Proof. These are the arithmetic-volume identities of Cassels and the restriction-of-scalars comparison of Milne [6, 19]. The ratio is formed with the period lattice, full point lattice, torsion, component factors, and finite Tate–Shafarevich group; the kernel and cokernel factors of an isogeny cancel in this combined ratio. The analytic \(L\)-function is isogeny invariant. Finiteness in the rational low-rank applications is supplied by Theorem 4; it passes to the products and restrictions used here. Thus no conjectural equality of either ratio with one is required. ◻

Proof. Negative odd fundamental discriminants in an admissible fixed congruence class have positive density. Prescribe splitting at \(2Np\), impose any additional splitting conditions, and exclude a finite set of quadratic fields. The zero-density exceptional set in Theorem 6 cannot exhaust these classes. We can therefore choose \(D\) so that \(a(E^D)\leq1\).

Keep \(D\) fixed. Choose \(D'\) prime to \(D\), subject to the same splitting conditions and, in addition, \(D'\equiv-1\pmod{|D|}\). Together with a sufficiently divisible modulus supported at \(2Np\) and any other prescribed primes, these conditions contain admissible classes of positive density. Since \(|D'|\equiv1\pmod{|D|}\), quadratic reciprocity gives the required Jacobi symbol. Apply Theorem 6 to the two fixed curves \(E\) and \(E^D\). The union of their exceptional sets still has density zero. Thus one can arrange both \(a(E^{D'})\leq1\) and \(a(E^{DD'})\leq1\).

For a fundamental discriminant \(b\) prime to \(N\), the root-number multiplier is \(\chi_b(-N)\). Our splitting and sign prescriptions make this multiplier \(-1\) for \(D\) and \(D'\), and \(+1\) for \(DD'\). The functional equation and the ranks at most one therefore give the ranks in (ii). The two quadratic factorizations of the \(L\)-functions over \(K\) prove the last assertion.

The three quadratic subfields of \(L\) have discriminants \(D,D',DD'\). The biquadratic discriminant formula yields \[|\operatorname{disc}L|=D^2(D')^2 =\operatorname{disc}(F)^2.\] The relative discriminant of \(L/F\) has norm one, so it is trivial. The field \(F\) is real and \(L/F\) is CM. Excluding the finitely many quadratic subfields responsible for additional roots of unity gives the last assertion of (iii).

The finite Galois extension \(\mathbb Q(E[p])/\mathbb Q\) has only finitely many quadratic subfields. A nontrivial intersection with the biquadratic field \(L\) would contain one of the three quadratic subfields of \(L\). Exclude these possibilities when choosing \(D\) and \(D'\): after \(D\) is fixed, prescribing either \(\mathbb Q(\sqrt{D'})\) or \(\mathbb Q(\sqrt{DD'})\) excludes only finitely many choices of \(D'\). The same argument excludes the CM field. Finite exclusions do not alter the preceding positive-density argument. ◻

Proof. Consider \(A=\operatorname{Res}_{K/\mathbb Q}E_K\). It is isogenous over \(\mathbb Q\) to \(E\times E^D\). Its point group and Tate–Shafarevich group are \(E(K)\) and \(\mathop{\mathrm{Sha}}(E/K)\), and its \(L\)-function is \(L(E/K,s)\). Its Neron period is \(A_E/\sqrt{d_K}\). To see the discriminant factor, use an integral quadratic basis for \(\mathcal O_K\) in the restriction of the invariant differential; the determinant of the dual Lie lattice has real Jacobian \(1/\sqrt{d_K}\). The differential stays minimal over \(K\): primes of bad reduction split, and good reduction stays good.

The finite component product for \(A\) is \(\prod_\ell c_\ell(E)^2\). At a bad prime, the two split places each contribute \(c_\ell(E)\). At a ramified prime of good reduction, the special restriction fiber is an extension of the good special fiber by a vector group and is connected, so there is no extra component factor. At the remaining good primes the component factor is one. These period and component calculations are the restriction-of-scalars arithmetic comparison [19, 24].

Let \(P_0\) generate the full free group \(E(K)/E(K)_{\rm tors}\), and write \(P_K=mP_0\) modulo torsion. The rank-one regulator for \(A\) is \(2H_{\rm abs}(P_0)\); hence \[\frac{2H_{\rm abs}(P_K)}{\mathop{\mathrm{Reg}}_A}=m^2, \qquad [E(K):\mathbb ZP_K]=|m|\,\#E(K)_{\rm tors}.\] Dividing (9) by the BSD arithmetic volume just computed gives the right side of (10). Proposition 5 identifies the same ratio with the product on its left. Taking \(v_p\) gives (11). ◻

Proof. Continuous cochains with free coefficients over an Artin ring \(A\) are flat: continuous functions are filtered colimits of finite free modules indexed by finite clopen partitions. Tensoring them with a finite scalar module gives the corresponding continuous cochains. A flat module over a local Artin ring is free. Indeed, lift a basis modulo the maximal ideal; nilpotence proves surjectivity, and flatness followed by nilpotence proves that the kernel is zero.

Split the residual complex into its cohomology and contractible disks, and lift the graded bases. Write \(d_0\) for the disk differential and \(i,q,h\) for the inclusion, projection, and contraction, normalized by \(h^2=hi=qh=0\) and \(1-iq=d_0h+hd_0\). For \(d=d_0+\epsilon\), the formulas \[ \begin{aligned} I&=(1+h\epsilon)^{-1}i,&Q&=q(1+\epsilon h)^{-1},\\ H&=h(1+\epsilon h)^{-1},& d_{\rm small}&=q\epsilon(1+h\epsilon)^{-1}i \end{aligned} \tag{13}\] are finite series in the nilpotent maximal ideal. Substitution, using \(d^2=0\), gives \(dI=Id_{\rm small}\), \(Qd=d_{\rm small}Q\), \(QI=1\), and \(dH+Hd=1-IQ\). The remaining ranks are the residual cohomology dimensions; hence they have the asserted bounds. Inserting \(I,Q\) transfers maps, and inserting \(H\) transfers their composition homotopies. Tensoring the contractions transfers the cup maps and their homotopies. Cones may be formed from transferred maps and then contracted.

Functions lift across coefficient quotients, so the contractions can be chosen compatibly. At a fixed precision the matrices belong to finite sets. Their ultrafilter values are therefore defined, and all finite matrix identities survive passage to the limit. Exact triangles survive by retaining their cone identifications; duality survives by retaining its maps and contracting the comparison cones. Because the models are free, ordinary tensor of these models is derived tensor. No inversion of \(p\) enters this argument.

For a fixed group, compare the limiting model at each precision with a contraction of its actual cochains. The two compositions are homotopic to the identities. These comparisons preserve the marked maps. Passing to compact inverse limits introduces no \(\lim^1\) term, by finiteness and compactness. When the diagram is enlarged, keep the old contractions, transfer the additional maps, and decrease the working precision along the ultrafilter if necessary. Any finite list of new identities is then retained at a cofinal sequence of precisions. ◻

Proof. Let \(G_c\) be the compact-support fiber of global localization. Poitou–Tate pairs \(G_c\) with the unrestricted dual global complex with shift \(-3\), and local Tate duality pairs \(L_v,L'_v\) with shift \(-2\). Dualizing \(G_c\to C(U)\to\bigoplus_vU_v\) and using the given local equivalences identifies the dual fiber with \(C(U')\). The same triangle identifies the boundary maps as adjoints.

At finite precision these are the cup, invariant, and compact trace diagrams of local and global duality [20, 21]. The top scalar local and compact cohomology groups, in degrees two and three, are the coefficient ring; their traces are compatible with reduction and corestriction. Retain these maps and the reciprocity homotopy in Lemma 10. Perfectness is checked on the residue field and lifted by contracting the comparison cone. This also proves compatibility after specialization. For split local complexes the required nullhomotopies are checked on the split cohomology; for unramified conditions they are supplied by the residue factorization above. ◻

Proof. Keep the inclusions of the finite models into actual cochains. At residual precision a bounded list of group elements detects invariants and detects whether the values of a model cocycle have the form \(g a-a\) for one \(a\) in the coefficient module. The bound comes from linear equations in the coefficient space and in the bounded model space. Mark the resulting evaluation map to \([M\to M^{\rm list}]\) in degrees zero and one. It is an isomorphism on residual \(H^0\) and injective on residual \(H^1\). Its cone has zero residual cohomology in degrees at most zero. A minimal model of the cone therefore has zero terms there. This remains true after every field base change and proves the required injection.

For every sequence of group elements, and every pair or finite tuple of sequences, the evaluation matrices have compatible limits. The cocycle and coboundary equations consequently hold on their product group. Conjugation identities hold on classes by the marked homotopies. The Selmer assertion follows from the localization sequence and the hypotheses on the local maps. ◻

Proof. The generic finite evaluation is nonzero, so the lower complex is generically acyclic. In the fiber, reciprocity against the nonzero opposite finite evaluation prohibits a transverse contribution. The two localization triangles therefore lower the dimension by one. Those same triangles identify the class tensors through the common lower tensor \(1\); the local isomorphism changes its volume only by a unit. At dimension one the minimal model is a free line in degree one, and the integral class has nonnegative coordinate valuation. ◻

Proof. Adjunction identifies the annihilator of the upper localization image with that same image, so the image is self-annihilating in the four coordinates. The two generic classes form a basis of the upper degree-one space. In the dual basis of \(H^2(C_-)\), write \(Y^*,(Y')^*\) for the functionals normalized to take value one on their respective upper basis vectors. These are dual basis vectors, distinct from the pairing functionals \(Y^\vee,(Y')^\vee\) used in the determinant tensors. Choose a rational complement \(x\) to \(f(Y)\) whose boundary is \((Y')^*\). The cross-pairing gives \(e(x,J s(Y'))\) a unit value, and \(s(Y')=Jf(Y)\) therefore makes \(f(Y)\wedge x\) a unit plane volume. The complement need not be integral: if both classes are multiplied by a uniformizer \(\varpi\), the normalized dual \((Y')^*\) and this complement are divided by \(\varpi\). Interchanging the planes gives the same volume. Thus both triangles identify the class-functional tensors with the same lower tensor, proving the equality. At a rank-two fiber the upper image is exactly the finite plane, giving the dimension drop. At the terminal fiber the model is two free lines with zero differential and unimodular pairing. The tensor is integral there. Compatible nullhomotopies on the lower comparisons are part of exact duality and are retained at all unchanged places. ◻

Proof. Use \(C_0\), strict at \(w\) and finite at \(\bar w\), and the triangles \(C_0\to C_F\to U_w\) and \(C_0\to C_*\to L_{\bar w}/U_{\bar w}\). The latter quotient is \(U_w^\vee[-2]\). Its boundary of \(e^*\) pairs with \(Y\) by \(a\). The two triangles therefore compare the tensors with a factor \(a^2\), and their local lattice volumes are dual. This gives the formula and the rank assertions; the converse follows by the same nonzero adjoint boundary. This is the determinant calculation of [24]. ◻

Proof. The measure of the identity-component points for the minimal differential is the nonsingular residue size divided by \(\ell\), namely \(P_\ell(1)\). Multiplication by the component number gives \(c_\ell P_\ell(1)\). Away from \(p\), the formal subgroup has pro-\(\ell\) order and contributes no \(p\)-primary valuation. At \(p\), a sufficiently small subgroup is identified by formal logarithm with its additive lattice. The \(p\)-primary part of the logarithm’s finite kernel has size \(p^{\tau_p}\), and its image has measure \(p^{-l}\); the prime-to-\(p\) kernel contributes a valuation unit. This gives the second identity. The real identity follows by translation of the minimal differential across the components. ◻

Proof. Derived Kummer reduction gives the usual finite Selmer groups; passage to divisible coefficients gives the usual Kummer sequence. Whole-Sha finiteness therefore identifies integral degree one with the compact Mordell–Weil module. The torsion in degree two is the finite quotient in the compact-to-divisible comparison and has length \(s_K\). Duality gives torsion length \(\tau_g\) in degree three, as well as a unimodular pairing on the free parts in degrees one and two. The torsion lengths in degrees one, two, three are consequently \(\tau_g,s_K,\tau_g\). Their inverse-determinant contribution is \(2\tau_g-s_K\), and the point and its pairing functional each contribute \(n_P\). This proves the formula, including under conjugate-field duality; compare [24]. ◻

Proof. Choose integers \(n_j\), one equal to \(1\) and the others with suitably large successive \(p\)-adic valuations, so that restriction to \(1+u_j=(1+x)^{n_j}\) separates a nonzero leading monomial of the reduction. Extend this to a group-power coordinate change. Then \(L\) is distinguished, up to a unit, in one tame coordinate. Applying this to a product handles a finite list. Prescribed characteristic-zero generic ranks and nonvanishings at \(t=0\) may also be retained: their leading homogeneous forms in logarithmic coordinates impose finitely many proper conditions on the integer choices, which can be avoided within the specified congruences.

Divide \(B\) by the monic Weierstrass polynomial of \(L\). At each high \(\theta\) its distinguished degree is unchanged, and its quotient module is \(p\)-torsion-free. Thus divisibility after inversion of \(p\) makes the integral remainder zero. Every coefficient of this remainder, considered as a bounded series in \(t\), vanishes at all high finite characters and is therefore zero by one-variable Weierstrass preparation. The argument also applies componentwise to finite coefficient orders embedded in products of DVRs when the same residual regularity is retained. It is the proof of [24], with \(p\) in place of two. ◻

Proof. Coefficientwise converging integral series converge uniformly on smaller closed disks. A zero of multiplicity \(m\) of a nonzero limiting denominator can be isolated on such a disk. Weierstrass preparation there retains its zero count on an ultrafilter-large set. If stage numerators are divisible by stage denominators after inversion of \(p\), the limiting numerator has at least that multiplicity. This pole test permits uncontrolled stage denominators because boundedness is imposed on numerator and denominator separately. Tame characters, when present, are fixed before this one-variable argument. ◻

Proof. Its prime generator is not divisible by \(p\). A weighted coordinate change makes it distinguished in one variable. Modulo its Weierstrass polynomial the norms of the finitely many nonzero functions to the remaining power-series ring are nonzero. Choose the other coordinates, in finite extensions and in their open unit disks, avoiding these norms; bounded-series avoidance gives such a choice. A root of the distinguished polynomial is also interior and belongs to a finite extension. It gives the required point. ◻

Proof. First prescribe power-residue data. In the rational case the radicands are the fixed primes themselves. In the imaginary case choose ideal-class generators away from the prescribed places, generators of their principal power relations, and generators for expressions of the prescribed prime ideals in those classes. Include the conjugates. For a split prime \(\mathfrak q\), take a class-number power \(\mathfrak q^h=(a_{\mathfrak q})\). The corresponding anticyclotomic residue exponent is, up to its common generator normalization, the Kummer exponent of \(a_{\mathfrak q}/\overline{a_{\mathfrak q}}\). These elements are independent by their valuations at the chosen split pairs.

Their Kummer image over the cyclotomic tower has full \(\mathbb Z_p\)-rank. One way to see the required restriction injectivity modulo \(p^a\) is to use a central cyclotomic element whose scalar differs from \(1\) by a unit; such an element exists since \(p\) is odd and splits in the fixed fields. A further fixed finite extension loses at most a bounded power of \(p\). A \(\mathbb Z_p\) relation annihilating the Kummer image would consequently give power relations modulo arbitrarily high \(p\)-powers, contradicting the independent valuations. Thus sufficiently deep prescribed exponents can be chosen independently.

To extend an inertia logarithm across the class group, write the fixed class-group relations in invariant-factor form. Require the power-residue exponents to be divisible by the fixed \(p\)-parts of these invariant factors, and divide the relations to assign the values on the ideal generators. Principal relations then define the whole character. Anti-invariance is imposed using the conjugate generators. The fixed divisions change neither the full rational span of the limiting exponents nor their independence. Finite Chebotarev conditions on roots of unity and on this radical list realize these power-residue prescriptions. Taking enough moving primes gives the independent row vectors in the assertion.

