away from the modulus by killing \(I_j\), which respects the class-power relations. It kills each old \(Q\); ray reciprocity gives the assertion and total ramification from the residue image. Roots for killing the finite global units can be included by the roots-of-unity congruence.

Evaluation injectivity with uniform auxiliary data. In the product evaluation on \(\prod_i G_L\) for global cohomology classes in these problems, restriction is injective to the exact joint kernel fixing all \(D_i\) and the constant Tate actions termwise. In fact set \(K_T\) the extension by the character and cyclotomic Tate actions, abelian over \(L\) (here \(T=T_\alpha\)). Its intersection with \(D_i^0:=L(\mu_{2^\infty}) B_i\prod_Q F_Q\) is just \(L(\mu_{2^\infty})\), by unramifiedness at these good odd places and separate total ramification of the cyclic factors. Indeed the inertias surject separately to those whole factors alone, which are jointly independent by the same ramification. The abelian part of \(D_i\) over \(L\) has bounded exponent over \(D_i^0\), by one fixed nonidentity cyclotomic action on its radical translation subgroup over \(D_i^0\). Consequently the Tate image over \(D_i\) still contains fixed powers (uniformly in \(i\)) of the image over \(L(\mu_{2^\infty})\). To see this extend an automorphism of \(K_T\) over the latter tower identically on \(D_i^0\) and then lift to the compositum with the abelian part; the bounded power kills that part, hence kills \(D_i\cap K_T\). This gives a lift fixing \(D_i\). The \(T\)-character on the cyclotomic kernel has infinite image by the CM character argument. Thus some sequence trivial on the \(D_i\) acts by a fixed infinite-order scalar on \(T\) (inverse on the dual), central in the product quotient by the joint kernel. This proves injection by the central scalar test in characteristic zero, using also finite-list evaluation injectivity of the diagram conventions, and proves global invariant vanishing. On that kernel the values of the two indicated classes to be tested are additive, and neither test is identically zero; take them simultaneously nonzero.

Chebotarev and precision margins. Keeping elements trivial on the \(D_i\), perturb the resulting choices by arbitrarily deep powers of the preceding scalar lifts, so as to approach Tate identity with \(\rho_T-1\) valuation tending to infinity but finite. Powers may preserve all finite old cochain congruences to increasing precision. Take \(A_i\) much larger still. Chebotarev at finite levels now gives \(Q_{\rm new}\) with the stated evaluations and splitting data including residue prescriptions, with \[ \begin{gathered} v_2(\mathrm NQ_{\rm new}-1)>A_i,\qquad A_i-v_2(F_i-1)\longrightarrow\infty,\\ v_2(F_i-1)\longrightarrow\infty,\qquad F_i=\rho_T(\mathrm{Fr}_{Q_{\rm new}}). \end{gathered} \tag{22}\] All fields and class prescriptions may use deeper finite precision in \(D_i\) as necessary after choosing \(A_i\). The Frobenius used on old evaluations is unaffected by a local inertia adjustment at the new place. This proves the required compatibility inductively; the new conductor adds just one prime ideal at a time in the norm over \(L\).

Derivative classes and bounded descent. For subsets \(I\) of places switched so far, use smoothed elliptic-unit classes evaluated first in \(T\) over \(H_I=B_i\prod_{Q\in I}F_Q\), with support \(Q\in I\) added to the base support and the corresponding elliptic-unit conductors. Apply to them \(\prod_{Q\in I}\mathfrak D_Q\), \(\mathfrak D_Q=\sum_{j=1}^{s_Q-1} j\sigma_Q^j\), with \(\sigma_Q\) the image of an oriented tame generator at \(Q\). At total splitting in all other fields the lower norm on deleting \(Q\) uses \(p_Q=1-F_i/\mathrm N Q\) (with \(F_i\) at this \(Q\)), so \(p_Q/(F_i-1)\to-1\). These are the elliptic-unit smoothing, twist and prime-ideal Euler distributions specified at [eq:C2]–[eq:C3]. The group-ring identity \((\sigma_Q-1)\mathfrak D_Q=s_Q-N_Q\) (\(N_Q\) the group sum) gives global invariance in reduced cohomology to increasing precision, by the norm relation and the margins.

Uniformly on \(H_I\), invariants in \(T/2^n\) and \(H^1(T)\)-torsion are killed by a fixed nonzero multiplier (disjoint Tate data by total ramification as above). Thus any rational-versus-integral torsion errors in exact norm comparisons can first be killed in integral cohomology, and another fixed multiplier makes the invariant reduced classes descend to \(B_i\) by inflation-restriction (obstruction in cohomology of bounded-exponent invariants). These choices are common across subsets; uniqueness loss is likewise bounded.

Weight-corestrict the descended classes from \(B_i\) by the horizontal character, modulo increasing Artin precisions. Denote the resulting limit classes by \(Y_I\), with \(Y_\varnothing\) the common multiple of \(Y\); they can be used successively with old choices retained, slowing working precisions if needed.

The exact finite–singular interchange. For \(Q\in I\) these have the relations in the limit \[ f_{\rm loc}(Y_I)=0,\qquad s_{\rm loc}(Y_I)=\pm f_{\rm loc}(Y_{I-\{Q\}}) \tag{23}\] in integral axes as above. Indeed before descent, with other derivatives applied and the common multiplier understood, use integral cocycles \(b_Q\) upstairs and \(b\) below (over \(H_{I-\{Q\}}\) and restricted up). They vanish on their respective local inertias above \(Q\): the local coefficients are unramified, the residue cardinality is still \(\mathrm N Q\), and \(F_i\ne\mathrm N Q\), excluding any nonzero inertia map to torsion-free \(T\). Use \(\sigma_Q\) lifted in inertia, so \(\rho_T(\sigma_Q)=1\) and \(b_Q(\sigma_Q^{s_Q})=0\). Take the residue lift \(\mathrm{Fr}\) trivial in all the stage finite cyclic fields here (adjust by inertia in the totally ramified factor). The exact norm identity on upper global cochains has \[ N_Q b_Q=p_Q b+d e,\qquad (F_i-1)e=s_Q b_Q(\mathrm{Fr})-p_Q b(\mathrm{Fr}) \tag{24}\] for some integral \(e\), by the cohomological distributions and unramified evaluations. We can work separately with \(Q\) outermost in the derivative (orders commute on classes). Also \((\sigma_Q-1)\mathfrak D_Q b_Q=s_Q b_Q-N_Q b_Q\) holds on cochains up there, since the extra inner action by \(\sigma_Q^{s_Q}\) fixes this cocycle. Match the upper restriction of the reduced descended cocycle with \(\mathfrak D_Q b_Q\) up to coboundary. Applying \(\sigma_Q-1\) identifies the coboundary of its value at \(\sigma_Q\) with \(-d e\) modulo precision (the mismatch coboundary is \(\sigma_Q\)-fixed). Invariant errors cost bounded precision, so its value tends congruentially to \(-e\). The exact displayed equation, taken before reduction, makes this agree up to sign in the limit with \(b(\mathrm{Fr})\) by the margins, which also calculates the lower descent value there (a coboundary adjustment costs a term divisible by \(F_i-1\)). On \(\mathrm{Fr}\) itself the upper derivative \(\mathfrak D_Q b_Q\) has value \(s_Q(s_Q-1)b_Q(\mathrm{Fr})/2\), tending to zero, and descent adjustment again disappears. Total splitting makes the same argument hold on all translates with their transported generators, since the cyclic fields are abelian over \(L\). Thus the weighted traces retain these identities. Unswitching conditions at other new places not in a given subset are unramified for the corresponding classes by support. The cohomology classes with the indicated local coordinates lift to the stated Selmer problems over the test DVR by the localization triangles.