We next check compatibility with the elliptic and fixed character conditions. Adjoin all \(p\)-power roots of elements of the base to the cyclotomic tower, and consider the joint compact \(p\)-adic analytic image of the Tate representation and the fixed characters. The common Lie quotient with the radical translation group is abelian. In the non-CM case it kills the \(\mathfrak{sl}_2\) factor of the Tate image, by Serre’s open image theorem [26]. In the CM case, after a fixed extension, the connected Tate image is a torus. Conjugation on the remaining abelian analytic quotient is trivial after a finite extension, whereas conjugation on radical translations is multiplication by the cyclotomic character. Choosing that character different from \(1\) proves that the common Lie quotient is zero. Hence fixing all radicals removes no Lie dimension from the analytic image.

Make the prescribed radical data sufficiently deep that an identity neighborhood remains available. Its determinant-one Tate projection contains an open special linear subgroup, or an open norm-one CM torus. The equation \(\det(V(\gamma)-1)=0\) is a proper analytic condition there. The weight character is nonconstant by its nonparallel labelled \(p\)-adic exponents, and the true anticyclotomic character is nonconstant by its inertia description. After shrinking the neighborhood, the torsion condition on the weight is the zero-logarithm locus. Avoiding these finitely many proper analytic loci gives the required Frobenius limits. The companion has the same Tate action on \(G_L\).

Only finite radical and coefficient tables are used at any stage. They can therefore be combined into finite Chebotarev prescriptions with accuracy tending to infinity. If a bound on \(v_p(r_{j,i}-1)-n_{j,i}\) is required, perturb by a sufficiently high power of a fixed near-identity element with nontorsion cyclotomic projection. Its radical geometric sums vanish to the required precision, while its cyclotomic displacement has valuation \(n_{j,i}+O(1)\). No boundedness of the unramified \(p\)-local degrees of the full conductor fields is used. ◻

Proof. At residual arithmetic precision, the dual of the localization kernel is the space of everywhere locally trivial constant \(\mathbb F_p\)-characters, by Poitou–Tate. Such a character is unramified everywhere and split at the support; it is therefore a class-group character killed by the chosen generating prime classes. Over \(\mathbb Q\) the same group is zero. The scalar representation and its cohomological comparison do not depend on the artificial variables. Tensoring its residual diagram with \(\mathbb F_p[[\mathbf u]]\) and then its fraction field preserves this injection. Equivalently retain the injectivity cone in Lemma 10. Thus a locally zero degree-two scalar class is zero, giving the primitive. ◻

Proof. Keep \(t=0\) and lift the residual test to the \((p)\)-DVR of \(\mathbb Z_p[[\mathbf u]]\). Non-\(p\) local complexes are contractible there, since their residual tests are acyclic. The remaining one-sided problem is perfect in degrees \(1,2\), with conjugate self-duality. It is generically acyclic. To check this last assertion, specialize centrally in characteristic zero. The moving-place singular determinants are nonzero because \(V(\gamma_j)-1\) is invertible. Inflation therefore gives the fixed-support problem. The simple-zero choices give one compact Selmer line from points, and its localization at either \(p\)-place is nonzero because the local logarithm has torsion kernel. At non-\(p\) places the rational local elliptic cohomology vanishes by the absence of elliptic invariants, local duality, and Euler characteristic zero. The strict–finite triangles of Lemma 15 now give strict acyclicity. A nonzero central determinant is nonzero generically.

We spell out the sign that controls specialization. On the induced coefficient, in its two \(K\)-coordinates, the pairing is \[ ((a,b),(a',b'))\longmapsto e(a,b')+e(b,a'), \tag{19}\] where \(e\) is the elliptic alternating pairing. This is alternating and Tate-equivariant, including under complex conjugation, which swaps the coordinates and reverses \(e\). Its compact cup gives a graded skew pairing of shift \(-3\). Split-coordinate local conditions are isotropic. Unramified conditions have the residue cup nullhomotopy, and contractible local conditions may be replaced by zero. At a newly allowed discriminant place the Tate module is unramified over \(K\); quadratic inertia on the induced coefficient swaps the two coordinates. Its invariants form the graph of a character unit. Restriction to this graph has the same residue nullhomotopy, since the residue group has cohomological dimension one and the quadratic index is invertible. The real Tate terms are zero.

Graded cup commutativity, the cone cup, and their homotopies are identities at finite \(p\)-power precision. Transfer them by Lemma 10. Antisymmetrization is allowed because \(2\) is a unit. On a minimal two-term model the resulting perfect pairing between degrees one and two identifies the differential with an alternating matrix: if \(P\) denotes the degree-three pairing, its chain identity gives \(P(x,dy)=P(dx,y)=-P(y,dx)\). The residual and generic nullities of an alternating matrix have the same parity. Generic acyclicity therefore gives even residual nullity.

This also explains the precise extension of the conjugate Bockstein argument in [24]. No characteristic-zero division is used in transferring the pairing to the residual field. For the later characteristic-zero localizations the same calculation applies with the constant-inertia unramified comparisons of Section 3; their nullhomotopies are again the residue ones. ◻

Proof. The primitive exists by Lemma 24. At each \(p\)-place one of the paired classes is strict, with the specified local nullhomotopy; away from \(p\) the residual local complexes are acyclic. At the later horizontal tests the paired unramified cups instead vanish through the residue complex of cohomological dimension one. Thus the cup is locally zero. Apply the arithmetic sum of local invariants to the degree-two cochain \[ x_1\cup d-\lambda\cup u, \tag{20}\] with the simultaneous sign convention making it closed. At old places \(\lambda\) is zero and the local condition removes the remaining cup. At the new place \(x,d,u\) are unramified. The Frobenius-minus-one operator is invertible, and \(A\) trivializes \(x\) in the residue complex. Subtracting this coboundary leaves the unramified scalar cochain \(u-A\cup d\). Cup with the tame direction has invariant equal, up to sign, to its Frobenius value. Here the Frobenius fixes the \(p\)-power roots of unity, and the direction is normalized by the power-residue exponent in the compatible root basis. The mixed residue–inertia cup is therefore the usual local reciprocity pairing. Over \(K\) the two places give the indicated difference. If several new directions are combined, the logarithm of one is unramified at the other new places and gives no additional mixed contribution.

We verify the residual interpretation explicitly. Mark the old global cochain inclusions and their values on the Frobenius sequences, then use the inflation homotopies in adjoining the new primes. Their restrictions to new inertia are zero, and the induced residue representatives have the Frobenius values just prescribed. The invertible residue operator removes their degree-one cohomology, leaving exactly the scalar residue cochain above. Mark its root basis and the mixed cup map too. All the coboundary corrections, restriction homotopies, and the compact invariant equation are finite matrix identities from actual arithmetic diagrams.

To use classes over \(k_r\), clear their finitely many denominators in \(\mathbb F_p[[\mathbf u]]\), including those in the solution for \(u\) and in the inverse residue operator. The resulting equations hold in the corresponding localized finite free models. Any equation which holds only after that localization becomes an equation over \(\mathbb F_p[[\mathbf u]]\) after multiplication by one more common nonzero parameter denominator. Now use the Artin rings \(A_b=\mathbb F_p[[\mathbf u]]/(\mathbf u)^b\), with the first-order new variables adjoined as needed. The prime \(p\) is zero throughout this residual calculation. The cochain inclusions and homotopies identify its equations with the corresponding actual stage arithmetic identities over \(A_b\); no lift of an arbitrary residual solution to \(\mathbb Z/p^n\) is asserted. Passing to the limit in \(b\) and then inverting the same nonzero parameter denominators proves the stated equation over \(k_r\). This does not divide by \(p\). It also shows that further field extensions preserve the test. For characteristic-zero field points one may instead clear fixed scalar denominators and increase the arithmetic precision, as in Lemma 10. Thus the primitive computation applies with its residual arithmetic identities established explicitly. ◻

Proof. Let \(J\) be the termwise joint kernel of the full Tate and old character actions, including \(L\) and the fixed splitting data over \(K\). It is harmless to retain the fixed character data as well. On \(J\) the evaluations of \(x,d\) are additive. They detect their classes: old inertia at one moving prime gives a central element in the quotient image whose scalar difference from \(1\) is invertible. Apply Lemma 12. Independent classes in a simple representation have evaluations spanning their full tuple, jointly for nonisomorphic representations.

In the rational case the opposite representations are already nonisomorphic on old inertia, which sees \(\chi\) and \(\chi^{-1}\). In the imaginary reducible case include the conjugate evaluations. The resulting four representations are distinct: the elliptic characters are distinct, and opposite infinite scalar twists cannot be identified. For an absolutely simple plane take \(d_j=x_j^c\) under conjugate duality. The two oppositely twisted planes are nonisomorphic for the same inertia reason.

Multiply a chosen Frobenius representative by a commutator \([v,w']\) with \(v,w'\in J\). Since the full Tate action is fixed, both elements fix the cyclotomic tower. Their radical actions are translations in an abelian group. The commutator therefore fixes all radicals, as well as the Tate and old character data. The additive evaluations of \(x,d\) on the commutator are zero, so the local correction involving \(A\) is unchanged. The primitive equation changes its scalar evaluation by \[ -e(x(v),d(w'))+e(x(w'),d(v)), \tag{21}\] up to the fixed sign convention.

Joint spanning and bilinearity now give elementary rectangular matrices in the rational problem and elementary square matrices in a character problem. For a plane over \(K\), take the conjugate difference. With \(d_j=x_j^c\), the transposed comparison has the opposite sign by Weil skew-symmetry; hence the changes span \(E_{ij}-E_{ji}\), the elementary alternating matrices. This uses linear span over \(k_r\), not an assertion that every vector in that span is a single Galois evaluation. Only finitely many commutators are needed to span the finite matrix spaces. Differences from the initial Frobenius test belong to the span of the resulting directions, so their linear span has the asserted size.

Realize the required Frobenius representatives by Chebotarev at increasing finite precision, preserving the radical and evaluation tables of Lemma 26. Normalize all root and tangent bases in the same diagrams. A common nonzero sign, or a nonzero scalar multiplying an entire tangent direction, does not change its span. For conjugate places use the transported marking, so the two entries are compared with the same normalization. ◻

Proof of Proposition 23. If a tested strict space is zero no new direction is needed. Otherwise Lemma 27 supplies a linear combination of directions whose necessary lift test has zero left kernel. In the plane case this uses Lemma 25: \(s\) is even and an alternating matrix with \(s/2\) invertible skew blocks is nonsingular.

Work in the enlarged residual ring, localized where the old variables have fraction field \(k_r\) and all new variables are zero. The new local complexes are acyclic at this point, so the specialized global diagrams are the old ones. Cancel their unit differential blocks. In the remaining minimal complex, the derivative of a lifted degree-one vector in a new tangent direction is the first-order differential. A vector in its kernel would pass every necessary cup test. The chosen test has zero kernel, so this first-order differential has full column rank in that direction. Some maximal minor of the actual differential has a nonzero initial term. It is therefore nonzero over the enlarged generic residual field, which kills that strict space.

Take the union of the finite direction lists for all simple constituents, both sides, and both imaginary curves. Each test retains its own slice, since every new local prime was chosen Frobenius-acyclic there. Long exact sequences for the residual composition filtration then give the desired vanishing for the original representation, including nonsplit extensions.

Over \(\mathbb Q\) the dual strict vanishing gives \(H^2=0\) for the primal full complex. Its Euler characteristic is \(-1\), so it has one residual degree-one line. Minimality first at \((p,t)\) and then at \((p)\) gives a free line over \(R_{(p)}\). Over \(K\), one-sided \(H^1\) and \(H^2\) vanish at the generic residual test. The square determinant consequently has nonzero reduction modulo \((p,t)\), on both sides and for the companion. This is [eq:source-1]. ◻

Proof. At the old residual test the new prime has unramified scalar Frobenius involving an old variable with nonzero exponent. Its Frobenius-minus-one determinants are nonzero; the local complex is acyclic over the residual fraction field. The localized full-to-unramified triangle multiplies the determinant by a nonzero residual singular factor. Hence specialization of each enlarged determinant remains nonzero at the old residual test, which proves the assertion. ◻

Proof. At a height-one test at least one old inertia scalar has infinite order: two independent variables cannot both specialize to roots of unity on a single prime divisor. Adjust the old Frobenius exponents of a new prime as above. On a line in an infinite-order coordinate, an elliptic eigenvalue exception excludes only boundedly many integer choices. A sufficiently large finite grid in all the old exponents therefore avoids the exceptions for the finite list of fields and both representations. Keep the nonzero-exponent requirement, all Tate and fixed character conditions, and the radical conditions making the new logarithms zero at old places. This gives simultaneous Frobenius acyclicity.

Exact orthogonality at the old unramified places uses Lemma 11. Their cup contributions and first-order contributions vanish through its residue nullhomotopy, since the new logarithm is locally zero. Scalar localization is obtained from the constant model and Lemma 24; it remains injective after these characteristic-zero specializations. The marked cup and primitive proof now gives the lift identity at each field. On the joint kernel, evaluations are detected by the central old inertia scalar, and the commutator formula (21) preserves all the imposed data. For any finite list of fields take the union of their direction lists. Each retains its own spanning slice; it is unnecessary to prescribe independent matrices at all fields with a single prime. Clearing the finitely many denominators and slowing precision preserves the comparison identities, also after field extension. ◻

Proof. Write \(E_0(\mathbb C)=\mathbb C/\Lambda_0\). The connected optimal kernel makes the map on integral homology onto \(\Lambda_0\). Quotienting by \(C_{\rm cusp}\) replaces \(\Lambda_0\) by the lattice generated by it and lifts of the cusp images. Relative paths between cusps supply exactly these additional generators. Consequently \(H_1(X_1(N)(\mathbb C),D_N;\mathbb Z)\to H_1(E(\mathbb C),\mathbb Z)\) is onto. The Farey triangulation generates the relative group by the Manin edges \(\xi\{0,\infty\}\). The invariant sublattice is saturated, so \(a\) is primitive; unimodularity of the intersection form now gives the asserted integral combination. Pullback by the map of pairs is integral, and relative Poincare duality gives the displayed etale map. This is the Betti and relative-cohomology argument of [24], before any choice of coefficient prime; no Hecke projector is involved. ◻

Proof. Further \(p\)-division gives a Cartesian product of the two lift sets: independence is automatic since the old tuple is already a basis at \(p\). At the old precision the inverse-root twists agree. Base change and the projection formula turn the trace of the cup into the cup of the traces, and distribution recovers the old class. The mark and lower root also agree when the base \(p\)-level grows. The functions and their inverses are integral away from the level. Indeed horizontal valuations follow from their divisors; on a smooth full-level model every geometric fiber component meets a cusp, where the Tate leading terms are signs, roots of unity, and powers of \(1-\zeta\) for nontrivial level roots. These are units away from the level. This also checks quotients of level after pullback. The relative open curve has lisse geometric cohomology away from this support, vanishing above degree one. Its arithmetic edge map lands in \(H^1(G_{\mathbb Q,S},H^1(\overline{Y(n_1,n_2)},\mathbb Z/p^j)(1))\). Edge maps commute with traces and maps of pairs. Finiteness in degree zero gives the Mittag–Leffler property needed for compact passage; the subsequent lattice and coefficient maps are integral.

At \(\ell\), start with the Cartesian division sets, subtract the tuples whose \(\ell\)-parts lie in a common line \(I\subset\mathcal E[\ell]\), and add back \(\ell\) times the double-prime-to-\(\ell\) lift. The zero pair was subtracted \(\ell+1\) times, which explains this multiplicity. Distribution on \(\mathcal E/I\) identifies the line sums with the good Hecke correspondence. Quotient isogenies raise the Weil roots to \(\ell\), giving \([\ell]\) and twist scalar \(\ell^{-1}\); the relative elliptic push has Hecke eigenvalue \(a_\ell(f)\). This last assertion holds for relative maps because pullback has zero endpoint-function coordinates and the equality of eigenforms also holds on path integrals. The added-back multiplication isogeny raises the roots to \(\ell^2\); its scalar \(\ell^{-2}\), multiplied by \(\ell\), gives the last term of (24).

For row filling, fix \(Q\). A discarded \(q_0^a\)-division lift of \(P\) has its \([q_0^{a-1}]\)-image in the order-\(q_0\) line \(I_Q\). For \(\rho:\mathcal E\to\mathcal E/I_Q\), these lifts are precisely the preimages under \(\rho[q_0^{a-1}]\) of the prime-to-\(q_0\) point \(\widetilde P\) satisfying \(\rho^\vee\widetilde P=P\). Express the other unit by distribution over \(\rho^\vee\widetilde Q=Q\). The points \(\widetilde Q\) have order \(n_2\), and their order-\(q_0\) lines differ from \(\ker\rho^\vee\). In reverse, the discarded sum is indexed by all lines other than the line of the source mark, since \(l=(Ns,1)\). Its branches at mark \(1/L_0\) are \(\tau\mapsto(\tau+i)/q_0\), \(i\bmod q_0\). Thus it is \(U_{q_0}\), acting on \(Af(A\tau)\) by \(a_{q_0}(f)\) because \((A,q_0)=1\). Root translation and inverse twist contribute \(\nu(q_0)/q_0\). This proves (25), including \(q_0=2\). The argument reproduces the geometric correspondences of [24]; every denominator displayed here is a unit at \(p\). ◻

Proof. Initially impose \(c,d\equiv1\pmod M\). Apply [15] with \(k=2\), \(r=r'=1\), and equal level orders \(M,M\). The prime-support containment is automatic, the range conditions are satisfied, and \(p\mid M\) selects the branch without an additional Euler operator. The symmetric power has degree zero. Its negative Tate moment is exactly the inverse determinant \(e_{p^j}(MP',MQ')^\vee\) on the basis coset of division tuples. The weight-one shifted and Fourier-dual forms agree under the fixed conventions [15]. The theorem therefore gives proportionality to the two brackets in (26). We determine the scalar by residues.

Put \(\Theta(t,q)=(1-t)\prod_{h\geq1}(1-q^ht)(1-q^h/t)\). The normalized Tate function is \[{}_c\theta(t)=(-1)^{(c-1)/2}q^{(c^2-1)/12}t^{-c(c-1)/2} \frac{\Theta(t,q)^{c^2}}{\Theta(t^c,q)}.\] The relation \(\Theta(qt,q)=-t^{-1}\Theta(t,q)\) verifies periodicity and the divisor; multiplication over \([s]\)-preimages verifies norm invariance for \((s,c)=1\). These norms fix the constant uniquely: the norms under \([2]\) and \([3]\) force its third and eighth powers to be one. For \(0<x<1\), \(cx\notin\mathbb Z\), put \[J_c(x)=c\lfloor cx\rfloor-\frac{c(c-1)}2 =c^2B_1(x)-cB_1(\{cx\}),\qquad B_1(x)=x-\tfrac12.\] At \(t=q^x\zeta\) the leading coefficient is \(\pm\zeta^{J_c(x)}\), and the order is \(xJ_c(x)+(c^2-1)/12-\lfloor cx\rfloor(\lfloor cx\rfloor+1)/2\). The last two terms are integers, since \((c,6)=1\).

On a determinant component choose the cusp \(P=q^{1/M}\), \(Q=q^{1/M}\zeta_M^u\), with \((u,M)=1\), over \(K_0=\mathbb Q_p(\mu_M)\). Set \(M_j=p^jM\) and \(K_j=K_0(\mu_{M_j})\). Each pair \(a_1,a_2\in[0,M_j)\cap(1+M\mathbb Z)\) gives a cusp orbit represented by \(P'=q^{a_1/M_j}\), \(Q'=q^{a_2/M_j}\zeta_{M_j}^u\). Translation by the old width eliminates the first root index; determinant primitivity makes cyclotomic Galois act freely on the remaining indices. There are \(p^{2j}\) such orbits, residue degree \(p^j\), and width ratio \(M_j/M=p^j\), accounting for \(p^{4j}\) division tuples. Prime-to-\(p\) roots contribute only fixed data. With \(x=a_1/M_j\), \(y=a_2/M_j\), the tame symbol has root exponent \(h\equiv\pm uJ_d(y)a_1J_c(x)\pmod{p^j}\). All signs have zero Kummer class because \(p\) is odd. The inverse twist uses \(\zeta_{p^j}^{\pm a_1u}\), so the twisted residue on \(G_{K_j}\) is \(\pm J_c(x)J_d(y)(\chi_{\rm cy}-1)/M_j\pmod{p^j}\). Abelian transfer and the exponential series give \[\frac{\chi_{\rm cy}^{p^j}-1}{p^jM} \equiv\frac{\log\chi_{\rm cy}}M\pmod{p^j}.\] Indeed \(\log\chi_{\rm cy}\in p^{v_p(M)}\mathbb Z_p\), and every term of degree at least two has the required valuation. Residue trace sums field transfers without a ramification multiplier. Bernoulli distribution gives \(\sum_{a_i}J_c(a_i/M_j)=J_c(1/M)\), and likewise for \(d\); hence the limiting residue is \(\pm J_c(1/M)J_d(1/M)\log\chi_{\rm cy}/M\). The dual exponential sends \(\log\chi_{\rm cy}\) to \(\pm1\): pair with Kummer classes of exponential units and use local reciprocity, with field trace in the de Rham pairing. The constant term of \(E_P\) is \(-B_1(x)\), so the two brackets have nonzero constant terms \(-J_c(1/M)\), \(-J_d(1/M)\) for the initial choices. Since \(dq/q\) has residue \(M\), the scalar is \(\pm M^{-2}\).

For arbitrary permitted smoothings use \((c_0^2-[c_0]^*){}_c\theta=(c^2-[c]^*){}_{c_0}\theta\). Changing a row by \(c\) changes its inverse-root basis by \(c^{-1}\), so the induced differential operator is \(c^2-c\,\mathrm{mult}_c^*\). Choose \(c_0\equiv1\pmod M\), cancel \(c_0^2-c_0\) rationally, and repeat on the other row. This cancellation asserts a rational identity only; it introduces no claim of integral divisibility. ◻

Proof. Apply the coefficient-prime-independent analytic trace identity of [24] to (26). Its inputs are square order, \(L_0\mid M\), mark \(1/L_0\), even \(\nu\), and \(c,d\equiv1\pmod{L_0}\), all imposed here. The two rescaled weight-one lattice sums cancel \(M^{-2}\). Row reindexing preserves the mark and gives \(\Delta_\nu\); Hecke unfolding has indices prime to \(M\) and gives \(L^{(M)}(f,\nu,1)\). The relative de Rham push has zero cusp-function coordinates, and its pairing with \(\overline\omega\) is \(\pm2i\operatorname{Im}(\int_{\Pi_*\xi\{0,\infty\}}\omega) \Delta_\nu L^{(M)}(f,\nu,1)\). Writing \(\delta_\xi=s_0\omega+\overline{s_0}\overline\omega\) gives \(s_0+\overline{s_0}=\delta_\xi(a)/\Omega_0\), which proves (2). These are algebraic differentials and algebraic trace and push maps; the same identity therefore gives the specified \(p\)-adic coordinate. Local duality uses field trace, with no character-orbit average. For rectangular order fill every missing row prime. Each factor \(1-a_{q_0}(f)\nu(q_0)/q_0\) is nonzero since \(|a_{q_0}(f)|\leq1\), and is precisely the additional omitted Euler factor at \(q_0\). Canceling these factors rationally proves (2) before filling.

The square assertion follows from Lemma 30. For the rectangular assertion take \(\xi\) with upper row \((1,0)\). The Fricke relation sends the degenerated path from \(0\) to \(p^k/(Ns)\) to the path from \(\infty\) to \(-s/p^k\), up to sign. Rohrlich’s theorem in [15], with prime support \(\{p\}\), gives a primitive odd character of sufficiently high \(p\)-power conductor with nonzero central twist. Odd characters exist even at \(p=3\), by adjoining a fixed odd tame character to deep wild characters. Primitive Gauss–Mellin then makes the weighted sum of these path integrals nonzero. If they were all real, complex conjugation would identify the integrals at \(s\) and \(-s\), and their sum against an odd character would vanish. Some imaginary part, and therefore the corresponding \(\delta_\xi(a)\), is nonzero. ◻

Proof. We verify the local hypotheses before constructing derivatives. In the first setting, dual local invariants vanish at a fixed non-\(p\) place by the exclusion of \(D_q\). At a moving place, invariance under inertia first forces \(u_j=0\); the remaining Frobenius equation is nonzero on that divisor because the limiting Frobenius has determinant one and trace different from two. At \(p\), choose an inertia element generating the cyclotomic quotient and a Frobenius lift trivial in that quotient. The first invariance equation is regular in \(t\); the second is independent of \(t\) and nonzero, by the fixed-place exponent condition. They have no common height-one component. In the second setting, nonconstant cyclotomic action gives the same vanishing. In the third, the imposed condition at \(p\) gives it directly.

At an old non-\(p\) place in the third setting, inertia acts through a constant representation. The unramified condition is residue cochains on inertia invariants. Its quotient is residue cochains on inertia coinvariants with twist \((-1)\), in degrees \(1,2\). Its differential has determinant \[D_{q,\eta}=\det\left(1-\frac{\chi_t(\mathop{\mathrm{Frob}}_q) \rho_{V\otimes\eta}(\mathop{\mathrm{Frob}}_q)}q \,\middle|\,(V\otimes\eta)_{I_q}\right).\] This is nonzero and equals one at a moving conductor where \(\eta\) is ramified. The quotient therefore has zero \(H^1\) over the DVR itself. This assertion is made before passing to its residue field, where the determinant may vanish. The conditions include all \(H^0\) and omit \(H^2\). Unramified cup products factor through residue cohomology, so are zero; local duality and the Euler characteristic give exact complementarity, including at the residue field. These are precisely the complex conditions needed by Lemma 13.

Global invariants vanish by the non-CM open-image argument [26]. We spell out its use on the limiting diagrams. In the product of the stage absolute groups, take the joint kernel of the Tate action and the abelian and radical data used in the characters. The quotient contains arbitrary sequences from a fixed deep principal subgroup of \(\operatorname{SL}_2(\mathbb Z_p)\), acting trivially on the latter data. Indeed those data have bounded derived length, whereas repeated Lie brackets retain \(\mathfrak{sl}_2\). The evaluation and diagonal-conjugation argument of Lemma 12 shows that restriction to this kernel detects a nonzero degree-one class. Its evaluations span the Tate plane by irreducibility. Both the primal and the opposite-condition spaces inject into unrestricted global cohomology, by the local invariant checks above. Poitou–Tate and the global Euler characteristic now give amplitude \(1,2\) and \[h=1+h^*,\] where \(h,h^*\) are their degree-one dimensions. This remains true after any of the line changes below.

Choose a fresh sequence of good primes \(\ell\to1\) in \(\mathbb Z_p^\times\), split to increasing precision on all old scalar data, whose Tate Frobenius tends to \[\tau=\begin{pmatrix}1&e\\0&1\end{pmatrix},\qquad e\ne0.\] The local degree-one cohomology over the testing DVR splits into the finite line \(T/(\tau-1)T\), measured by Frobenius, and the singular line \(\ker(\tau-1)\), measured by tame inertia. The notation here means scalar extension to the testing ring; \(e\) is a unit there because \(p\) has been inverted. The conjugating power in the inertia relation tends to one. The finite line is isotropic by inflation, and a pure singular cocycle has values in the Weil-isotropic invariant line. The mixed pairing is perfect. Taking all \(H^0\), the respective line in \(H^1\), and no \(H^2\) therefore gives the exact complementary conditions.

A nonzero primal class and a nonzero dual class can both be made nonzero on the finite line. On the joint kernel their evaluations are additive and span the plane. Each prohibited zero is a proper affine condition; finitely many such conditions can be avoided in characteristic zero. Chebotarev realizes the resulting finite tables at increasing precision.

The order of the further choices is relevant. At each stage fix the cyclotomic base \(K_0=\mathbb Q(\mu_M)\) first. Perturb the desired unipotent element within the available deep special linear subgroup so that \(v_p\det(1-\rho_T(\mathop{\mathrm{Frob}}_\ell))\) is large but finite. Next choose \(A_i\) with \[A_i-v_p\det(1-\rho_T(\mathop{\mathrm{Frob}}_\ell))\longrightarrow\infty, \qquad \ell\equiv1\pmod{p^{A_i}}.\] Require exact splitting in \(K_0\), all earlier derivative fields, and the \(p^{A_i}\)-th roots of all earlier derivative primes. These conditions make the new degree-\(p^{A_i}\) conductor-\(\ell\) field and the earlier derivative fields mutually split at their conductors. They are compatible with the finite Tate and evaluation conditions by the same open-image argument. A Frobenius lift trivial in the new derivative field is obtained by an inertia adjustment.

For a set \(I\) of derivative primes, norm the system class to the compositum \(K_I\) of their degree-\(p^{A_i}\) fields with \(K_0\), choose inertia representatives \(\sigma_\ell\) generating their own cyclic factors and trivial on the other factors, and apply \[D_I^{\rm der}=\prod_{\ell\in I} \sum_{j=1}^{m_\ell-1}j\sigma_\ell^j, \qquad m_\ell=p^{A_i}.\] The good Euler norm at \(\ell\) is \[p_\ell=\ell^{-1}\det(1-\rho_T(\mathop{\mathrm{Frob}}_\ell)).\] There is one fixed nonzero \(p\)-power \(L_*\) which clears all torsion ambiguities and descent obstructions used in a finite switching argument. To see uniformity, the commutator image for a non-CM curve bounds the exponent of \((V/T)^{G_F}\) over all abelian fields \(F\) in question. This bounds \(H^1(T)\)-torsion, and also the invariants of \(T/p^nT\) that occur in inflation–restriction. The obstruction and ambiguity groups are killed by the same bounded exponent. The identity \[(\sigma_\ell-1)D_\ell^{\rm der}=m_\ell-N_\ell, \qquad N_\ell=\sum_{j=0}^{m_\ell-1}\sigma_\ell^j,\] then makes the derivatives invariant modulo \(p^n\), for \(n\to\infty\) sufficiently slowly. Descend to \(K_0\), use unnormalized character corestriction, and pass to the marked models. We obtain integral classes \(Z_I\), with \(Z_\varnothing=L_*z_{\rm sys}\).

Here is the exact local derivative comparison. Put \(m=m_\ell\). Let \(b,b_\ell\) be the lower and upper integral cocycles after all other derivatives, and put \(F=\rho_T(\mathop{\mathrm{Frob}}_\ell)\). They vanish on local inertia: good reduction and the Weil bound exclude the eigenvalue \(\ell\) in the relevant inertia cohomology. Put \(x=b(\mathop{\mathrm{Frob}}_\ell)\), \(x_\ell=b_\ell(\mathop{\mathrm{Frob}}_\ell)\). The norm identity gives a coboundary \(dy\) with \[\sum_{j=0}^{m-1}\sigma_\ell^jb_\ell=p_\ell b+dy, \qquad (F-1)y=mx_\ell-p_\ell x.\] After adjusting the descended cocycle, its restriction is exactly the derivative of \(b_\ell\). Its finite value tends to zero because the derivative coefficient sum is \(m(m-1)/2\). Applying \(\sigma_\ell-1\) shows that its singular value is \(-y\), up to the uniformly bounded loss already cleared. Finally \[y=m(F-1)^{-1}x_\ell-\ell^{-1}\operatorname{adj}(F-1)x.\] The first term tends to zero by the chosen margin on \(A_i\). A lower coboundary \((F-1)w\) disappears after applying the adjugate in the limit. The calculation holds on every translate before character corestriction. Therefore \[ f(Z_I)=0,\qquad s(Z_I)=\operatorname{adj}(\tau-1)f(Z_{I\setminus\{\ell\}}). \tag{27}\] The map between the two lines is multiplication by \(\pm e\), and hence is a unit comparison at the testing DVR. The classes lift to the complexes with the new conditions; the inclusion of \(H^0\) and the vanishing of the old singular quotients’ \(H^1\) give the claimed lifts over the DVR itself.