Reversing the switches. This is the unit line isomorphism on evaluations needed in the line switch, with classes integral at the test DVR. By the nonzero primal and dual tests, each such switch drops \(h\) by one on its fiber, preserving the generic line and its nonzero derivative by the triangles from the lower condition. Iterate with the new primes as above. At \(h=1\) integrality gives nonnegative order of the derivative coordinate there (one-line minimal complex). Reversing the determinant equalities proves \(\operatorname{ord}_{\mathfrak p} U\ge0\), since the common multiplier is a unit here. ◻

Proof. An abstract Selmer class and a transverse tame direction.

Parity gives odd functional sign. We will construct a tame exponential series \(Z(v)\) with \(Z(0)=0\) and prove \(Z'(0)\ne0\). Corollary 93 will then make the Hilbert trace nontorsion, so Gross–Zagier will detect the analytic simple zero. The argument for \(Z'(0)\ne0\) has two parts: a nonzero Selmer Bockstein gives a simple zero of the finite Selmer determinant, and elliptic-unit comparisons make the ordinary determinant coordinate a unit at the same characteristic-zero center.

Choose a nonvanishing split companion \(K'\) and the spectral data of Corollary 93. At this stage no simple analytic zero or rational point spanning the Selmer line is assumed. Use a single tame prime sequence \(r_i\) there with variable \(v\), order \(2^{m_i}\to\infty\), action by \((1+v)^{\lambda_i}\) or its inverse. We impose additionally [eq:P2]. More explicitly take \(x,\beta,k\) as in the unknown-center test [eq:P2]–[eq:P3], on primal and polarization-dual actual Tate over \(\mathbb Q\) and scalar twist (1) respectively, unramified-data cochains outside fixed support \(S\), integral up to fixed scalings. Here \(x\ne0\) in rational Selmer, \(\beta=\operatorname{pol}(x)\), \(dk=-\beta\cup x\), which exists by Poitou–Tate and local Kummer isotropy as there. Identify rationally the dual Tate with \(M^*\) after extending coefficients. On the kernel fixing full Tate as well as \(F_0\), roots and prime radicals of the Frobenius and height construction, the \(x\)-values span the plane: the argument there (central homothety over \(F_0\), exclusion of factoring the remaining values through cyclotomically conjugated translations, normality and absolute irreducibility) needs only nonzero cohomology of \(x\).

Choose \(g\) fixing the required auxiliary data and with determinant one and nonroot eigenvalues as there, and adjust by a joint-kernel commutator to achieve \[ k(g)-\beta(g)g_T(g_T-1)^{-1}x(g)\ne0 . \tag{25}\] The adjustment works by the same cochain identity. Choose \(r_i,\lambda_i\) by approximation to this \(g\), with all requirements of the CM horizontal sequence above, including splitting in \(L,F_0\), increasing residue-root/radical precision and killing fixed support and smoothing norms.

The ordinary and finite limiting complexes.Use the perfect limiting ordinary global complex \(C\) over \(R=\mathcal O[[v]]\) on \(T_\alpha\) over \(L\) with this scalar action, allowing fixed support and both places above \(r_i\); equivalently by Shapiro use \(M^*\) over \(\mathbb Q\). Include the smoothed elliptic-unit class \(Y\), whose exponential coordinate at the split variable specialization local at 2 (that is, the scalar action there is trivial for the whole variable) is the bounded-denominator series \(Z\) obtained by the exponential-coordinate construction, applied to this sequence. Smoothing is fixed nonzero as in that calculation. All comparisons here use the ordinary finite free and cup diagrams specified earlier. At fixed places the variable is trivial; at the two old moving places there is primitive inertia exponent, and at \(v=0\) the determinant-one Frobenius prescription and norm limit 1 make both ordinary local complexes rationally acyclic with unramified terms also acyclic. Thus unramified inflation there recovers fixed-support global cohomology rationally at the center.

The order-one Bockstein.Over the DVR \(R_{(v)}\) impose further at 2 the constant Kummer line on the induced representation, giving \(C_F\). Local cohomology there is the constant plane in degree one, with singular quotient measured by \(\exp^*\). At \(v=0\) Poitou–Tate gives just cohomology lines in degrees 1 and 2 for \(C_F\), by corank one and invariant vanishing (odd local terms acyclic). Its single minimal differential has order one. To prove this, suppose instead that its order were at least two (including the case of zero differential). Then \(\beta\) lifts on induced coefficients modulo \(v^2\), with finite localization at 2. This gives the same closed-cochain test as [eq:P3] over \(\mathbb Q\), now with no supplementary primes. Namely after a common multiplier, stage representatives modulo \(v^2,2^{n_i}\) to slower cofinal precision have constant term the fixed multiple of \(\beta\); their derivative \(z_1\) gives \(z_1\cup x\) plus the corresponding multiple of \(\lambda_i\cup k\) closed. Here identify commensurable lattices by maps with bounded denominators. The lift and adjustment are by the contractions and constant-term coboundary lift as in [eq:P3], retaining the dyadic localization and cup with \(x\) on constant local models there. Thus its invariant tends to zero at 2; other fixed places contribute zero after limit by bounded coboundary denominators (at infinity up to bounded torsion), and outside support the cochain is unramified. At \(r_i\), trivializing unramified \(x\) by bounded-denominator \(w_i=(g_{i,T}-1)^{-1}x(g_i)\) as there, \(g_i\) arithmetic Frobenius, gives the prescribed nonzero invariant limit from \(\lambda_i\cup(k-\beta\cup w_i)\), up to orientation and nonzero multiplier. This uses exactly the primitive compatible residue exponent and Frobenius convergence, and contradicts global reciprocity. This proves the order assertion, without assuming a global point spans.

Specialization at high-order characters. Let \(J\) be the constant dyadic singular quotient, a free line in degree one. The order-one differential makes \(C_F\) generically acyclic, so the triangle \(C_F\to C\to J\) identifies the generic cohomology of \(C\) with this single line. Write \(U\in\operatorname{Frac}R\) for the determinant coordinate of \(Y\), using a generator of \(\mathcal D(C)\). For all fixed roots of unity \(\zeta\) of sufficiently high 2-power order we have \[ v_2 U(\zeta-1)=v_2\Delta , \tag{26}\] with the fixed smoothing. Indeed nonzero pivots for the generic cohomology dimensions remain nonzero at these evaluations by one-variable preparation. At such a fixed root the actual character at stages is eventually allowed and the corresponding matrices converge with bounded pivot denominators, so those stage complexes have the same rational concentration (Euler characteristic \(-1\)). Their coordinates then satisfy the valuation test by [eq:C2] separately, increasing dyadic powers of the modulus if needed, and use the same \(\Delta\) since the smoothing norm is killed. Determinant base change and coefficient limits now give the assertion, including generic nonvanishing of \(U\). This uses the one-character determinant theorem as specified above, not an integral comparison across the whole tame group.

The determinant coordinate at the center. The Bockstein calculation and (26) verify respectively generic concentration and nonvanishing of \(Y\). Lemma 98 therefore applies to this very horizontal sequence. It gives nonnegative order for \(U\) at every height-one prime not over two.

Here is the one-variable consequence of these two comparisons. Write, by Weierstrass preparation, \[U=\pi^a\varepsilon(v)\frac{P(v)}{Q(v)}, \qquad \varepsilon\in R^\times,\] where \(P,Q\) are coprime distinguished polynomials and \(a\in\mathbb Z\). Horizontal nonnegativity forces \(Q=1\). For roots of unity of sufficiently high two-power order, the leading term of \(P\) gives \[v_2 U(\zeta-1) =a\,v_2(\pi)+\deg(P)\,v_2(\zeta-1).\] Equation (26) makes this expression constant as \(v_2(\zeta-1)\) tends to zero. Hence \(\deg(P)=0\), and in particular \(\operatorname{ord}_v U=0\).

Detection of the simple zero. Over \(R_{(v)}\) the triangle from \(C_F\) to \(C\) now gives \[ 0=-1+\operatorname{ord}_{v} Z, \tag{27}\] using the simple differential test and the constant singular line. Hence \(Z'\ne0\). Corollary 93 now shows that the projected Hilbert trace is nontorsion. The Gross–Zagier formula [eq:GZ], with the nonzero companion central value, gives \(\mathop{\mathrm{an}}(E)=1\). Corollary 95 then gives \(X(E)=0\). ◻

Proof. Suppose \(E\) is non-CM with \(W=E[2]\) having image \(S_3\) and imaginary quadratic subfield \(L\). Use a CM newform \(f_0\) lifting \(W\) on \(M^*\) as constructed in the CM comparison. Quadratic twists retain this lifting property; take \(L(f_0,1)\ne0\), then take imaginary \(K\) of odd fundamental discriminant \(k<-4\) splitting \(2N_E N_{f_0}\) with \(L(f_0/K,s)\) having a simple zero at 1. These choices use the nonvanishing theorems in Section 2, with their compatible sign and local splitting prescriptions [25, 9]. If desired, the comparator may first be twisted by a quadratic character ramified at every prime of \(2N_E\) where the form’s original level was good, locally trivial at primes already dividing that level (squareclass approximation). Those good local representations become ramified. Use a subsequent coprime twist split at that level with archimedean sign chosen for functional sign plus, for value nonvanishing. Thus all splitting primes required for derivative nonvanishing can be included in the actual level. Use the primitive form after twisting. Since \(L\) ramifies within its level, \(K\ne L\), \(W|_{G_K}\) still has full image. ◻

Proof. We construct \(h\) by successive fresh twists. The two curves \(E,E^k\) have opposite signs and no rational two-torsion, hence opposite finite two-Selmer dimension parities by 2-parity and Cassels alternation. Twist each time by a fresh positive good prime local square at all old relevant places, splitting in \(K\) and on \(W\); signs and conditions away from the new prime are unchanged. There on both curves the unramified plane switches to a transverse plane (Kummers of invariant two-torsion on the ramified twist evaluate surjectively on inertia, untwisted halves unramified and twist inertia negating them). By reciprocity and exact Kummer orthogonality the dimension change is \(\le 2-2j\) on either curve for old evaluation rank \(j\) there: new singular contributions annihilate the finite evaluations and the kernel is old strict there. Frobenius evaluations can be prescribed on a basis of the span of both Selmers in \(H^1(\mathbb Q,W)\): restriction to the joint kernel of \(W\) and abelian congruence data detects classes by the normal \(C_3\) without invariants, and additive joint evaluations span by absolute simplicity (the ground field is just \(\mathbb F_2\)). Use Chebotarev; the simultaneous residual evaluation statement is also the one in [40]. First evaluate with rank two on the odd group until it has dimension one. Then whenever the even group has dimension at least two evaluate with rank two on it and rank one on the odd group simultaneously (possible even for a line inside the even group). The resulting dimensions \(0,1\) and quadratic isogeny with parity give the assertion. ◻

Proof. For \(M^*\) the limiting local tame/Frobenius cochains split into \[U_{s,j}=[M\xrightarrow{v_j}M]\ (0,1),\qquad Q_j=[M\xrightarrow{\pm v_j} M]\ (1,2), \quad v_j=(1+u_j)^{\alpha_j}-1,\quad \alpha_j\in\mathbb Z_2^\times\] with \(M\) the scalar-extended plane locally. Use the two procyclic resolutions as in our diagram conventions (\(r_j^2\to1\)); \(\alpha_j\) uses tame generators \(\sigma_j\) generating also full relative cyclic inertia of order \(r_j+1\). At the center \(U_{s,j}\) imposes the pure singular plane as for [eq:K], i.e. for coordinates \(s_{\rm loc},f_{\rm loc}\) by evaluation at \(\sigma_j,\gamma_j^2\) it sets \(f_{\rm loc}=0\), includes degree zero and omits degree two. Using all the \(U_{s,j}\) instead of full there on the CM problem gives \[D=\mathrm{unit}\cdot D_s\prod_j v_j^2 . \tag{I1}\] The modified complex too has square amplitude 1,2 by residual duality (pure plane orthogonality as for [eq:K], including at residual precision as checked in the square switch). Also \(u_j\mid b^\pm\) on that problem: set \(u_j=0\), trace over the inert conductor step at high \(t=\theta\) in [eq:M2]. Each point is a descendant from omit-\(r_j\) conductor by this inert prime with transported level (canonical 2-subgroup preserved); the relative trace is Hecke with multiplier zero, and remaining weights descend. Bounded-series testing proves the assertion. ◻

Proof. Proceed inductively; Proposition 107 will construct primes satisfying both evaluations. Before switching a new prime, its unramified condition recovers the old diagram by inflation. Initially \(H^1(C_{F,\varnothing})\) is the Heegner line with nonzero logs by [eq:C-pair]. The finite problem has ordinary and conjugate self-duality over the fraction field, since the fixed odd local complexes are acyclic. These are the initial rank and duality assertions needed for the induction.

For these derivatives one applies \(D_I^{\rm der}=\prod_{j\in I}\sum_{a=1}^{r_j}a\sigma_j^a\) to the upper projected point, descends its Kummer class modulo increasing precision, and traces the Hilbert field to \(K\). Equation [eq:K] applies on full actual abelian Tate before projection and \(\lambda_A\): the Frobenius-square divisibilities hold on the whole CM variety, all CM Hecke eigenvalues here are zero, and descent uses fixed base local ramification. This gives [eq:I2], finite Kummer at two, pure singular conditions on \(I\), and unramified conditions at the unswitched auxiliary primes. Increase \(H\) on all compared classes together if further switches require it; bounded invariants remove the descent ambiguities, as in Proposition 69.

Inductively let \(Y_I\) span \(H^1(C_{F,I})\) with nonzero dyadic logs. The conjugation-stable conditions make this line an eigenspace. Relaxing both dyadic conditions adds two local quotient lines, exchanged by conjugation. Their boundary onto \(H^2(C_{F,I})\) is dual to the nonzero finite localization. Its kernel therefore supplies a class \(z\) of the opposite sign; the isotropy choices may be made equivariant by averaging over the fraction field.