If \(h>1\) generically, choose the two nonzero evaluations above. Equation (27) makes the derived primal class nonzero on the singular line. Its pairing with the old dual class is then nonzero at this prime and zero at all other places, contradicting reciprocity. Thus \(h=1\) generically. At a DVR where the fiber has \(h>1\), use the primitive reduction of the generic line and a nonzero fiber dual class. The exact line-switch Lemma 13 reduces \(h\) by one and preserves the determinant valuation, because the comparison in (27) is a unit. After finitely many switches the complex is a free line in degree one and its class is integral. The factor \(L_*\) is a unit at this DVR. Reading the equalities backwards proves [eq:source-4]. ◻

Proof of Proposition 34. At \((p)\), [eq:source-1](a) identifies \(G\) with a free line in degree one. The class is integral and the normalizing factors are units there. This proves nonnegative order at the vertical divisor once generic nonvanishing is known.

For that nonvanishing, use the second setting in Lemma 35. Rohrlich’s finite-support twist nonvanishing, in the form [15], applies to the fixed weight-two newform and characters with conductor supported at \(p\). Infinitely many sufficiently ramified even characters have nonzero central value. The period formula [eq:source-2] therefore makes the fixed-support square class generically nonzero: a torsion class over the one-variable power-series ring would vanish at all but finitely many such characters. Lemma 35 gives one generic line. Setting all \(u_j=0\) in the full diagram adds acyclic singular quotients, since the moving Frobenius limits have \(\det(1-g_j)\ne0\), where \(g_j\) is the limiting Frobenius at the \(j\)-th moving prime. The good norm relations multiply the class by the nonzero constants \(2-\mathop{\mathrm{Tr}}(g_j)\). Thus the full diagram also has a nonzero class on a single generic line. The first setting of the lemma now treats every horizontal divisor outside the fixed factors.

It remains to remove the fixed factors. Express \(U=g_0/f_0\) using integral determinant pivot formulas, with \(f_0\ne0\), and retain integral stage expressions \(g_{0,i},f_{0,i}\). Specialize the tame variables to finite nontrivial values \(u_j=\eta_j-1\), avoiding identically zero denominators and the prohibited Frobenius values at \(p\). These specializations still detect bounded series: repeated one-variable Weierstrass preparation permits avoidance of any finite collection of nonzero conditions. The conditions also hold at sufficiently precise original stages.

Fix such a stage and a horizontal prime of \(\Lambda_\eta\). Use a rectangular system with unfilled first row, with \(b=N_{p'}\), and \(M\) supported only at \(p\) and the active tame conductors. Choose a degeneracy \(A=p^k\) and a symbol with nonzero evaluation. Smoothing integers \(c',d'\) can be chosen so that \(\Delta'\) is a unit at this prime. Indeed preparation realizes its \(t\)-value as an algebraic point of the open unit disk. Use CRT to choose integers tending to one \(p\)-adically, congruent to one on the required fixed symbol orders, and with a prescribed nontrivial \(\eta\)-value at one active conductor and trivial values at the others. All coprimalities can be maintained. At the selected \(t\)-value, the cyclotomic factor tends to one whereas the chosen constant weight is nontrivial, so the smoothing factor is eventually nonzero. Fix one such choice while applying the line comparison.

Call the resulting class \(z'\). It has primitive Euler factors away from \(p\): the only other omitted primes are ramified conductors of \(\eta\), where the twisted local factor is one. The same nonvanishing theorem makes \(z'\) generically nonzero. Lemma 35 gives its integral determinant order with unramified old conditions. On changing those conditions to full, the order changes by \(-\sum_{q\ne p}v(D_{q,\eta})\).

In the common generic full cohomology line the ratio of the square and rectangular classes, apart from their nonzero constant symbol ratio, is \[ \frac{\Delta\prod_{q\in S_f\setminus\{p\}}P_q} {\Delta'}. \tag{28}\] To verify the identity rather than merely its character values, evaluate the two classes at the infinitely many sufficiently ramified cyclotomic characters with nonzero central value. Formula [eq:source-2] gives exactly the displayed ratio; omissions at active conductors contribute one. Clearing the generic denominators and applying one-variable preparation proves the identity. Squaring first gives the same valuation conclusion if the common orientation sign has not been fixed. Consequently the tested quotient \(g_{0,i}/f_{0,i}\) has no horizontal pole, that is, \[g_{0,i}(t,\boldsymbol\eta-1)/f_{0,i}(t,\boldsymbol\eta-1) \in\Lambda_\eta[1/p].\] There is no assertion of a uniform denominator for these stage quotients.

Every remaining possible pole of \(U\) divides a factor regular in \(t\). For such a factor \(H\), take a distinguished representative and divide the integral numerator by \(H^a\), where \(a=v_H(f_0)\). At each permitted finite tame specialization, the limiting numerator contains all the zeros of the limiting denominator with their multiplicities. This follows from Lemma 19: on a smaller closed disk about a zero, the initially integral numerator and denominator converge uniformly, and Weierstrass preparation preserves the number of zeros with multiplicity. Thus the specialized division remainder is zero. Finite-character detection makes the remainder zero in \(R\). There are therefore no remaining height-one poles. Since \(R\) is factorial, \(U\in R\), proving the proposition. ◻

Proof. Let \(G_y\) be the extension of \(E\) by \(\mathbb G_m\) defined by the Poincare biextension at \(y=\operatorname{pol}(P)\). Choose a rational lift \(B_P\) of \(P\). The fiber is a line torsor, so this lift exists. Splitting its underlying Tate modules gives an upper-right Galois block \(\beta(g)g_T\); \(\beta\) is the Kummer cocycle of \(y\), up to the fixed polarization sign. At finite order this follows by identifying division points of \(y\) with splittings of the extension and using the Weil pairing to compare two splittings. Write the Kummer cocycle of \(B_P\) as \((k,x)\). Then \[dk=-\beta\cup x.\] Include all ramification and nonintegrality primes of these data in the finite killed list.

Adjoin to a fixed normal splitting field all cyclotomic \(p\)-power roots and all \(p\)-power radicals of the killed primes. Its relative Galois group has bounded derived length, so the Tate image over this field still contains a deep \(\operatorname{SL}_2\). On the additional Tate kernel the values of \(x\) span the plane. Indeed nonzero Kummer classes remain nonzero over a fixed finite extension; a central homothety in the open Tate image detects them on its kernel. The further radical quotient has scalar cyclotomic conjugation, so cannot support the irreducible Tate plane as a quotient. Choose an element \(g\) fixing the splitting and radical data, with determinant-one Tate action of the required trace, and \[ k(g)-\beta(g)g_T(g_T-1)^{-1}x(g)\ne0. \tag{29}\] For existence, on the joint kernel the commutator of two elements has \(x=\beta=0\) and \(k\)-value \(\pm2e(x(h_1),x(h_2))\), where \(e\) denotes the Weil form. Full span supplies a nonzero value. Multiplying the chosen \(g\) by such a commutator changes the left side of (29) without changing its Tate action. Chebotarev and compatible power-residue bases now give the asserted \(r_i,\lambda_i\).

We use the biextension construction of [24], verifying its \(p\)-power version here. Over an extension containing a ring class field of positive \(r_i\)-power conductor, the normed character is unramified also above \(r_i\). Indeed the split order class sequence identifies the two residue-unit factors modulo the diagonal and the global units \(\{\pm1\}\). For units at one specified split place, the other component is one. The local norm subgroup therefore has residue in \(\{\pm1\}\), which the even character kills. Sum the local character valuations of a rational biextension lift relative to its rigidified integral fibers. Reciprocity gives independence of the lift, and the two biextension laws give bilinearity. On a Jacobian this is the Deligne-pairing intersection formula for disjoint generic supports, with the common orientation sign.

To define the same height downstairs, note that \(\#E(\mathbb F_{r_i})\to2-a_*\ne0\) \(p\)-adically. One fixed \(p^b\) therefore removes every finite \(p\)-primary reduction part. After multiplying the first argument by \(p^b\), divide it by \(p^{m_i}\) in the uniquely \(p\)-divisible local factor. For lifts \(\mathcal B,\mathcal B'\) before and after division, apply the local character to \[\mathcal B/([p^{m_i}]_1\mathcal B').\] Changing \(\mathcal B'\) changes this scalar by a \(p^{m_i}\)-th power. The two laws give bilinearity; comparison with integral lifts gives the unramified construction after extension. Norming lifts in the second argument proves trace compatibility. Passing to limits and dividing by \(p^b\) defines the stated rational heights.

For the fixed lift \(B_P\), all places except \(r_i\) contribute zero. Put \(w_i=(g_{i,T}-1)^{-1}x(g_i)\), with \(p^bw_i\) integral after enlarging the fixed \(b\). Correcting division points by this coboundary leaves the fiber cocycle \(p^b(k-\beta\cup w_i)\). Its Frobenius value is the power residue character of the scalar ratio above. Thus the limiting height is, up to sign, (29), and is nonzero. Trace over \(K_b/\mathbb Q\) contributes its degree two. ◻

Proof. At stage \(i\), write a cocycle modulo \((p^{m_i},u^2)\) as \(z_0+uz_1\). Then \(dz_1=-\lambda_i\cup z_0\). The good norm relation and one fixed clearing factor permit the adjustment \[z_0=\alpha_i\beta,\qquad \alpha_i=\pm\left(1-\frac{a_f(r_i)}{r_i}+\frac1{r_i}\right) \frac{\log_\omega z_b}{\log_\omega P}.\] The clearing is uniform because the degree-one torsion has bounded exponent. The scalar two-cocycle \(z_1\cup x+\alpha_i\lambda_i\cup k\) is closed. Its local invariant at \(p\) tends to \(Z'\log_\omega P\), by the logarithm–dual-exponential pairing. At the other fixed places the character is killed and the rational local class is a coboundary. At \(r_i\), use \(dw_i=x\) to obtain \(\alpha_i\lambda_i\cup(k-\beta\cup w_i)\). Its invariant is \(\pm\alpha_i h_i^\mathbb Q(P,P)\), by the scalar-ratio computation in Lemma 37. The sum of local invariants is zero. Passing to the limit, the Euler factor tends to \(2-a_*\), giving [eq:source-6]. ◻

Proof. The proof evaluates the same finite differences of kernel coefficients in two ways. Projection onto the \(f\)-block expresses them through \(Z'\); their intersection interpretation expresses them through \(h_\lambda^{K_b}(Q_X,Q_X)\). Matching the two evaluations, with their constants, gives the identity. We first specify the finite-character formulas and then prove their interpolation in the present \(p\)-power direction. They are the complex coefficient and geometric intersection identities in [24]. Their hypotheses are an odd fundamental discriminant \(-D<-4\), split Heegner level \(N\), a prime \(r\nmid ND\) split in the Hilbert class field with \(r\equiv1\pmod{ND}\), an even prime-conductor character \(\chi\) with \(\chi^2\ne1\), and indices \(nr^h\) with \((n,ND)=1\) and odd valuation at the chosen inert prime \(\ell\). None of these identities has a coefficient-prime hypothesis.

Here are their precise normalizations. Choose integral ideal representatives \(C\) of \(\mathop{\mathrm{Pic}}(\mathcal O_{K_b})\), prime to \(D\), and put \(a_C=\mathrm NC\). Let \(e(x)=\exp(2\pi i x)\) and \[\theta(z)=\frac12\sum_C\sum_{x\in C}e(\mathrm Nx\,z/a_C).\] Its positive coefficient is \(\rho(n)=\sum_{d\mid n}\varepsilon(d)\). If \(\theta_\chi\) is its Fourier twist, define \[E_{\chi,s}(z)=\frac12 \sum_{\substack{(c_1,d_1)\ne(0,0)\\ND\mid c_1}} \varepsilon(d_1)\bar\chi^2(c_1) \frac{\operatorname{Im}(z)^s} {(c_1z+d_1)|c_1z+d_1|^{2s}}, \qquad I_\chi=\left.\frac{\sqrt D}{2\pi} \mathop{\mathrm{Tr}}_{NDr^2/Nr^2}(\theta_\chi E_{\chi,s})\right|_{s=0}.\] Characters are extended by zero at nonunits; the trace is unnormalized. The continuation is holomorphic. Poisson summation pairs the odd row weights, eliminates the nonholomorphic modes, and permits pairing with cusp tests.

For every tested index \(n\) and \(h\ge1\), regard the coefficient at \(nr_i^h\) as a polynomial \(D_{i,h}(n;u)\) in \(\mathbb Z_p[u]/((1+u)^{p^{m_i}}-1)\): in the following sum \(r=r_i\) and \(\chi(x)=(1+u)^{\lambda_i(x)}\). \[ D_{i,h}(n;u)=\frac12 \sum_{\substack{C,\ y\in C,\ b>0\\ a=\mathrm Ny/a_C>0\\ a+Nb=nr^hD}} \sum_{\substack{kl=b\\\gcd(k,l,D)=1}} W(k)\chi(a/D)\bar\chi^2(Nk), \tag{30}\] where \[W(k)=\prod_{\substack{q\mid D\\q\mid k}}(l/q) \prod_{\substack{q\mid D\\q\nmid k}}(-a_CNk/q).\] Write \(\epsilon_q(x)=(x/q)\). The local partial Fourier transform at \(q\mid D\) has scalar \(\kappa_q=q^{-1}\tau_q\epsilon_q(-a_CD/q)\), with \(\tau_q=\sum_{x\bmod q}\epsilon_q(x)e(x/q)\). Multiplying the partial traces and the Poisson factor \(-2\pi i/D\) leaves common scalar one, because the primitive Gauss sum of \(\varepsilon\) is \(i\sqrt D\). This explains in particular the factor \(1/2\) in (30). For a trace matrix \(\gamma=\left(\begin{smallmatrix}a_\gamma&b_\gamma\\ c_\gamma&d_\gamma\end{smallmatrix}\right)\in\Gamma_0(Nr^2)\), translation of the theta twist multiplies its weight by \(\chi(a_\gamma^{-2})\), canceled by the Eisenstein row weight \(\chi^2(a_\gamma)\). Thus these finite-character computations apply to our characters as soon as \(\chi^2\ne1\), which holds for every nontrivial \(p\)-power character.

Use the same marked Artin limits as before to define \(D_h(n;u)=\lim_i D_{i,h}(n;u)\) and \(D'_h(n)=[u]D_h(n;u)\). Only terms with \(a\) a unit at \(r\) occur in (30). Since \(r\mid nr^h\), this is equivalent to \(k,l\) being units there. The identity \(W(l)=-W(k)\) cancels the divisor-independent terms in the derivative of the character weight. Hence \(D'_h(n)\) is the limit of the masked sum with weight \(-2\lambda_i(k)\).

For \(h\ge0\), define \(B_{i,h}(n)\in\mathbb Z/p^{m_i}\mathbb Z\) by the same sum with the mask removed and character weight replaced by \(-2\lambda_i(k/r_i^{v_{r_i}(k)})\). Write \(B_h(n)\) for its Artin limit, put \(B_h(n)=0\) for \(h<0\), and set \(C_h(n)=B_h(n)-B_{h-2}(n)\). Stripping a common power \(r^j\) from \(a,b\) shows that, for \(h\ge2\), \[ C_{h+1}-2C_h+C_{h-1}=D'_{h+1}+2D'_h+D'_{h-1}. \tag{31}\] For completeness, when the primitive index is positive the \(C_h\)-multiplicities are \(1,4,8,12,\ldots\); their second differences are \(1,2,1\). At primitive index zero they are affine and the second difference is zero. The off-\(r\) weights do not change, since the primes over \(r\) are principal and \(r\equiv1\pmod D\). This proves the identity for \(\mathbb Z/p^{m_i}\)-valued weights and then on their limits, with no alteration of constants.