By hypothesis, the evaluations of \(Y_I\) and \(z\) at the next prime are nonzero and lie in opposite \(J_j\)-eigenlines. The identity [eq:I2] and the square switch preserve ranks one and one, with \(Y_{I+j}\) spanning degree one. Cup reciprocity against \(z\) forces a nonzero dyadic localization of \(Y_{I+j}\): the mixed cup at the new prime is nonzero and every other odd contribution vanishes. Conjugation stability of the new line then gives nonzero logarithms at both dyadic places. Equation [eq:D] restores strict acyclicity. In particular the final modified determinant satisfies \(D_s(0,\mathbf0)\ne0\).

For the exact valuation, the left side of [eq:I3] is the class-functional valuation with the assigned dyadic volumes. Their comparison triangles are compatible with the separate odd switches; the adjunctions needed here are over the fraction field. At the new prime the mixed pairing is a unit times \(e(x,J_jy)\). Thus [eq:I2] and the square switch, with the integral plane-coordinate volumes, preserve this class-functional valuation. Its initial value is the right side of [eq:I3], including the common multiplier \(H\). ◻

Proof. We construct the primes inductively, retaining three properties of the chosen tuple:

1. The first cohomology of the central CM finite problem is a line generated by \(Y_I\), with nonzero dyadic logarithms, and its strict determinant is nonzero.

2. The first tame variable has nonzero Frobenius exponent at one place above each prime of \(S_f\).

3. Each chosen prime is inert in both \(K\) and \(L\), has the required CM involution limit, and has nonzero limiting trace on \(E\).

The initial central assertion is [eq:C-pair]. At each new prime we impose the two central evaluations of Lemma 106 to preserve it. After the first prime, we also impose a residual rank-two evaluation whenever the residual dimension is positive. We first show that these requirements force termination, and then prove that they can all be met simultaneously.

Dimension reduction. Choose the first prime \(r_0\) with the nonzero fixed-place exponents in \(u_0\). At \(t=0\), consider the DVR at \((\pi)\) of \(\mathcal O[[\mathbf u]]\). Write \(h\) for the degree-one dimension of the full-at-added-primes strict problem on its residue field. The central invariant and [eq:I1] make the generic problem acyclic. Fixed odd local complexes are acyclic over this DVR: their nonconstant unramified scalar has no invariants or dual invariants against the fixed residual matrices, and the odd local Euler characteristic is zero. The added inert local complexes are acyclic by their inertia scalars; the dyadic degree-zero terms vanish by the Frobenius test. Thus conjugate duality is perfect.

To apply [eq:H1] after inducing over \(\mathbb Q\), allow the quadratic ramification primes with the unramified conditions in that lemma. The coefficient is good there and the scalar unramified over \(K\); unramified inflation recovers the same global problem. At the acyclic odd places use the zero replacement allowed in that lemma. Corollary 101 now makes \(h\) even. The curve has the identical residual problem and hence the same dimension.

If \(h\ge2\) add a variable with evaluations of two independent strict classes of rank two in the finite plane at that new place (initial \(u_j=0\), residue generic field of the old variables with \(t=0\)). Over that field localization and duality are exact with either pure plane: the local action at this new place is trivial in the limit, and conjugate transport with pure cups and mixed pairing is as at [eq:K]. Thus changing to the pure singular plane drops \(h\) by two as in the square switch (the upper image is exactly the finite plane). This switched problem is a specialization of the problem using \(U_{s,j}\) before setting \(u_j=0\); with \(u_j\) also generic that condition and full give the same cohomology. Hence the new generic residual dimension is at most \(h-2\) by the perfect models. Even parity at each step then terminates with \[D|_{t=\pi=0}\ne0.\]

It remains to justify the simultaneous choices used in this induction. The first prime must supply the fixed-place exponents. Every later prime must supply the residual rank-two evaluation when \(h>0\). At every step the two central evaluations preserve the first invariant, and the non-CM trace must stay nonzero. The following constructions impose these requirements in that order without losing the earlier ones.

Ring-class characters and the first variable. In \(\operatorname{Pic}(\mathcal O_K)\) take cyclic factor generators represented by prime ideals \(I_d\) at distinct split primes outside the support and away from use, by Chebotarev. Write \(I_d^{h_d}=(x_d)\) for their order relations. For the chosen \(v\)’s put \((y_v)=v\prod I_d^{-n_{vd}}\). At inert \(r_j\to-1\) with \(2^{m_j}\mid r_j+1\), use the \(2^{m_j}\)-power residue symbol on \(\mathbb F_{r_j^2}^\times\) with compatible roots at the chosen place; it kills rational residues and maps onto the cyclic roots of that order. Require zero exponents on \(x_d\); then by the order class sequence the residue character extends killing the \(I_d\)’s on the ring class group, with exponents at \(v\) given up to conventions by \(y_v\). This produces the specified independent conductor variables (their relative inertia groups for the different primes multiply, using units just signs).

Take \(\gamma_j\) approaching elements \(g_i=c\eta_i\) (\(i\) indexes stages) where \(\eta_i\in G_K\) fixes \(W,\mu_{2^\infty}\). Thus \(g_i\) is also in the inert \(L\)-coset. Exponents on the above numbers of \(K^\times\) are read modulo \(2^{m_j}\) from the Kummer translations by \(g_i^2\), i.e. translations by \(\eta_i\) minus their values by \(\eta_i\) on the conjugate numbers. At the first step we can prescribe zero on \(x_d\) and all differences nonzero on \(y_v\) with fixed values as required. Indeed the numbers and their conjugates are jointly multiplicatively independent by valuations, and Kummer images on the specified kernel are open in the full tuple space (use valuations after the finite extension and rational restriction injectivity on passing to the cyclotomic tower by a central nonidentity cyclotomic scalar). Take these initial \(\eta_i\) constant. At subsequent steps instead require \(\eta_i\) trivial also on all the radicals, giving zeros on the class relations again.

Residual evaluations. In the residual tests for those subsequent steps use the evaluation groups \(\mathcal G_K=\prod_i G_K,\ \mathcal G=\mathcal G_K\rtimes\langle c\rangle\) of the diagram conventions. Degree one in question injects into unrestricted crossed classes (in particular by dyadic invariant vanishing just checked). Take the termwise product kernel \(\mathcal H\) over \(K\) of \(W\), ring-class and cyclotomic data and the radicals just used (including conjugates). Restriction here detects the classes. Before radicals the quotient has a normal product of the \(C_3\)’s (other data abelian over \(K\)), whose diagonal \(C_3\) kills its invariants and abstract \(H^1\); and a lift fixing the cyclotomic data acts trivially by conjugation on radical translations, so these give no equivariant homomorphisms either. The conjugate induction is absolutely simple since \(W\) is absolutely simple over \(K\) and the two scalar twists differ in determinant (already by the infinite-order image using \(u_0\)). Joint evaluations on \(\mathcal H\) thus span both copies of the induction for the two classes, by abstract Shapiro and the simple evaluation test. Evaluation on \((c\eta)^2,\ \eta\in\mathcal H\) acts via \(1+c\) and projection back to the first summands, additively and still spanning both tested planes. Some \(\eta\) therefore gives rank two by the quadratic determinant test.

Nonzero traces on the elliptic curve. Now perturb either choice of \(g\) by products of squares within \(\mathcal H\), preserving the residual test as well as any exact radical prescriptions. We can thus ensure \(a_{j,*}\ne0\) on \(E\), since these kernels contain uniformly an open determinant-one piece on the curve’s Tate data (non-CM open image with bounded-derived-length excluded data as in the evaluation discussion). Products of squares there still contain such a piece, so the coset trace is not identically zero even on limiting matrices. Preserve the nonvanishing by keeping henceforth \(E[2^a]\) actions fixed to sufficient fixed depth.

The two characteristic-zero evaluations. Lastly use squares within the deeper kernel fixing also \(E[2^a]\) and full CM Tate of \(A\). For the two central rational classes \(Y_I,z\) of the required opposite signs, evaluation still detects each by restriction here. Indeed these Selmer classes with finite or relaxed dyadic conditions inject into abstract crossed classes as before. A constant element \(h_0 c h_0 c\), with \(h_0\in G_{K L(E[2^a])}\) of cyclotomic value a fixed integer \(m>1\), acts by homothety \(m\) on CM Tate and \(M^*\) (diagonal CM characters with product cyclotomic, swapped by \(c\)), trivially on ring fields and the other designated finite data. It is central modulo the pre-radical kernel. And it acts on radical translations by the integer \(m^2\); so even abstract equivariant homomorphisms from the additional kernel to the plane vanish. Thus evaluations of either detected class on the deeper kernel span the simple characteristic-zero plane over \(K\). Multiplying \(g\) by squares there adjusts evaluations at \(g^2\) for a sign-\(\epsilon\) class by \(2(1+\epsilon J_j)\) applied to its kernel evaluations, additively: conjugation by \(g\) uses the eigenlaw on classes, with coboundaries trivial here. These projected changes span the required eigenline for each. Hence in characteristic zero the two nonzero tests can be met together without changing previous requirements.

Chebotarev at increasing precision now gives the primes, retaining the previous cochain evaluations and all stated field and Frobenius prescriptions, including the exact class-relation zeros to cofinal symbol depth. Local generators can be taken from inertia with the needed relative cyclic action. This validates [eq:I2]–[eq:I3] and the residual-dimension induction simultaneously (central and residual cocycles use their respective finite diagrams, extended before limits). ◻

Proof. We retain the descent multiplier throughout the coefficient calculation. At \(t=0\) use the disk-center formula of [eq:M1]–[eq:M3] with \(\Psi_{\mathbf u}\) the tame weights. Euler operators all act by translations, and opposite original-level sums differ up to sign and unit translation by Atkin–Lehner as there. The projected point at conductor \(\prod r_j\) in the sum is denoted \(X_I\).

To check the positive-orientation formula with additive congruences, first multiply the stage formula by \(H G(\Psi_{\mathbf u}(\mathrm{Fr}_w)^{-1})\), where \(G\) is a fixed Frobenius polynomial with \(G(1)\ne0\) sending all points on the actual abelian factor over dyadic unramified local fields into the small formal-log range. Use unramified Néron base change, component order, the abelian and toric Frobenius relations and a characteristic power for unipotent reduction, then a fixed integer in the formal range as in the unramified log comparison. Here \(\mathrm{Fr}_w\) uses the corresponding CM class action. Thus the weighted logs are now of translates of \(G(\mathrm{Fr}_w) H X_I\), uniformly bounded and with log congruences for division up to the same fixed loss. All coefficients with a variable dropped from the weighted sum vanish by the inert Hecke traces.

Work modulo increasing precision with each \(u_j^2=0\) (group powers map there, absorbing fixed Euler denominators). Write the labels as Hilbert representatives times \(\prod_j\sigma_j^{k_j}\). In the mixed coefficient only weights \(\prod_j\alpha_{j,i} k_j\) contribute (\(\alpha_{j,i}\) stage exponents): expanding the product of exponents of the full labels, all other terms drop at least one relative weight and use trace zero. Euler multipliers contribute just \(e_S\). The resulting additive sum without these unit exponents and Euler multipliers is the log on the Hilbert-label sum of translates of \(G(\mathrm{Fr}_w)H D_I^{\rm der}X_I\). Its limit is \(G(1)\log_\omega Y_I\). Indeed restriction back up of the traced descended class is exactly the corresponding Hilbert sum before \(G\) in Kummer modulo growing precision. This comparison takes place on full actual abelian Tate after the common multiplier and before final lattice projection/\(\lambda_A\); the descended trace classes at \(w\) are represented by local Kummers as in the descent proof. Thus the point congruence after restriction is modulo division in the upper unramified local fields, so transfers to a uniform additive congruence after \(G\). On the base-field local point \(G\) acts by \(G(1)\); base logs of such finite-level representatives converge to the localization log along the cohomology models at the fixed group (Kummer injection and compactness), with the stated polarization convention. Passing precisions therefore computes the mixed coefficient. On the series side multiplying by the group polynomial likewise uses only \(G(1)\) in that coefficient, by the \(u_j\)-divisibilities. Cancelling yields [eq:I4], also at the opposite orientation (same Euler constants). More explicitly, [eq:I3] and the initial value of \(D_{\rm pre}\) give \[v_2(D_{\rm strict,I}) =2v_2(\log_\omega Y_I)-2v_2(H)+2v_2(e_S).\] By [eq:I4], this is the valuation of \(\widetilde B(0,\mathbf0)\), and it is finite because the central derivative has nonzero logarithm. Equation [eq:I1] identifies \(\widetilde D(0,\mathbf0)\) with \(D_{\rm strict,I}\) up to a unit. The two scalar values therefore have the same valuation. This proves the stated central ratio, retaining every power of two in \(H\). ◻

Proof. We first establish integral divisibility, then use the central scalar ratio to prove that the quotient is a unit. At high finite \(t=\theta\) use \(\mathscr A=\mathcal O(\theta)[[\mathbf u]][1/2]\), where \(\mathcal O(\theta)\) is the integer ring after extension; test at its height-one DVRs as in [eq:B1], [eq:D2]. We spell out the applicability here.

1. The split dyadic Kummer lines and weighted log identification over extended constants of the local comparisons there apply. The \(\mathbf u\)-actions are unramified with some nonzero Frobenius exponent by construction. Thus the substitution from the unramified Shapiro tower (after splitting \(\theta\) locally as the ramified-field character and unramified part) is flat as discussed there. Local Heegner fields of the combined conductor contain the fixed ramified character field and arbitrarily large fixed unramified layers eventually, since they cut \(\theta\) and have the growing unramified orders. High \(\theta\) is nontrivial on inertia after the indicated fixed reduction extensions. These are precisely the hypotheses for bounded Kummer/log transfer also on projected abelian Tate and several-variable cyclic quotients.

2. Fixed odd singular determinants removed in imposing the characteristic-zero unramified conditions multiply to \(D_{\rm odd}=\prod_{q\in S_f,\ q\ne2}P_q(\chi(\mathrm{Fr}_v))P_q(\chi(\mathrm{Fr}_v)^{-1})\) up to units, \(\chi=\theta\Psi_{\mathbf u}\), \(v\mid q\). Use the local coinvariant computation of [eq:B1] (for the CM form this is the usual good/bad local Euler factor on the induced character realization; inertia has finite image and the Hecke \(L\)-function gives the Frobenius polynomial on its unramified part). None vanish on a whole \(u_j=0\), by the all-zero tame specialization and Frobenius weights. The added inert locals are acyclic on the test except possibly on the CM branch at primes \((u_j)\). Indeed away from these divisors tame inertia alone suffices; on the curve branch the Frobenius-square limit minus one is also invertible by \(a_{j,*}\ne0\), determinant \(-1\) on the original Frobenius limit, and all the scalar weights trivial on its square.