The symbols \(D'_h,B_h,C_h\) will also denote these coefficient functions of \(n\). The Hecke polynomial \(\mathcal A\) acts on them by the weight-two coefficient rule \((T_s a)(n)=a(sn)+s a(n/s)\), with the second term zero unless \(s\mid n\). All its primes are away from \(NDr_i\). The finite set of tested indices was chosen to contain the indices needed to evaluate this polynomial at \(n=\ell\).

We next project its holomorphic side. At level \(Nr^2\) use \(f_j=r^jf(r^jz)\), \(0\le j\le2\), and write \(\tau(\chi)=\sum_{x\bmod r}\chi(x)e(x/r)\). The finite-character Fricke and unfolding identities are \[ \frac{(I_\chi,f_2)_{Nr^2}}{(f,f)_{Nr^2}} =\frac{C_0}{r+1}\frac{\tau(\chi)}{\tau(\bar\chi)} L(f,\bar\chi,1)L(f\otimes\varepsilon,\bar\chi,1), \qquad (I_\chi,f_1)_{Nr^2}=0. \tag{32}\] The exact constants arise from the kernel factor \(\sqrt D/(2\pi)\), the Fricke factor \(r\), the Mellin integral \(1/(4\pi)\), and the level index \(r(r+1)\). At primes of \(N\), use the primitive degree-at-most-one polynomial and \(\varepsilon(q)=1\); at primes of \(D\) the twisted local factor is one; at \(r\) both twisted factors are one. The total nebentypus is trivial. The conductor-\(r\) new cuspidal block is therefore Steinberg, with \(U_r=\pm1\), and its complementary raises have eigenvalue zero. The conductor-\(r^2\) cuspidal block has \(U_r=0\). In a ramified Eisenstein pair both characters are ramified at \(r\), so its \(U_r\)-eigenvalue is also zero. Thus \(U_r^h(U_r^2-1)\) kills these blocks for sufficiently large \(h\). The remaining old blocks come from level \(N\); the fixed operator \(\mathcal A\) kills all of them except the \(f\)-block.

Write \(a_h=a_f(r^h)\) and \(a=a_f(r)\). The normalized Gram matrix of \(f_0,f_1,f_2\) has entries \(g_{|i-j|}\), where \[g_0=1,\qquad g_1=\frac a{r+1},\qquad rg_2=ag_1-1.\] The coefficient vector at index \(nr^h\), after removing its \(n\)-factor, is \((a_h,ra_{h-1},r^2a_{h-2})\). The coefficient of the Gram row for \(f_2\) is \[s_h=\frac{r^2a_{h-2}-g_1ra_{h-1}}{1-g_1^2}.\] Its denominators have bounded \(p\)-valuation because \(a_*\ne\pm2\). Set \(\gamma_h=\lim_i(a_f(r_i^h)-a_f(r_i^{h-2}))\). The Hecke recurrence gives \[\lim_i\frac{s_h}{r_i+1} =\frac{\gamma_h}{(a_*-2)(a_*+2)}.\]

All factors in (32) have bounded group polynomial realizations. The Gauss ratio is \(\tau(\chi)^2/r\); its coefficients are integral and fixed by \(p\)-Frobenius because \(\lambda_i(p)=0\), and its limiting augmentation is one. The first \(L\)-value factor is \(Z_i/d_0\). For the companion, use the even additive modular-symbol sum multiplied by \(\tau(\bar\chi)/r\). Its denominators are bounded by the fixed relative period lattices. Before Gauss multiplication its augmentation is \[r\sum_{r\mid j}\frac{a_{f\otimes\varepsilon}(j)}j -L(f\otimes\varepsilon,1) =(a_f(r)-2)L(f\otimes\varepsilon,1),\] by the Hecke recurrence and Mellin continuation. The trivial Gauss sum is \(-1\), so its limiting normalized augmentation is \((2-a_*)L(f\otimes\varepsilon,1)/\Omega_{\rm tw}\). Multiply the finite-character identity by \((1+u)^2-1\) to remove the excluded character values, pass to bounded-series limits, and cancel the nonzero multiplier. Since \(Z_i(0)=0\), only that companion augmentation contributes to the derivative. We obtain \[ \begin{split} [\mathcal A(D'_{h+2}-D'_h)](\ell) ={}&\pm\mathcal A_fa_f(\ell) \frac{\gamma_{h+2}-\gamma_h}{(a_*-2)(a_*+2)}\\ &\hspace{4mm}\cdot C_0\Omega_0\frac{Z'}{d_0} (2-a_*)L(f\otimes\varepsilon,1). \end{split} \tag{33}\] This interpolation uses bounded denominators, not a coefficient-prime restriction; in particular it applies when \(p=3\).

For the geometric side, the coefficient-prime-independent intersection identity cited above applies at \(q\nmid6NDnr^h\). Its local divisor sum, with weight \(-2v_q(k)\) and the original factor \(1/2\), is zero unless \(q\) is inert, \(d_q=v_q(b)\) is odd, and \(\varepsilon_{K_b,s}(-bNa_C)=1\) for all \(s\mid D\), where \(\varepsilon_{K_b,s}\) is the local quadratic norm character of \(K_b/\mathbb Q\) at \(s\), distinguished from the residue symbols \(\epsilon_q\) above. When these conditions hold it equals \[-\frac{1+d_q}{2}\rho(b/q) 2^{\#\{s\mid D:s\mid b\}}.\] Its negative is the intersection of the oriented ordered Hilbert-class sums, based respectively at \(\infty\) and \(0\), with the second acted on by \(T_{nr^h}\), weighted by residue degree over \(q\). The generic supports are disjoint by the odd inert \(\ell\)-valuation. The remaining fixed primes have character zero. This yields all the required height contributions except those at \(r_i\).

The difference \(C_h=B_h-B_{h-2}\) keeps cyclic paths in the second Hecke divisor; its second difference eliminates the two continuing maximal-order paths. Every other orbit is a trace from an extension containing a positive-\(r\)-conductor ring class field. After pulling up the first divisor, intersections at \(r\) are zero. Indeed ordinary CM reduction gives linear or conjugate-semilinear maps over the unramified quadratic \(\ell\)-adic algebra, whose norm degrees have even \(\ell\)-valuation; they cannot produce the prescribed odd index. Cusps remain disjoint. The missing contributions may therefore be added as zeros. Lemma 37 now identifies the resulting intersections with full trace heights: the normed characters are unramified over exactly the extensions just obtained.

Applying \(\mathcal A\) at \(\ell\) inserts the projector \(\phi^*\phi_*/\delta_\phi\), with scalar \(\mathcal A_fa_f(\ell)\). The Hecke second difference acts on the second point with eigenvalue \(\gamma_h\). Thus the geometric side of (31), after the additional \(h\mapsto h+2\) difference, is \[\pm\mathcal A_fa_f(\ell)(a_*-2)(\gamma_{h+2}-\gamma_h) \frac{h_\lambda^{K_b}(Q_X,Q_X)}{\delta_\phi}.\] All projector denominators and the fixed first-argument multiplier \(p^b\) cancel in these rational identities. Cusp torsion has zero rational height. On the holomorphic side, the coefficients \(1,2,1\) in (31) insert \(a_*+2\) into (33). Cancel \(\mathcal A_fa_f(\ell)\) and \(\gamma_{h+2}-\gamma_h\). The latter is nonzero for some sufficiently large \(h\), since the limiting eigenvalues are distinct and not roots of unity. The remaining scalar \((a_*+2)(2-a_*)/((a_*-2)(a_*+2))=-1\) gives [eq:source-7]. ◻

Proof of Proposition 36. The companion has analytic rank zero. Hence \(E(K_b)\otimes\mathbb Q=E(\mathbb Q)\otimes\mathbb Q\), and we may write \(Q_X=t_PP\) modulo torsion, with \(t_P\in\mathbb Q\). The exact split Gross–Zagier formula of Theorem 8, for this single Hilbert-class trace, is \[ C_0L'(f,1)L(f\otimes\varepsilon,1) =\frac{2t_P^2H(P)}{\delta_\phi}. \tag{34}\] In particular \(t_P\ne0\). Trace compatibility gives \(h_\lambda^{K_b}(Q_X,Q_X)=2t_P^2h_\lambda^\mathbb Q(P,P)\). The same factor two occurs in (34). Substituting both identities into [eq:source-7] gives \[Z'=\pm d_0(2-a_*)\frac{L'(f,1)}{\Omega_0H(P)} h_\lambda^\mathbb Q(P,P).\] Compare with [eq:source-6]. The factors \(2-a_*\), \(h_\lambda^\mathbb Q(P,P)\), and \(\log_\omega P\) are nonzero. Their cancellation proves [eq:source-5], with no unit ambiguity other than the common orientation sign. ◻

Proof. Compute first at an actual stage with trivial characters and support \(S'=S\cup\{r_1,\ldots,r_k\}\). Put \[O^j=H^j(G_{\mathbb Q,S'},T),\quad s_0=v_p(\#\mathop{\mathrm{Sha}}(E/\mathbb Q)),\quad \tau_g=v_p(\#E(\mathbb Q)_{\rm tors}),\quad \tau_v=v_p(\#E(\mathbb Q_v)[p^\infty]).\] Let \(p^l\mathbb Z_p=\log_\omega E(\mathbb Q_p)\). The singular quotient \(J_p\) of local degree-one cohomology by compact Kummer is a free line, perfectly dual to compact Kummer modulo torsion. At non-\(p\) places Kummer fills compact degree one. Integral Poitou–Tate therefore gives \[ \begin{split} O^1\longrightarrow J_p\longrightarrow\mathop{\mathrm{Sel}}_{p^\infty}(E/\mathbb Q)^\vee \longrightarrow O^2\longrightarrow \bigoplus_{v\in S'_f}H^2(\mathbb Q_v,T) \longrightarrow E(\mathbb Q)[p^\infty]^\vee\longrightarrow0. \end{split} \tag{35}\] Real Tate terms vanish because \(p\) is odd. The local degree-two groups have lengths \(\tau_v\). Also \(O^0=0\), \(\mathop{\mathrm{rank}}O^1=1\), and \(\mathop{\mathrm{length}}O^1_{\rm tors}=\tau_g\).

In rank zero, put \(j=\mathop{\mathrm{length}}(J_p/\mathop{\mathrm{im}}O^1)\). Equation (35) gives \[\mathop{\mathrm{length}}O^2=s_0-j+\sum_{v\in S'_f}\tau_v-\tau_g.\] If \(z_i^\circ\) is the central stage class and \(\exp^*z_i^\circ=\alpha_i\omega\), logarithm–exponential duality gives its valuation in \(J_p\) as \(v_p(\alpha_i)+l\). Its valuation in \(O^1/O^1_{\rm tors}\) is therefore \(v_p(\alpha_i)+l-j\). Adding degree-one torsion and subtracting degree-two torsion gives raw determinant order \[ v_p(\alpha_i)+l+2\tau_g-s_0-\sum_{v\in S'_f}\tau_v. \tag{36}\]

In rank one, let \(P\) generate the full free Mordell–Weil group modulo torsion. The map from \(J_p\) to the dual of the divisible global point subgroup is injective, with cokernel of length \(v_p(\log_\omega P)-l\). Consequently \[\mathop{\mathrm{length}}O^2=s_0+v_p(\log_\omega P)-l +\sum_{v\in S'_f}\tau_v-\tau_g.\] Finite \(\mathop{\mathrm{Sha}}\) identifies \(O^1\) with compact global points. The free coordinate of the class has valuation \(v_p(\log_\omega z_i^\circ)-v_p(\log_\omega P)\). The same signed torsion sum therefore gives \[ v_p\left(\frac{\log_\omega z_i^\circ} {(\log_\omega P)^2}\right) +l+2\tau_g-s_0-\sum_{v\in S'_f}\tau_v. \tag{37}\] The logarithm is nonzero by [eq:source-5]. When restoring the moving omissions, the good norms multiply the fixed class by \(\prod_r(1-a_f(r)/r+1/r)\), with exactly these factors.

Lemma 16 gives the local Haar identities for the minimal differential: \[\tau_q=v_p(c_qP_q(1))\quad(q\ne p),\qquad \tau_p-l=v_p(c_pP_p(1)).\] They include all reduction types: the nonsingular reduction measure is its point count divided by the residue cardinality, and the formal logarithm accounts for the finite torsion kernel at \(p\). Substitute [eq:source-2] into (36), or [eq:source-5] into (37). Every omitted Euler factor cancels against its Haar factor. The result in either case is \[X(E)+v_p(\Delta(0)).\] In rank one the use of a generator of the full free group makes \(H(P)=\mathop{\mathrm{Reg}}_E\). The ratio \(\Omega_E/\Omega_0\) is a power of two and has zero \(p\)-valuation. Thus neither a Mordell–Weil index nor a real-component factor remains.

Finally pass from the stage calculation to the central value of the limit. The moving local torsion lengths are bounded, because their Frobenius traces tend to values different from two. The displayed cohomology formulas bound the remaining torsion lengths. Hence determinant pivots of the same ranks survive the limit by Remark 21. The raw valuation is unchanged. Each \(D_q/P_q\) has central value one; division by \(\Delta(0)\) leaves \(v_p(U(0))=X(E)\). Proposition 34 now gives the inequality. ◻

Proof. Work first on a fine full prime-to-\(p\) level over \(\mathcal V=W(\overline{\mathbb F}_p)\). The ordinary elliptic moduli and Serre–Tate theory [8, 16] provide the canonical connected subgroup and the Igusa tower trivializing the multiplicative formal group. The dual of the Kodaira–Spencer differential, expressed using the square of the trivialized invariant differential, defines \(d_{\rm mod}\); on Tate charts it is \(q_{\rm Tate}\partial_{q_{\rm Tate}}\). For quotient maps we use the target differential whose pullback has the corresponding degree factor. These are precisely the ordinary-moduli conventions used in the cited lemmas.

The initial algebraic pullbacks have bounded denominators on the quasi-compact ordinary formal region, including its Tate charts. Integrality can then be tested by successive division by the uniformizer and expansion at cusps. The special-fiber determinant components of the fine tame curve are irreducible, and every component of a finite trivialization level surjects onto its ordinary component. Thus expansions at all trivialization lifts of a multiplicative cusp detect zero reduction on every component. Modulo any fixed precision, a section on the completed tower uses a finite level, so the same test applies there.

The \(p\)-depletion has no terms whose index is divisible by \(p\). For \(n\in\mathbb Z_p^\times\), the numbers \(n^{(p-1)p^j-1}\) tend uniformly to \(n^{-1}\). The weights \(2(p-1)p^j\) tend to zero on \(\mathbb Z_p^\times\). Consequently the differentiated expansions converge integrally on every trivialization. The expansion test just proved gives convergence on the tower. The limit is invariant under the full \(\mathbb Z_p^\times\)-action on differential trivializations, so torsor descent gives \(F_0\) on the ordinary formal curve at the fixed fine tame level. Its reduction modulo \(p\) is consequently a regular function on that curve’s ordinary special fiber. The Hecke recursions give the displayed fully depleted expansion of \(F\). This construction uses the canonical multiplicative \(p\)-structure, so it also applies when \(p\mid N\).

Mahler duality identifies the integral expansion in \(z-1\) with a bounded measure. The trace of \(F\) along the canonical quotient is zero: on the parameter it is the finite flat map \(z\mapsto z^p\), reducing to Frobenius, and at a Tate cusp the trace kills every prime-to-\(p\) exponent. Expansion detection therefore gives \[\sum_{\zeta^p=1}F(\zeta z)=0.\] In the measure transform this sum multiplies the restriction to \(p\mathbb Z_p\) by \(p\) and kills its complement. Uniqueness of transforms and torsion-freeness show that this restriction is zero.

It remains to identify the constant in the antiderivative. The map to \(E\) extends by the Neron mapping property on the smooth ordinary charts carrying the canonical \(p\)-subgroup. Such a chart has the same deformation germ as its tame chart: a subgroup lifting the connected special subgroup has zero image in the etale quotient and must be the canonical multiplicative subgroup. In a Neron residue tube the formal group logarithm, translated by the value at the center, is analytic. Multiplication into a smaller formal neighborhood and division by that integer gives the same analytic logarithm throughout the interior disk. This proves analyticity of \(G\), including when the elliptic target has bad reduction.