3. Formula [eq:M2] on these conductors and the weighted-log identification give [eq:B1] in the form \(B(\theta,\mathbf u)=\mathrm{unit}\cdot p_0^2 D_{\rm odd}\) over extended 2-inverted constants. Here \(p_0\in\mathscr A\) is the localized weighted Heegner class coordinate at \(w\) on the free Kummer line, up to common point/descent multipliers (using the unnormalized class sums). Membership and log comparisons are on finite cyclic quotients before the unramified limits exactly as in the local comparison, with no division by a growing degree. In the exceptional \((u_j)\) tests on \(f_0\), [eq:I1] has order exactly two at \(u_j\) for high \(\theta\), since \(D_s(0,\mathbf0)\ne0\). The \(u_j\)-divisibilities of both \(b^\pm\) hence show \(p_0^2/(D(\theta)/D_{\rm odd})\) has no pole there (order in \(u_j\) is unchanged by extending constants).

4. At every other test, if \(B(\theta)\ne0\), apply the square switch proof of [eq:D2] with that primitive divisor \(D(\theta)/D_{\rm odd}\) (nonzero by residual regularity). All local lines/conditions and duality at old places apply by the preceding comparisons. Degree zeros vanish by Tate absolute irreducibility over \(K\) (even before scalar character, on either plane), and Selmer \(H^1\)’s inject into unrestricted crossed classes including at surplus fibers. The new derivative primes for this test are additional inert primes, tending to conjugation on full Tate data as at [eq:K], not changing the variables. That proof applies on the modular abelian factor also, with fixed dyadic conductor and old moving conductors unramified at the required fixed places, high-character Kummer membership as above. For each surplus test the conjugate induction on the fiber is absolutely simple by Tate simplicity and distinct twisted determinants (high dyadic inertia in \(\theta\)). Restriction to product kernels fixing full Tate and ring-class data detects evaluations by the same central homothety trivial on ring fields used at [eq:D2]; hence squares of elements \(c\eta\) with \(\eta\) there provide rank-two evaluations including one of a primitive reduction, as in that proof. Thus [eq:K] and exact plane comparisons give the nonnegative class-functional valuation at the DVR by switching to minimal fiber rank with integral classes (fixed multipliers are units). This says \(p_0^2/(D(\theta)/D_{\rm odd})\) has no pole here either.

Consequently normality and [eq:B1] give \(B(\theta)\in(D(\theta))\) after inverting two in the extended series ring; this is automatic if \(B(\theta)=0\). Residual regularity at \(t=0\), a tame Weierstrass change, and the high-character remainder test give \(D\mid B\) integrally for both planes.

Let \(U_0=B_0/D_0\) be the integral CM quotient. Canceling the common \(\prod_j u_j^2\) gives \(\widetilde B=U_0\widetilde D\), so its constant coefficient is the scalar ratio computed in Lemma 108. It is a dyadic unit, and hence \(U_0\) is a unit. Finally the congruences \(B_E\equiv B_0\) and \(D_E\equiv\epsilon D_0\pmod\pi\), together with \(D_0\bmod\pi\ne0\), identify the quotient reductions by cancellation. Thus the elliptic quotient is also a unit. ◻

Proof of Proposition 102. Use the unit identity of Proposition 109 first at \(t=0\) on a group-power tame line retaining generic nonvanishing and nonzero dyadic Frobenius exponent, localized at its characteristic-zero origin. The rank-detection argument in the proof of Proposition 73 applies there, with the following local checks: dyadic conditions use the unramified Kummer and log comparisons at that DVR, specializing to usual Kummer. All odd local complexes are contractible there including the inert moving ones by \(a_{j,*}\ne0\). Thus the all-dyadic-finite problem has special rank one by \(s_2(E/K)=1\) and inflation, with only the paired lines in special cohomology. The \(t=0\) log formula gives the square of the weighted trace localization up to units there (all Euler terms nonzero at the origin by their weights); again one may first multiply by the Frobenius bound of the unramified comparison. Strict generic acyclicity and nonzero log then give the generic line assertions by [eq:D], and zero valuation of the class-functional tensor by the unit equality. Hence the global class specializes nontrivially exactly as in the rank implication there. This specialization is the Heegner Hilbert trace on the curve times \(\prod_j a_{j,*}\) (and any fixed nonzero class multiplier), by the inert Hecke relations. By [eq:GZ-E] the curve’s split product therefore has a simple zero and hence finite Tate–Shafarevich over \(K\).

At the all-zero center, let \(F_j\) be the limiting Frobenius matrix on the curve at the \(j\)-th inert prime. It has \(\det F_j=-1\) and \(\operatorname{tr}F_j=a_{j,*}\), so \[\det(F_j^2-1)=\det(F_j-1)\det(F_j+1)=-a_{j,*}^2.\] The full-to-unramified singular block is in degrees \(1,2\), with Frobenius square and residue size tending to one. Its determinant therefore has valuation \(2v_2(a_{j,*})\). Inflation identifies the remaining strict complex with the original strict complex. On the numerator [eq:M3] the inert traces in each orientation give exactly the same valuation multipliers. Cancelling therefore gives the equality of valuations for the true strict complex and paired conductor-one log used at [eq:E], now with support \(S_f\) (the calculation there includes extra split good primes just the same way by Haar). Thus [eq:E] and [eq:G] give \(X(E)+X(E^k)=0\), with the general split odd discriminant \(k<-4\); here the split Gross–Zagier normalization and isogeny comparison are as specified at [eq:GZ-E]. Restoring the original notation, this proves \[\mathop{\mathrm{an}}(E^h)+\mathop{\mathrm{an}}(E^{hk})=1,\qquad X(E^h)+X(E^{hk})=0.\] The product \(h\) is an odd positive fundamental discriminant, or \(1\), prime to \(2N_E\); the odd negative discriminant \(k\) is prime to \(2hN_E\), and every prime dividing \(2hN_E\) splits in \(K\). These are precisely the requirements of Proposition 4. ◻

Proof. The character has no conductor at infinity, since its order is three. It kills rational ideles: anti-invariance makes its value on a rational idele equal to its inverse, and an element of order dividing both two and three is trivial. At split primes the two conductor exponents agree. At a ramified odd prime a positive conductor exponent is even. Indeed conjugation acts trivially on residue units and by \((-1)^j\) on the last contributing principal-unit layer of index \(j\); anti-invariance forces \(j\) to be odd, so the conductor exponent \(j+1\) is even. If \(2\) ramifies in \(L\), wild inertia in the \(S_3\) image contains a transposition. Its normalizer has order two, so the decomposition group has order two and the cubic character is unramified there. If \(2\) is unramified in \(L\), the character exponents at two are at most one, by tameness.