The differentials of \(F_0\) and \(P_p(V_p)G\) agree. Indeed differentiating \(V_pG\) introduces a factor \(p\), and differentiating \(V_p^2G\) introduces \(p^2\). Their difference is therefore constant on the disk. The noncanonical quotient sources of a target \(B\) give, by the \(T_p\) or \(U_p\) correspondence, \[\sum_{\zeta^p=1}G(\zeta z) =a_pG(B)-\epsilon_pG(V_pB).\] Cuspidal differences are torsion [18, 9] and disappear under the logarithm. When \(p\mid N\), the noncanonical quotients retain the required canonical level, so the same identity uses \(U_p\). It follows that \(P_p(V_p)G\), like \(F_0\), has trace zero. The \(p\) sources lie in the same residue disk. The trace of their constant difference is consequently \(p\) times that difference, which is zero in characteristic zero. This proves [eq:source-9]. Equivalently the same normalization is the elliptic weight-two case of [23]; the preceding argument specifies its extension to a potentially bad target and to the canonical \(p\)-structures used here. ◻

Proof. Finite Fourier summation on \((\mathbb Z/p^m)^\times\) multiplies the inverse moment of a measure by the displayed Gauss sum. Transporting the frame in disk \(\mathfrak a\) changes the root exponent by \(\delta_{\mathfrak a}\). The correction \([t_{\mathfrak a}-\eta_\Gamma(\delta_{\mathfrak a})]\) therefore makes its weight exactly the Artin character \(\chi\). The class-kernel description identifies the resulting sum over roots and disk centers with the sum over \(\mathop{\mathrm{Gal}}(H_{p^ms}/K)\).

Apply [eq:source-9]. For sufficiently large \(m\), the \(V_p\) and \(V_p^2\) terms factor through lower-conductor orbits, so their sums against the primitive character vanish. Each remaining quotient \(V_q\) translates the orbit by \(\sigma_q\), with its level choices retained. This proves [eq:source-10], with no averaging factor.

Atkin–Lehner at the non-\(p\) part of the primitive level reverses the non-\(p\) orientations. On the newform quotient it acts by a sign; the change of base cusp is torsion. The raw logarithmic sum therefore changes only by that sign and a translation character. The same assertion holds when a subset of the non-\(p\) orientations is reversed. The translating ideals are prime to \(p\), so these translations can be chosen compatibly in the \(p\)-tower. Applying [eq:source-10] at all high characters, followed by uniqueness for bounded series, gives [eq:source-11]. At the moving stages the group-like translations likewise have coefficientwise limits.

At the trivial character, integration is evaluation at the disk center. Formula [eq:source-9] now retains the \(p\)-Euler factor, because \(V_p\) acts by translation on the conductor-one centers. The sum is taken once over the class group. The two orientation sums are translates of the same Hilbert trace up to sign. Recalling \(G=c_E^{-1}\log_{\omega_E}\phi\) gives [eq:source-12]. ◻

Proof. The good Hecke correspondence at \(r\), after subtracting its two horizontal split translations, is the trace from conductor \(rs\) to conductor \(s\). Its action on the character sum in [eq:source-10] is [eq:source-13]. Transport the frames by polarization. Their exponents are \(r_i^{\pm1}\), which tend to \(1\) modulo every fixed \(p^m\). Thus the same identity holds at each fixed sufficiently high \(\theta\), simultaneously in the group rings of the tame variables that remain. The expressions defining \(b_s\) are bounded as group polynomials. We can consequently pass to limits in those group rings; bounded character projectors are not required.

For the arithmetic determinant, unramified inflation restores the full pair over each removed prime. The two singular determinants are the unramified Tate Euler polynomials at that pair. Since \(r_i\to1\), their product is the square of [eq:source-13] up to an integral unit. This is an identity of the marked integral determinant triangles, not a comparison after rationalization. Iterating with the old tuple fixed proves all the stated specializations. The factor need not be a unit at the central specialization. ◻

Proof. Extend arithmetic unramified Frobenius \(\varphi\) at \(w\) to fix the cyclotomic \(p\)-power roots. At every finite stage, [eq:source-10] identifies its action on a raw CM sum with translation by a fixed local Artin element: the ring-class extension is abelian over \(K\), so this action commutes with all labels. The Gauss coordinates are fixed. Passing to limits and testing at high finite characters gives, for each series in question, \[\varphi(b)=\kappa b,\] where \(\kappa\) is group-like of augmentation \(1\). On \(\mathcal V^*\), \(\varphi\) is Witt Frobenius. There is a unit \(e\), with constant coefficient \(1\), satisfying \(\varphi(e)=\kappa e\). Solve recursively in total degree: at each step the remaining equation is \(\varphi(a)-a=b_0\). It is soluble modulo \(p\) because the residue field is algebraically closed, and successive lifting solves it in \(\mathcal V^*\). Then \(b/e\) has coefficients fixed by Witt Frobenius, hence in \(\mathbb Z_p\). Products give the assertion for \(B_w\). We continue to use the actual analytic series for value formulas and the descended generators for height-one ideal comparisons. ◻

Proof. We first establish the independence statement in the elliptic case. Let \(k=\overline{\mathbb F}_p\), let \(x\) be an ordinary maximal-order CM point with endomorphism ring \(\mathcal O_K\), and let \(\mathscr R\) be its algebraic local ring on the prime-to-\(p\) modular tower. Thus a finite set of elements of \(\mathscr R\) comes from a common finite tame level; we do not replace \(\mathscr R\) by its completion. For \(a_1,\ldots,a_h\in\mathbb Z_p^\times\), consider \[\Phi:\mathscr R^{\otimes_k h}\longrightarrow k[[z-1]],\qquad f_1\otimes\cdots\otimes f_h\longmapsto \prod_{i=1}^h f_i(z^{a_i}).\] We will prove that \(\Phi\) is injective when the \(a_i\) have pairwise inequivalent ratios under \[\{\alpha/\bar\alpha: \alpha\in K^\times,\ \alpha\text{ a unit at both primes over }p\}.\]

Suppose the prime kernel of \(\Phi\) is nonzero and choose \(h\) minimal. Let \(X\) be its irreducible relation locus in the product of the tame towers. Its projections to every proper product of factors are dominant by minimality. The common CM stabilizer preserves the kernel: its scalar Serre–Tate substitutions commute with all the \(a_i\). Weak approximation in \(K\) makes the ratios of its two local unit actions dense in \(\mathbb Z_p^\times\). The local rigidity and normalization argument of [13] therefore gives a finite tame-level model and a normalized branch over \((x,\ldots,x)\) whose completion is a formal subtorus. This branch contains \((z^{a_1},\ldots,z^{a_h})\). Write \(Y\) for the normalization of the irreducible model. Dominance onto \(h-1\) factors and properness give \(\dim Y=h-1\).

Here is the global constraint on that formal subtorus. Pull back the product of the universal elliptic schemes to \(Y\). The global canonical-coordinate map of [7] has, on a normal base, a cotorsion-free smooth kernel. Its annihilator \(\mathscr N\) is a smooth cocharacter subsheaf. Its rank can be evaluated on the chosen formal branch, where it is \(h-1\). The proof of [7], applied to the normalization and this branch, consequently gives a dense smooth open \(U\subset Y\) on which the same rank is \(h-1\). The elliptic-factor idempotents of the product abelian scheme place \(\mathscr N\) inside the direct sum of its \(h\) diagonal Serre–Tate cocharacter lines. Each coordinate projection of \(\mathscr N\) is nonzero. Otherwise that elliptic deformation would be formally constant on \(U\), contrary to the dominance of the corresponding projection of \(X\). This construction uses the canonical-coordinate relation sheaf on a normal base; the ranks of two-coordinate projections alone do not determine its rank.

Descend \(Y\), \(U\), the elliptic schemes, and their canonical-coordinate sheaves to a finite field. Suppose first that the \(h\) generic elliptic factors are pairwise nonisogenous over an algebraic closure. Choose an odd prime \(\ell\ne p\). After a finite etale cover of a smaller open \(U\), all generic endomorphisms are defined and the joint arithmetic algebraic \(\ell\)-adic monodromy group is connected. The semisimplicity and isogeny theorems apply to its finitely generated function field [34]. Each nonconstant elliptic factor has derived monodromy \(\mathrm{SL}_2\). The Lie algebra of the derived group is a subdirect product of these simple nonabelian Lie algebras. The Lie-algebra Goursat argument decomposes it into diagonal blocks: the kernel of a projection is an ideal, and a simple factor is either independent or identified with one of the preceding simple factors. A block meeting two factors identifies their standard two-dimensional representations on the derived group. The connected center acts on these representations by scalar characters whose squares agree, because their determinants are equal. The two characters therefore agree. The intertwiner is equivariant for the whole connected group, and the isogeny theorem over finitely generated fields of characteristic \(p\) makes the two generic elliptic curves isogenous. Thus no such block is possible, and the derived monodromy group is \(\mathrm{SL}_2^h\). Passing to the connected group has also removed possible finite quadratic-twist correlations. These are the endomorphism and monodromy inputs used in [13].

Chebotarev gives a closed point \(u\in U\) at which the elliptic Frobenius traces \(c_1,\ldots,c_h\) have pairwise distinct squares: the equations \(c_i^2=c_j^2\) define proper closed subsets of the monodromy group, whose \(h\) derived factors vary independently. Let \(\alpha_i\) be the \(p\)-adic unit root at \(u\), and put \(q_u=\#k(u)\). If \(\alpha_i^2=\alpha_j^2\), then \[c_i=\alpha_i+q_u/\alpha_i =\pm(\alpha_j+q_u/\alpha_j)=\pm c_j,\] a contradiction. Arithmetic Galois descent Frobenius acts on the \(i\)-th Serre–Tate character line by \(\alpha_i^2\), since that line is the tensor square of the etale Tate line; it acts on the cocharacter line by \(\alpha_i^{-2}\). These are distinct eigenvalues. The stable subspace \(\mathscr N_u\otimes\mathbb Q_p\) must therefore be a sum of coordinate axes: each eigenprojection is a polynomial in Frobenius. Its nonzero projection to every coordinate forces it to contain every axis, contradicting its rank \(h-1\).

There is consequently a generically isogenous pair. Extend its isogeny over a normal finite cover of \(Y\); homomorphisms between abelian schemes extend over a normal base. At a point above the chosen CM branch its specialization is a nonzero \(\alpha\in\operatorname{End}^0(x)=K\). Serre–Tate functoriality and the formal curve in that branch give \(a_i/a_j=\alpha/\bar\alpha\), up to simultaneous inversion. One may see this before inverting \(p\) by clearing denominators in the two local actions of \(\alpha\) and comparing exponents in \(k[[z-1]]\); the map \(b\mapsto z^b\) from \(\mathbb Z_p\) is injective. Since \(a_i/a_j\) is a unit, the two valuations of \(\alpha\) above \(p\) agree. Rational scaling makes both zero without changing their ratio. This contradicts the stipulated inequivalence and proves injectivity of \(\Phi\). In particular, a sum of functions from distinct slots is constant only if every slot function is constant.

We now work at \(s=1\), with one of the two non-\(p\) orientations. The functions here are regular weight-zero ordinary functions at a maximal-order CM point. The weight-zero torsor descent in Proposition 41 places their reductions on the tame ordinary special fiber, not merely on a finite Igusa cover. The canonical multiplicative \(p\)-structure has the same deformation germ as this tame curve. Their germs therefore belong to the algebraic prime-to-\(p\) stalks just considered.

Suppose \(b_1\) is zero modulo \(p\). Restrict its measure to the open subgroup \(\eta_\Gamma(\mathbb Z_p^\times)\) of \(\Gamma\). For each contributing class choose \(\lambda_{\mathfrak a}\in\mathbb Z_p^\times\) with \(\eta_\Gamma(\lambda_{\mathfrak a})=t_{\mathfrak a}\). Pushing the measure through \(\eta_\Gamma\) folds it over its kernel \(\mu_{p-1}\). On transforms the assumed vanishing becomes \[\sum_{\mathfrak a}\sum_{\zeta\in\mu_{p-1}} F_{\mathfrak a} \bigl(z^{\zeta\delta_{\mathfrak a}/\lambda_{\mathfrak a}}\bigr) =0\pmod p.\] The class sum includes exactly those classes that contribute to the indicated open subgroup. Pull back by the ideal isogenies that transport the frames to a common prime-to-\(p\) level. This absorbs \(\delta_{\mathfrak a}\) into the pulled-back function, so the exponent whose slot is to be tested is \(\zeta/\lambda_{\mathfrak a}\). Combine equivalent slots before applying the injectivity of \(\Phi\). An equivalence \(\alpha/\bar\alpha\) is realized by the germ of a CM quasi-isogeny. Since this ratio is a \(p\)-unit, rational scaling makes the isogeny prime to \(p\).

We describe the slot equivalent to \(1\). Put \(h=h_K\) and \(\mathfrak a^h=(x)\). The class-sequence description gives \[\lambda_{\mathfrak a}^{\,h} =(x/\bar x)\xi\] up to simultaneous inversion, where \(\xi\) is a torsion unit. If \(\zeta/\lambda_{\mathfrak a}=\alpha/\bar\alpha\), raise to a power killing \(\xi\) and \(\zeta\), and compare ideal valuations. The group of fractional ideals is torsion-free; cancellation of that power and of \(h\) shows that a principal scaling of \(\mathfrak a\) is conjugation invariant. This is cancellation in the fractional ideal group, not division by \(h\) in \(\mathbb Z_p\). Such a class is represented by a product of ramified prime ideals, each used once or omitted.

Conversely, choose those ramified products as representatives. Their squares are rational principal ideals, so their images \(t_{\mathfrak a}\) in the torsion-free group \(\Gamma\) are zero; we may take \(\lambda_{\mathfrak a}=1\). Only \(\zeta=\pm1\) can belong to the slot of \(1\), since a torsion element of \(K^\times\) is then a sign. The ramified products represent these classes twice, paired by complementary subsets. To check the multiplicity, if a ramified product is \((\alpha)\), then \(\alpha/\bar\alpha=\pm1\). The plus sign makes \(\alpha\) rational and forces the subset to be empty; since the discriminant is odd and fundamental, the minus sign makes it a rational multiple of \(\sqrt{\operatorname{disc}(K)}\) and forces the subset to be full. The scaling between complementary products has conjugate ratio \(-1\). Thus the two expansions with exponents \(\pm\delta_{\mathfrak a}\) are exactly the two pullbacks indexed by the complementary ramified products.

This combined function is nonconstant. Indeed pass to a common tame level and choose a Tate cusp at which all these prime kernels are multiplicative. They occur at distinct primes, disjoint from the oriented split level, so the choices can be made simultaneously; retain multiplicative type for those oriented cyclic groups too. The identity translate has first positive Fourier coefficient \(1\). Every other ramified product has degree greater than one, and its quotient expansion begins at a larger index. No cancellation of that coefficient is possible. The injectivity of \(\Phi\) now contradicts the displayed zero sum. Hence the conductor-one series is nonzero modulo \(p\) for each orientation.