These local statements give conductor \(b\mathcal O_L\). Triviality on the local units at the corresponding ray depth, together with triviality on rational units, implies triviality on the units of the order of conductor \(b\). Thus \(\psi\) factors through its ordinary ring class group. Lift \(\psi\) to characteristic zero and use the conductor formula for induction. The inertia-invariant dimensions agree after reduction, including for an order-two subgroup; the odd wild-invariant dimensions agree as well. This gives the asserted odd Artin conductor.

Chebotarev supplies \(\ell_0\): the prescribed \(3\)-cycle and the condition \(\ell_0\equiv3\pmod4\) are compatible because \(L\) is real. Formula (28) preserves the squareclass of \(d_L\). The conductor of \(\mathbb Z[\sqrt n]\) contains all required odd factors and also contains two when \(d_L\) is odd. These are exactly the remaining possible character-conductor factors. ◻

Proof. First take an odd fundamental signed twist, prime to \(2nN_E\), with functional sign plus; use a split negative discriminant if the sign must change. Since \(W\) has no rational invariants, the two-Selmer dimension is even by the parity comparison of Section 2. While this dimension is at least two, take a positive good prime \(p\equiv1\pmod4\), locally square at all earlier support, with trivial Frobenius on \(W\), at which two independent old classes have rank-two joint finite evaluation. Such a prime exists: restriction to the kernel of \(W\) and of the abelian congruence data detects the classes by the normal \(C_3\) without invariants. The joint evaluation image is additive over \(\mathbb F_2\), and absolute simplicity and independence make it simultaneously full. Chebotarev then imposes the required evaluation and congruences.

At the new prime the old unramified plane and the new twist Kummer plane are transverse. The Kummers of the two-torsion points give the two inertia coordinates: untwisted halves are unramified, whereas twist inertia negates them. Reciprocity makes the new ramification images annihilate the old evaluations. The remaining kernel is the common strict Selmer group, so the dimension drops by two. Iterating produces \(E_0=E^h\) with trivial two-Selmer group.

Choose \(d\) by Chebotarev and quadratic reciprocity. The residue conditions expressing splitting of all primes of \(2nhN_E\) in \(K\) force \(d\) to split in \(L\), which is compatible with a \(3\)-cycle on the abelian intersection. Genus theory gives odd class number, and \(d\equiv7\pmod8\) excludes extra units. The local conditions for \(E_0\) and \(E_0^{-d}\) agree at every finite place: at the old support this follows from splitting, and at \(d\) the local cohomology vanishes by the cycle Frobenius. At infinity one Kummer line can change. The twist’s two-Selmer dimension is therefore at most one; the coprime split twist sign and parity make it exactly one. Quadratic induction now gives \(s_2(E_0/K)=1\). ◻

Proof. The Betti exponential sequence and its 2-adic period lattice identify this group with the components of \(J(\mathbb C)^\sigma\), or its quotient by norms, compatibly with Kummer. We will assign a weight to each real circle of \(X\), show that the resulting divisor count is a Hecke eigenfunctional, and evaluate it on the Hilbert trace.

Weights on the real circles.

A \(\sigma\)-fixed interior point gives an anti-linear isogeny \(S\) of the elliptic torus, of degree \(n\) and with the specified cyclic kernel: compose the quotient map with the anti-complex Fricke identification. Since \(S^2=n\), the rational period lattice is a line over \(L\), with \(\sqrt n\) acting by \(S\). Identifying this line with \(L\) gives a fractional ideal \(I\subset L\), well-defined up to \(L^\times\), with multiplier order \(O\supset O_0:=\mathbb Z[\sqrt n]\).

The ideal \(I\) is locally invertible over \(O\). Indeed, locally scale a quadratic-order lattice into the maximal order, with both minimal valuations zero in the split case. It can then be scaled by a unit to contain 1, avoiding at most two proper residual subspaces. A lattice between \(\mathbb Z_p\) and the maximal order that contains 1 is an order. This gives the asserted invertibility over the multiplier order, and hence a class \(a=[I]\in\mathop{\mathrm{Pic}}(O)\). No odd prime \(p\) divides \([O:O_0]\): otherwise the form of nested quadratic orders gives \(p\mid n\) and makes \(S\) scalar modulo \(p\), hence zero, contradicting its cyclic kernel.

Use the extension map \(\mathop{\mathrm{Pic}}(O_0)\to\mathop{\mathrm{Pic}}(O)\) to test whether \(\psi\) descends to \(O\). Give the point weight zero if it does not; if it does, continue to denote the descended character by \(\psi\) and give the point weight \[\theta=\psi(a)+\psi(a)^{-1}\] in characteristic two. Replacing \(S\) by \(-S\) conjugates the ideal class and therefore preserves this weight. There is no order-two elliptic stabilizer modulo signs: multiplication by \(i\) cannot preserve the level at \(\ell_0\). Multiplying \(S\) by a possible order-three elliptic automorphism, up to sign, conjugates it by a level automorphism up to sign, since the anti-complex isogeny inverts these scalars. Thus these choices also preserve the weight.

The weights are constant on each real circle. On a compact local uniformizing chart, trivialize the integral period lattice. The anti-linear maps of fixed degree have bounded norm, so only finitely many integral matrices occur. Each branch therefore has constant ideal-class data; at a branch meeting the possible choices have the same weight by the preceding sign and automorphism calculation. No cusp is fixed: its denominator divisor of \(n\) is unchanged by \(c\) and exchanged with its Fricke complement by \(w_n\), whereas \(n\) is nonsquare.

From circle weights to a component functional. There exists a real point: on the chosen Hilbert orbit \(\sigma\) acts on the conductor-one labels by \(z\mapsto a_0z^{-1}\) for a fixed \(a_0\). The class number is odd, so \(z^2=a_0\) has exactly one solution. Denote the corresponding fixed CM point by \(x\). Every invariant line bundle class therefore has a real divisor: use the fiber at \(x\) to lift the real action with square one, then apply real Riemann–Roch after adding a sufficiently large multiple of \(x\).

For such a divisor, count its points on each real circle modulo two, multiply by that circle’s weight \(\theta\), and add. A principal invariant divisor is the divisor of a function that can be scaled to be real; its parity on each circle is even. Norm divisors also have zero count. The weighted count consequently descends to degree-zero divisor classes and vanishes on the identity component, which consists of norms. This defines the required component functional.

The Hecke eigenlaw and the Hilbert trace. We can compute the Hecke action at generic real points, since small motions preserve the component counts both before and after the correspondence. Nonreal pairs contribute zero. On the unbranched locus without automorphisms modulo signs, real lifts under \(T_p\) correspond to \(S\)-stable \(p\)-lines. At inert \(p\) there are none. At split \(p\), the descended anti-isogenies give the two prime-ideal translates of \(I\), with the same multiplier order. If \(\psi\) descends to this order and \(\mathfrak p\mid p\), their weights add to \[\bigl(\psi(\mathfrak p)+\psi(\mathfrak p)^{-1}\bigr) \bigl(\psi(a)+\psi(a)^{-1}\bigr).\] If \(\psi\) does not descend, all these weights are zero. This proves \(\Theta T_p=\operatorname{tr}W(\mathrm{Fr}_p)\Theta\).

It remains to compute the weight of \(x\). The character \(\psi\) factors through its multiplier order \(O\). The only case not already settled by the odd-index argument is nonzero character conductor at two when \(n\) and \(d_L\) are odd. Here \(S\) interchanges the two parts of the split \(\mathcal O_K\otimes\mathbb Z_2\)-lattice invertibly, by anti-linear scalar conjugation. It cannot be scalar modulo two, so the overorder index is odd in this case as well.