Finally apply Lemma 43. Each new factor [eq:source-13] remains nonzero modulo \(p\): its true \(\Gamma\)-exponent is nonzero by the tuple construction. Its powers therefore vary nontrivially in \(\mathbb F_p[[t]]\), and the displayed Laurent polynomial cannot vanish identically. After setting the new variables to zero we obtain a product of nonzero reductions. The full series consequently has nonzero reduction, proving [eq:source-14]. ◻

Proof. Let \(P_v/K_v\) be the finite totally ramified extension cut out by the inertia part of \(\theta\), and separate its unramified part. Write \(G_v=\mathop{\mathrm{Gal}}(P_v/K_v)\), \(\mathcal O_\theta=\mathbb Z_p[\theta]\), and \(\Delta=\mathop{\mathrm{Gal}}(D_\infty^{\rm loc}/K_v)\simeq\mathbb Z_p\), where \(D_j^{\rm loc}/K_v\) is the unramified extension of degree \(p^j\). Choose its arithmetic Frobenius generator \(\gamma_{\rm ur}\). The Shapiro coefficient ring before finite-character projection is \[\mathcal O_\theta[G_v][[\Delta]][1/p].\] Project by the character idempotent corresponding to the chosen cohomological character. The resulting scalar ring is \(\Lambda_\theta=\mathcal O_\theta[[X]][1/p]\), with \(1+X=\gamma_{\rm ur}\). If \(e_{vj}\in\mathbb Z_p\) are the tame Frobenius exponents at \(v\), its map to the comparison ring is \[\Lambda_\theta\longrightarrow A_\theta,\qquad 1+X\longmapsto \alpha_v\prod_{j=1}^d(1+u_j)^{e_{vj}}, \quad \alpha_v=\theta_{\rm ur}(\mathop{\mathrm{Frob}}_v).\] The exponent vector is nonzero. Divide it by its largest common power \(p^a\), then complete the resulting primitive vector to a \(\mathrm{GL}_d(\mathbb Z_p)\) coordinate change. In the new group-power coordinates the image is \(\alpha_v(1+y_1)^{p^a}\). The equation \[(1+y_1)^{p^a}=\alpha_v^{-1}(1+X)\] is monic and distinguished of degree \(p^a\) in \(y_1\). It makes the one-variable substitution finite flat; adjoining the remaining variables preserves flatness. The marked local models can therefore be base-changed through this map.

We first control lattices uniformly in \(j\), with constants allowed to depend on \(\theta\). A sufficiently small additive lattice exponentiates because the ramification is fixed. For an upper bound pass to a fixed extension with semistable reduction. The component groups are bounded after adjoining \(P_v\) and making unramified base changes. Choose \(\theta\) nontrivial on sufficiently deep inertia over this fixed field. After multiplication by the bounded component exponent, inertia acts trivially on the special points of the identity component. The Neron base-change map is surjective on these geometric semiabelian points: its map on rational auxiliary-adic Tate modules identifies the inertia invariants, which are unchanged under finite extension once inertia is unipotent. Thus an inertia difference takes points into the formal kernel. A further fixed multiple takes them into a small logarithm lattice. On the projected eigenspace that difference is a nonzero constant, giving the upper bound. Use the unnormalized integral projection to obtain the same bounded-exponent assertion for the torsion logarithm kernel. The argument applies to the inverse character as well. Trace-dual bounds follow from the bounded different of the fixed ramified extension.

Integral Kummer and local duality on the Shapiro tower give an exact sequence with ends the compact points and the dual of compact points; transitions are norms and dual inclusions. Compactness makes the inverse limit exact. One can identify both ends explicitly from trace-compatible integral normal generators in the unramified tower. Choose their successive traces compatibly and their total trace equal to \(1\). Modulo \(p\), a unit augmentation generates the cyclic group-algebra module, so these are normal generators integrally as well. For the fixed ramified factor use a rational normal basis and its bounded trace dual. Before projection, the additive inverse limit is the regular rank-one module over \(\mathcal O_\theta[G_v][[\Delta]][1/p]\). Apply the finite-character idempotent to this additive module. The projected logarithm lattice bounds identify the projected compact point module with it after inverting \(p\); the projected torsion kernel is killed by the fixed bound already proved. Thus the compact point endpoint is free of rank one over \(\Lambda_\theta\). The bounded trace-dual comparison gives the same conclusion for its dual endpoint. Applying the displayed flat substitution yields the two free lines over \(A_\theta\). The torsion bounds kill higher cohomology; invariants vanish too. The full local complex consequently splits. Shapiro compatibility of the cup product, finite Kummer isotropy, and local duality give exact orthogonality, including the nullhomotopies in these split complexes.

Form the logarithmic group polynomials with the unnormalized character weights before making the substitution. The normal resolvent on the ramified factor is nonzero; on the unramified factors the compatible resolvent is a unit because its total trace is \(1\). Thus logarithm is the stated line isomorphism. To identify transferred point classes, specialize first to bounded cyclic quotients. Freeness with coset bases identifies the substitution with the corresponding finite unramified layer. At sufficiently large stages the local ring-class fields contain that layer, by the nonzero Frobenius exponent, and contain the fixed ramified character field. Trace the stage classes to these fixed local fields and apply the finite-level logarithmic calculation. Membership in the Kummer line is detected on all these finite quotients of its free complementary quotient. The logarithmic expressions are detected in the same way. The fixed-ramification bounds remain valid through the unramified growth at \(p\), so passage to the bounded limits gives exactly [eq:source-10]. ◻

Proof. Choose the Frobenius sequence \(\gamma_i\to\tau\) on the full Tate module. Then \(a_f(\ell)\to0\), while \(\gamma_i^2\) acts trivially on the ring-class fields by the dihedral relation. The old characters are unramified at \(\ell\). In the cyclic inertia resolution the residue conjugating power tends to \(1\). Thus the limiting full local cohomology is split free. Evaluation on \(\gamma_i^2\) and on a tame generator gives the finite and transverse coordinates. Retain invariants in the lower condition and omit \(H^2\) in the upper condition, as required by the square-switch lemma. Conjugate transport through \(\gamma_i\), together with local reciprocity, gives mixed pairing \(\langle x,Jy\rangle\). Pure scalar cups vanish and the mixed scalar cup is a unit. These are exactly the Weil-plane hypotheses of that lemma.

Here is the derivative calculation, following [24], with the odd-\(p\) precision specified. The relative ring-class order at an inert prime is \(\ell+1\), since the only order units are signs. For its generator \(\sigma_\ell\) use \[D_\ell=\sum_{j=1}^{\ell}j\sigma_\ell^j.\] Writing \(\operatorname{Tr}_\ell=\sum_{j=0}^{\ell}\sigma_\ell^j\), the group-ring identity is \[(\sigma_\ell-1)D_\ell=(\ell+1)-\operatorname{Tr}_\ell.\] If \(Y\) is the upper point and \(X\) the lower point, the Hecke trace gives \(a_\ell X\), after the common cusp multiple, and supersingular reduction gives \(\widetilde Y=F_\ell\widetilde X\). Choose growing precision \(p^a\) slowly enough that \(p^a\mid \ell+1\), \(p^a\mid a_\ell\), and \(\gamma_i^2=1\) modulo \(p^a\). The group-ring derivative identity then makes the upper Kummer class invariant to that precision.

Inflation and restriction descend it after a fixed multiple. The needed uniform bound on invariant torsion follows from Bogomolov’s homothety theorem [3]: take \(h\in G_K\) scalar sufficiently close to \(1\), with square different from \(1\). The element \(h\tau h\tau\) kills every ring-class action and acts on the Tate module by that scalar squared. It bounds torsion over all fields in use. At the old \(p\)-places, descent is unramified and retains the Kummer condition after a fixed component multiple. Lang’s theorem handles the connected special points; the formal-group assertion follows from induced additive successive quotients and passage to the limit. Component groups remain bounded under unramified base change above the fixed ramified extensions. The same argument retains unramified conditions at the old non-\(p\) support. Include all these multiples in one scale for every switched class.

To verify the coordinates, suppress that scale and choose \(p^aQ=X\), \(p^aW=D_\ell Y\). If \(b\) is the descended upper cocycle, the crossed identity makes \((\sigma_\ell-1)W-b(\sigma_\ell)\) fixed by the upper-field subgroup. Multiplication by \(p^a\) gives \((\ell+1)Y-a_\ell X\), so it differs from \[((\ell+1)/p^a)Y-(a_\ell/p^a)X\] by bounded-exponent torsion. After reduction, inertia is trivial. Thus, to the retained precision, \(b(\sigma_\ell)\) gives \[(a_\ell-(\ell+1)F_\ell)\widetilde Q =(F_\ell-a_\ell)(F_\ell^2-1)\widetilde Q,\] where the equality is Eichler–Shimura. The lower Kummer cocycle, evaluated on the residue Frobenius \(\gamma_i^2\), has finite coordinate \[f_{\rm loc}(Z_I)=(F_\ell^2-1)\widetilde Q\] at the retained precision. The displayed value of \(b(\sigma_\ell)\) is the transverse coordinate of the upper class. Consequently it equals \((F_\ell-a_\ell)f_{\rm loc}(Z_I)\). Passing to the prescribed limit gives \(F_\ell-a_\ell\to J\), which proves the asserted transverse identity. For the upper finite coordinate, \(\widetilde{D_\ell Y}\) has an \(F_\ell^2\)-fixed \(p^a\)-division obtained using \(\ell(\ell+1)/(2p^a)\) times \(F_\ell\widetilde X\); the ambiguity is killed by \(F_\ell^2-1\). Reduction detects prime-to-\(\ell\) torsion, proving the zero finite coordinate. The calculation applies on each translate before weighted corestriction, with any earlier derivative factors already present. It therefore proves the two identities simultaneously at all previous switches and with the identical common scale. ◻

Proof of Proposition 46. We show that [eq:source-16] is nonnegative at every height-one test. If fiber degree one has dimension greater than one, choose two independent classes, one being a primitive reduction from the generic line. They admit a rank-two evaluation on the finite plane at a suitable inert prime.

We spell out this detection because the same assertion must hold after previous switches. Lemma 12 realizes the fiber classes on the product of the stage Galois groups. Restriction to the joint kernel of the full Tate and ring-class actions detects them: on the image quotient the central element \(h\tau h\tau\) just used has scalar difference from \(1\) invertible. Global invariants vanish. The Selmer classes inject into unrestricted cohomology by the local hypotheses, and the complexes have amplitude \(1,2\) by duality. Let \(\kappa(\mathfrak p)\) be the testing residue field, and write \[W=(V\otimes\kappa(\mathfrak p))\chi,\qquad W^c\simeq(V\otimes\kappa(\mathfrak p))\chi^{-1}.\] Conjugate transport makes these the two summands of the induced representation after adjoining diagonal \(\tau\); the element \(\tau\) exchanges them through the elliptic involution \(J\). They are absolutely simple and nonisomorphic: their scalar twists have distinct determinants because the inertia character has sufficiently high order. For the two independent classes and their conjugates, joint evaluations on the kernel therefore span \(W^2\oplus(W^c)^2\). For an element \(\eta_b\) of that kernel, evaluation on \((\tau\eta_b)^2\) is additive in the kernel evaluations. On a tuple \((a_1,a_2;b_1,b_2)\), the operator \(1+\tau\) followed by projection to \(W^2\) gives \[(a_1+Jb_1,\ a_2+Jb_2).\] This map is surjective, so its evaluations still span both copies needed for a rank-two test. An additive subgroup contains the integer grid generated by any finite spanning set; in characteristic zero this grid cannot lie in a proper rank-defect locus. A rank-two choice follows. Finite Chebotarev realizes these evaluations to increasing precision by primes whose Frobenius agrees with \(\tau\eta_b\).

Apply Lemma 14, using Lemma 48. The fiber dimension drops by two; the generic ranks and the valuation of the point/class-functional tensor are preserved. The fiber dimension cannot become zero: upper semicontinuity bounds it below by the generic rank one. Thus the repeated two-dimensional drops end at dimension one. There the complex is homotopy equivalent to two free lines with zero differential and perfect duality. Its integral point tensor has nonnegative determinant valuation. All fixed scales are units at this test. Thus [eq:source-16] is nonnegative. The height-one criterion in the UFD \(A_\theta\), and its coefficient extension, gives \[p_0^2/(L_w/D)\in A_\theta',\] which is [eq:source-15].

For the association assertion, keep the global character unchanged and perform the same comparison at \(\bar w\). At conductor \(p^ms\), for large \(m\), the canonical connected quotient lowers the Serre–Tate parameter order by \(p\). It is the unique ascending edge in the CM isogeny graph. The required fixed number of cyclic \(p\)-level edges therefore gives the same level along the class orbit for the two choices of \(p\)-adic place. Their points agree up to the non-\(p\) Atkin–Lehner operations and translations already considered. The weighted classes differ only by units in \(A_\theta\); the cuspidal differences are killed. Hence the all-Kummer complex and its tensor in [eq:source-16] are the same up to units at every height-one test. The two quotient valuations coincide, and the quotients are associates. Contracting the equal principal ideals along the finite faithfully flat coefficient extension gives the same association over \(\widetilde A_\theta\). The high-conductor condition is essential here; this argument does not compare two arbitrarily normalized logarithmic lines at a low-conductor specialization. ◻

Proof. By [eq:source-1], \(L_w\) is nonzero modulo \((p,t)\). Lemma 18 therefore applies to Proposition 46. For clarity, make \(L_w\) distinguished in one tame variable by an integral group-power change and divide \(B_w\) by it integrally. At every sufficiently high character of \(t\), the remainder is zero after inverting \(p\) and extending coefficients to \(A_\theta'\). Faithful flatness descends this vanishing to \(\widetilde A_\theta\); the monic quotient has no \(p\)-torsion, so it is zero integrally there. Each bounded coefficient of the remainder vanishes at all those characters and hence vanishes identically by one-variable Weierstrass preparation. This proves integral divisibility. Since \(p\nmid B_w\), the integral quotient has no factor \(p\).

Lemma 43 restores the same squared factor in numerator and denominator on setting each new variable to zero. That factor is nonzero, and the determinant comparison unit is integral. Cancelling in the fraction field proves the asserted specialization identity. Integrality of the larger quotient then makes the specialization integral as well. There is no loss of precision in this step. ◻

Proof. The ray-character construction allows its additional conductors at arbitrarily chosen sufficiently deep split places: imposing that the ratio of a principal idele to its conjugate be \(1\) at such a place removes the finite unit obstruction, after which the character extends over the finite ray-class quotient. The same construction gives the weight and inverse unary splitting characters. Neat level may also be imposed at a new split place. Carry out these choices twice, avoiding the finite set already used.

The required spherical tests at an unmarked split place can be seen in matrix coordinates. Choose integral \(e,B_1,B_2\) columns, and scale the \(f\) column dually to the additive conductor. Their mixed position lattice contains the primitive norm-one orbit. Conjugate the hyperspecial to these bases if necessary. On a split unary line, the integral lattice test is torus-unit invariant and its diagonal Mellin integral is the nonzero central Tate integral. The absolute-value half shift in the oscillator and temperedness give its nonzero value; fixed volume and uniformizer powers are nonzero. Thus these local tests may be retained in the unary nonvanishing construction. At nonsplit places use the matched hyperspecial tests. Refining a metaplectic multiplier or cusp level for an auxiliary calculation does not impose a new target Hecke control at that place.

Start with a nonzero compact vector having the required invariances. Translate at a free split place until its restriction to \(T\) is nonzero. More explicitly, the torus supplies determinants, and strong approximation for \(\mathrm{SU}(A)\) away from that split place places the other components in their prescribed compact opens. Fourier expansion of this nonzero function on the compact torus quotient gives \(d_0\). Its unramifiedness at the retained compact places follows from invariance. A fixed diagonal translation at \(p\) makes the vector invariant under the integral lower unipotent without making its torus period zero.

Apply the simultaneous unary nonvanishing lemma of [23]. Its inputs here are coherent matched line pairs, unramified data at nonsplit finite places, self-dual critical unary characters of parallel infinity weights \(d_j+c_j\) with nonnegative remaining type, a reduction prime greater than two split completely in \(L\), and a split degree-one variation prime. Those primes can avoid all the fixed sets just prescribed. Inverse variation of the two characters preserves their product. The split-line calculation above retains the prescribed unramified tests; the one-dimensional local Hom space identifies them with the nonzero local functional. At \(p\) use the unit-character and unit-cut tests of that lemma. Its congruence choice gives the asserted sum of weights. The analyticity branches depend on powers of \(\mathcal C\) and on fixed CM scales, not on the new finite-order coefficients; hence they can be narrowed, including the \(\Gamma\) condition above, before making the nonvanishing choices. ◻

Proof. The cohomological conditions are open conditions on the divisor: they are the nonvanishing of minors of the finite free models. Their generic validity was proved for the horizontal tests in Sections 3 and 6. The asserted absence of the two jumps is another such open condition.