The CM action of \(\sqrt{-d}\) anticommutes with \(S\). On \(L\) it therefore has the form \(\beta\iota\). Its image \(\beta I^\iota\subset I\) is an \(O\)-submodule of index \(d\). The prime \(d\) splits in \(L\) and is prime to the order conductor, so this image is \(\mathfrak d I\) for one of the primes \(\mathfrak d\) above \(d\). Taking ideal classes gives \[[\mathfrak d]a=[I^\iota]=a^{-1},\qquad [\mathfrak d]=a^{-2}.\] The choice of \(d\) gives \(\psi(\mathfrak d)\ne1\), whence \(\theta(x)=1\). All other points in the Hilbert orbit occur in nonreal pairs. Since the detector prime \(\ell\) has cycle Frobenius and \(\ell+1\) is even, applying \(T_\ell-\ell-1\) leaves this count equal to one. This proves [eq:R0]. ◻

Proof. Rational plane traces of group elements belong to \(A\) reducing to traces of \(W\), by density. Componentwise reductions have semisimplification \(W\) after residue extension by trace and determinant, hence are absolutely irreducible. Choose four group elements spanning the residual matrix algebra (by absolute simplicity); their trace Gram determinant is a unit. They therefore span all the Galois matrices integrally, a free rank-four algebra acting on \(M\), reducing to the matrix algebra. Lift a primitive residual matrix idempotent by completeness. Its associated left ideal is free rank two and the algebra acts as all matrices on it by Nakayama. This gives the assertions including the decomposition (matrix Morita equivalence); \(D\) is free over \(\mathbb Z_2\), not required free over \(A\).

All occurring primitive forms are non-CM (else on an imaginary quadratic field the Tate reductions semisimplify reducibly, impossible for \(W\)). Their levels away from \(2,\ell_0\) are exactly the given one since residual conductor is a lower bound (inertia invariants, and odd wild invariants). At \(\ell_0\) exponent one is impossible by cycle Frobenius versus Steinberg with trivial nebentypus. Also at \(2\mid n\) no old form of good level occurs. Otherwise its modular abelian variety of good reduction at two would yield a finite flat model of the local \(W\): it occurs as a global two-torsion subquotient, by the rational Tate realizations and lattice independence of reduction semisimplification (the chosen coefficient embedding reduces to a scalar extension of \(W\)). Take closures and quotients over the DVR. The unique local line would give a finite flat subgroup of order two, thus \(\mu_2\) or \(\mathbb Z/2\) over \(\mathbb Z_2\) (order-two finite flat classification). The nontrivial generic point of the quotient extends by properness, so lifting it gives the residual quadratic radical by a torsor under that subgroup, a unit or unramified radical. This contradicts \(v_2(d_L)=3\) with decomposition of order two. Thus \(2\) is purely Steinberg then. The only multiplicity increase is possible by \(\ell_0^2\)-raising a primitive form unramified there. ◻

Proof. Choose an odd imaginary discriminant \(k'\) splitting \(2N_gN_{E_0}\) with \(L(g\otimes\chi_{k'},1)\ne0\), by the split sign and twist nonvanishing as at the outset. This last form has zero Selmer rank there: use [eq:K0] with a further imaginary split Heegner companion giving simple twist zero by derivative nonvanishing. The Tate absolute irreducibility required holds over any such imaginary field (residual \(W\) with real quadratic \(L\)). This uses the split Gross–Zagier input to [eq:K0].

Over the field of \(k'\) use one full and one strict condition at each split pair over \(2N_gN_{E_0}\), comparing a Tate lattice for \(g\) and the elliptic \(T_2E_0\) at a common scalar DVR. Use [eq:H1] on induction including good unramified conditions at the odd discriminant places. These are perfect self-dual problems in degrees \(1,2\) (residual invariant vanishing, injectivity for conditions in degree zero and duality), with identical residual diagrams. Thus the characteristic-zero strict-pair dimensions have equal parities. Odd local factors are acyclic in characteristic zero by invariant vanishing for the abelian Tate realization and the odd Euler formula. At the split dyadic pair the full/strict plane and the plane of finite Kummer lines meet in a line, with no local invariants; both are Lagrangian in degree one for the conjugate pairing, which is symmetric (cup sign and alternating determinant form, with conjugate local invariants identified). For the form the finite lines are exact orthogonals by abelian polarization and Hecke projection, with Lie dimension one. Thus passing to the ordinary Selmer problem flips parity by the symmetric-plane comparison of the arithmetic diagram conventions. The ordinary dimension on \(E_0\) there is odd by quadratic induction, elliptic parity and split sign. Hence also on \(g\); decomposing by quadratic induction with the twist Selmer rank zero gives the claim. ◻

Proof. The complex \(U_j\) and its degree-two-shifted dual, and the two analogues for a complementary line, are Koszul models of \(R/(\alpha_j,m_j)[-2]\) (regular sequence even over the order). Pairing morphisms are thus determined on \(H^2\), which can be checked on \(H^2,H^0\) pairing after reduction to that quotient (last-term cohomologies commute with reduction here). Then the local \(K\)-action is trivial, conjugate transport via \(\gamma_j\) is by \(\Phi_j\) up to scalar unit, and \(e(\Phi_j l,\Phi_j l)=0\) with the opposite entries for complementary lines units by the determinant form \(e\). Indeed the last term gives the tame-residue class up to unit trace normalization by the procyclic cochains and local duality at each quotient precision; Shapiro cup uses the invariant over \(K_v\), not twice that map. This proves nullhomotopy and quotient duality over \(R\).

Thus \(C\) has perfect self-duality and a minimal square model \(\delta_C:F\to G\) in degrees \(1,2\) by residual invariant vanishing over \(K\), injectivity of the local condition maps on \(H^0\), and duality. The full strict complex also has square amplitude \(1,2\) as before (dual invariant vanishing, and strict at \(w\) removes the global Euler contribution). ◻

Proof. Frobenius weights at fixed places. At \(q_0\) require that the Frobenius exponents in \(u_0\) at every fixed place over \(S\) are nonzero. Indeed take principal generators \(x_v\) for the \(h(K)\)-th ideal powers of one chosen place at each split prime. Along inert \(q_0\to-1\) the power symbol of order \(2^a\mid q_0+1\), \(a\to\infty\), kills rational residues and reads the translations on \(2^a\)-roots by \(\gamma_0^2\). Thus by ring class reciprocity and odd \(h(K)\) it detects the exponents via \(x_v\). Take lifts approaching \(c\eta,\ \eta\in G_K\) fixing the cyclotomic data and with cycle image giving \(W(c\eta)\) a cycle (compatible on the abelian intersection). On its square the translations are differences on conjugate numbers. Vary \(\eta\) within the deeper kernel of \(W\); the Kummer tuples for the \(x_v\) and their conjugates there with cyclotomic fixed have open image by independent valuations even after the finite extension (rational Kummer restriction survives the cyclotomic tower by a central nonidentity scalar). Thus all differences can be kept nonzero; apply Chebotarev. At the residual generic field in the \(u\)’s with \(t=0\) this gives vanishing local \(H^0\) at both dyadic places and acyclicity at fixed odd allowed support.