At a fixed split good prime the additional exceptional conditions are zeros of finitely many Laurent polynomials in its unramified scalar parameter. These are the spanning denominators of [23] and the Frobenius separation resultants of [23]. They are nonzero at trivial scalar and weight zero. The parameters at the two \(F\)-places are powers or inverse powers of the single parameter at the rational prime. Two different rational primes cannot both give exceptional identities on \(\mathfrak p\): their independent exponent vectors make equations fixing both parameters to algebraic values have codimension at least two. One may verify this after a finite flat power change of multiplicative coordinates, which diagonalizes the two exponent vectors. At most one rational prime is therefore identically exceptional, and one of the two disjoint auxiliary sets avoids it. Avoid the remaining finitely many zero loci at the point.

For a moving prime, the inverse fake Euler factor is a polynomial of degree at most two in the character value of a uniformizer. Changing the uniformizer by a unit leaves the cleared formula unchanged: the only term using the fake factor as an actual Euler factor is multiplied by the unit norm sum, on which the character is trivial. If the tested unit character is nontrivial, it has at least three values, since \(p\) is odd. A unit change avoids the two possible roots. If it is identically trivial on the divisor, its own coordinate is \(u_j=0\), and the trace condition in the moving-prime construction makes the limiting Euler determinant nonzero. The same trace condition gives the Frobenius separation resultants in this case. In the ramified case choose a unit with nontrivial tested value; it separates the elliptic head from the two trivial inertia eigenvalues. Preserve these finitely many nonvanishings.

The interior algebraic points are dense in each nonempty affinoid open subset of a characteristic-zero divisor. The bounded-series specialization argument therefore supplies \(z_0\) outside the listed proper zero loci. Enlarge its finite field of definition for the fixed data only. ◻

Proof. The multiplicative subgroup of an ordinary \(p\)-divisible group is canonical at every finite level. Cartier duality identifies its subflags and frames with quotients and markings of a finite etale character group. Once the rank-three member is this subgroup, the remaining level data are exactly these etale data. The full upper radical is what removes any choice of a lift of the etale quotient. The corresponding generic-fiber choices form an open and closed branch after normalization. Idempotents extend over the ordinary completion; normalization commutes with that completion by excellence.

At a rank-one boundary the semiabelian object is an extension of a binary abelian object by a torus. The preimage of its binary multiplicative torsion is of multiplicative type. Its character group and flags are constructed before division by the Raynaud period lattice, so they do not involve the center Fourier parameter. The resulting chart covers are finite etale covers of the abelian extension-data charts. Pointing such a cover over a strict henselian boundary base makes it an abelian scheme and the covering an isogeny, by lifting addition through the etale cover. This is the geometric construction in [23], applied separately at depth \(s\); no numerical bound in its proof depends on \(s\).

The hyperspecial base has ordinary cusps on every component. Each connected finite etale component maps onto its base component and consequently meets its cusp charts. The expansion principle on these charts detects a zero reduced section. Repeated division by a uniformizer proves saturation at every precision. These are statements about reduction and etale covers, so finite ramification of the coefficient field changes neither the assertion nor the valuation normalized by \(v_p(p)=1\). ◻

Proof. The upper shifts are indexed by \(M_{j,4-j}(\mathbb F_p)\). On the support of the test the head has rank two modulo \(p\), so the expanding columns can become divisible by \(p\) in exactly \(p^{(j-2)(4-j)}\) ways. None of these equations changes the head function, including its determinant character. The split Weil action supplies \(p^{4-j}\). Their product is \(p^{(j-1)(4-j)}=p^2\) for \(j=2,3\). The computation uses the full upper radical and is independent of its opposite depth and of the head character’s conductor. ◻

Proof. We first prove a scalar estimate. If a cut or test has values in a group algebra, write it in the group basis and apply the scalar argument to each coefficient. Coefficients of products are sums of products, so the ultrametric inequality retains the same bound, independently of the group order. In particular we take squares of scalar coefficient jets, rather than infer coefficientwise integrality from a square in the group algebra.

Fix a finite list of scalar cuts and polynomials. All their conductors and degrees are finite. Choose the initial integral position frame with first vector in the multiplicative direction at the fixed CM object, and use this same finite marking in the level construction. For these unary theta functions use the ordinary Hilbert modular Igusa tower with the fixed tame Schwartz level and its multiplicative differential trivializations, as in [23]. At each split \(p\)-factor the multiplicative height is one. Marking its character group at a deeper level is finite etale, and does not mark a lift of the etale quotient. Over a toroidal cusp it leaves the Fourier parameter unramified. To see which cusps detect this branch, include the toroidal boundary in this construction. At a Tate cusp its multiplicative group is the Tate torus, and its character-group markings are finite etale unit markings of that torus. A connected component of this finite etale cover maps onto its connected base component and hence has a point over the chosen ordinary Tate cusp. Choose its adelic cusp representative integral at \(p\), using Iwasawa decomposition and approximation in the cusp parabolic, and realizing this level marking. Relative to the fixed initial position frame, its marked connected vector is \(ue\) modulo \(p^s\) with \(u\) a unit, at each Hilbert \(p\)-factor. Its integral symplectic comparison matrix \(g\) therefore has lower-left entry divisible by \(p^s\). In a two-dimensional factor, write \(g=\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)\). Since \(a,d\) are units and \(c\in p^s\mathbb Z_p\), explicitly \[g= \begin{pmatrix}d^{-1}&b\\0&d\end{pmatrix} \begin{pmatrix}1&0\\c/d&1\end{pmatrix}.\] The second matrix is congruent to the identity in the lower direction. Take \(s\) at least the stabilizer depth of the finite list of tests. The second matrix then fixes those tests. Thus every component has a detecting cusp representative in \(B(\mathbb Z_p)\operatorname{Stab}(c)\) relative to the original position frame. No preliminary symplectic transformation of the cut is being made. The Fourier interchange of the two directions would mark the etale direction instead and does not describe this canonical branch. The scalar Hilbert expansion map on this canonical branch is therefore saturated, by the same reduction argument as in Lemma 52. This is the scalar Hilbert tower, rather than the unitary target of that lemma. We may compare the finite list on one sufficiently deep such level. We shall remove every denominator introduced in making that comparison, rather than bound it in terms of the level.

For integral positions the upper integral unipotent acts through the additive quadratic phase and is in the stabilizer with our unit self-dual normalization. The diagonal stabilizer can be made as deep as the local constancy of the cuts requires. Smoothness supplies a lower stabilizer of some finite depth; equivalently one may choose that depth after bounding the Fourier-transform support of this individual finite list. These statements put all the tests on the canonical multiplicative levels just described.

Here is the effect on moments of these compatible frame changes. The preceding construction compares each detecting cusp directly with the fixed CM position frame through \(B(\mathbb Z_p)\operatorname{Stab}(c)\). Their effective changes are therefore upper triangular: they rescale positions by units, permute the tests, and multiply by unit phases. For a diagonal unit \(u\), the test reads \(\phi(ux)\), while the multiplicative marking changes the differential by \(u^{-1}\). The normalized degree-\(d\) moment therefore reads \((ux)^d\) in the same position coordinate. An upper unipotent contributes a phase common to the polynomial moments. Thus both the cut and the polynomial are transformed in the same coordinate. We use no uniform bound on the Fourier transform of fine cuts at \(p\); a Fourier interchange with the opposite position direction is not among these compatible changes.

Consequently the coefficient at index \(\beta\) of a polynomial combination at such a cusp is \[ \sum_{Q(x)=\beta}\phi(x)c(x)P(x). \tag{38}\] The sum is finite, and \(Q\) has unit quadratic coefficient in the normalized position frame. It has absolute value at most the fixed bound for \(\phi\) times \(\|P\|_{\mathrm{pos}}\). The same calculation holds for several positions. Additive transforms away from \(p\) have normalization a residue-characteristic power or a root of unity and cost only the fixed denominators already included in \(C\). Moving determinant characters have unit group coefficients. The bound in (38) is thus independent of every changing depth.

We justify that saturation applies to these analytic jets. At any individual set of degrees and levels, enlarge the tame multiplier level, including the level at two, so that products of two base half-weight theta functions are algebraic integral-weight forms. Leibniz and Gauss–Manin express products of their normalized jets as rational expressions in jets of the base products. There are no horizontal poles: locally in characteristic zero the square of a base theta has even divisor, and after removing that divisor an etale square-root frame makes the connection and its fixed iterates regular. Cross-products fix the signs and descend these products. The logarithmic cusp calculation is the same. Projection to the ordinary unit-root splitting therefore makes the square of every finite linear combination an element of the ordinary tower ring with \(p\) inverted. At an ordinary CM object this splitting agrees with the CM splitting. These algebraicity and comparison facts are the finite-level facts proved in the unary-moment argument of [23]; they require finiteness of the chosen list, not a uniform bound for its depth.

Let \(G\) be such a square, multiplied by the fixed clearing factor predicted by (38). Its expansions are integral at all compatible cusps. If \(\varpi^rG\) is integral with \(r\) minimal and positive, every expansion of \(\varpi^rG\) vanishes modulo \(\varpi\). By the scalar Hilbert expansion principle just established, its reduction is zero on every component. Division by \(\varpi\) contradicts minimality. Thus \(r=0\). Evaluation and taking the square root give the claimed bound for the original combination. This argument removes its whole initial vertical denominator, even when that denominator depends on the cut. It is unchanged over a ramified coefficient extension with the valuation normalized by \(v_p(p)=1\).

For a general cut, separate the parity pieces of the finite tests and polynomial jets with \(2^{-r}\sum_{\varepsilon\in\{\pm1\}^r} \varepsilon^e[\varepsilon]\). These projectors have norm at most one because \(p\) is odd. In each coordinate write the degree as \(e_0+2j\), with \(e_0\in\{0,1\}\); the corresponding base unary theta has weight \(1/2+e_0\), and its \(j\)th normalized jet has position factor \(x^{e_0}Q(x)^j\). Carry out the preceding square argument for these parity bases; their cross-products supply the common descent. Each expansion still has the form (38). Thus the estimate applies to arbitrary cuts, including coset cuts obtained by decomposing a matrix determinant character; it does not require invariance under the separate signs of its entries.

With the cut equal to one, the estimate gives a bounded linear functional on all polynomial parity pieces, and polynomial density extends it to a bounded measure. For the additional identification of a cut with multiplication of this fixed measure, assume that \(c\) is sign invariant. Approximate it uniformly by sign-invariant polynomials, using symmetrization. Apply the same bound to the difference between a cut moment and its polynomial approximation. In (38) that difference is exactly \(\phi(x)(c(x)-P_n(x))P(x)\) in the same frame. Its norm tends to zero. Hence the measure with the cut is \(c\) times the uncut measure. All these arguments are coefficientwise for finite group algebras, so their bounds do not involve the group order. ◻

Proof. Choose canonical cusp representatives integral at \(p\) and preserving the multiplicative member modulo the chosen deep level. In the mixed expansion the \(f\) column is Fourier-dualized to the position \(x\). It remains invariant under integral translations: a change of that column affects the head of [eq:source-18] only at depth \(s_i\), and preserves the trailing integrality tests. The Fourier transform therefore has integral support and introduces no nonunit normalization at \(p\). Partition the remaining position tests by cosets and decompose them into products of unary tests. Their coefficients retain the same supremum bound, so Lemma 54 applies to their monomial moments.

The column-degree filtration of the differential bundle identifies these moments with the mixed coefficients. Binary columns have the normalized unary moments, and the hyperbolic column contributes integral position powers. Logarithmic frames and prime-to-\(p\) Jacobi torsion evaluations split the required differential sequences integrally. These are the mixed-expansion identities of [23]; the argument above checks the only changing input, the increasing \(p\)-depth.

At moving and fixed finite places choose the boundary representatives by approximation, with the required integral conditions and the deep \(p\)-condition simultaneously. Detecting Jacobi torsions remain prime to \(p\). Source integration has a fixed volume denominator because the determinant equivariance cancels the character on the fixed source compact, including at \(p\). Thus all expansion coefficients have one fixed denominator. Saturation on the canonical branch, supplied at every depth by Lemma 52, gives the assertion for the sections themselves. This concerns the canonical ordinary lattice; it makes no assertion about an unrelated generic-level lattice. ◻

Proof. An ordinary deformation is the extension of the etale group by the multiplicative group classified by its Serre–Tate unit \(z\). For \(z=\zeta_{p^b}^j\), pulling back on the etale generator by \(p^b\) makes the extension split. Relative to that rational splitting, the compatible roots \(\zeta_{p^{a+b}}^j\) of \(z\) shift an etale lift at level \(a\) by \(j/p^b\) in the connected coordinate. This gives the displayed lattice and parameter; changing the compatible root basis or reciprocity sign gives the fixed unit \(c\).

The lattice intersects the connected rational line in precisely \(\mathbb Z_pe_p\), and its image in the etale rational line is precisely \(\mathbb Z_pf_p\). Thus the rational comparison transports the connected lattice integrally, not by multiplication by a power of \(p\). One can also realize it by canonical connected quotients followed by etale quotients with the chosen roots; dividing the resulting isogeny by the common scalar gives the same integral connected comparison. Its multiplicative differential changes by a unit. Canonical levels in \(W_0\) remain canonical, and the rational CM splitting agrees with the ordinary unit-root splitting by the CM idempotents. In particular no factor \(p^{bm}\) appears in a weight comparison.

Substitute the measure expression for \(F\) into the left side of (39). The finite sum over \(j\) is \(p^b\) when \(x\equiv c\delta y\pmod{p^b}\) and zero otherwise. There is exactly one such unit \(x\) if \(y\) is a unit, and none if \(y\) is not. Division by \(p^b\) gives the right side. This proves the equality and the bound without estimating the Fourier coefficients separately. ◻

Proof. First keep \(i\) and \(b_p\) fixed. The CM/Katz comparison and Proposition 41 identify the weight limit for the first form with \(F_0=d_{\mathrm{mod}}^{-1}f^{[p]}\) in integral Igusa frames. The CM representatives at the moving tame conductor can be chosen with \(p\)-component one. Transport reference algebraic CM frames through their toric calculations and use the corresponding Igusa frames for the derivative limit. All degree-dependent comparison factors are then fixed unit powers. The translations in [eq:source-19] preserve those frames by Lemma 56. Thus the branch conditions are compatible; no new nonunit raised to the weight is introduced.

For the disk indexed by \(\mathfrak a\), a prime-to-\(p\) isogeny changes the root parameter to \(\zeta_{p^{b_p}}^{c\delta_{\mathfrak a}j}\). The unit action \(a(u)n(j/p^{b_p})\) changes the index to \(uj\). CM reciprocity therefore expresses the unit character in [eq:source-19] through the exponent map \(\eta_\Gamma:\mathbb Z_p^\times\to\Gamma\) of Section 6. Formula (39) gives the inverse moment label \([-\eta_\Gamma(y)-\eta_\Gamma(\delta_{\mathfrak a})]\), up to the common group-like unit coming from \(c\). The represented torus class contributes \([t_{\mathfrak a}]\psi_s(\mathfrak a)\). Summing the disk expressions is exactly the formula for the undepleted \(b_s^0\) in Proposition 42. Changes of basepoint, root basis, or non-\(p\) orientation introduce only its fixed sign and translation units. This establishes the identification including the global scalar labels.

For the second form the identical local Fourier calculation suffices for a bound. Its fixed toric tests away from \(p\) are finite expressions of translates of the primitive form with fixed differential constants. Its inverse-derivative functions on the ordinary CM disks give bounded measures. Unit restriction, prime-to-\(p\) translations, and the raw conductor sums preserve a common bound. The first form has the same bound from the explicit \(b_s^0\) formula. The two limits are therefore uniformly bounded group polynomials.

There is no assertion here that inverse Fourier transform at all stages has uniformly bounded norm. At fixed \(i\), its denominator \(p^{b_p}\) is fixed, and weight convergence may be taken to any precision before letting \(i\) grow. Equivalently, characterwise convergence in the fixed characteristic-zero group algebra implies coefficientwise convergence with a finite stage-dependent loss, which is paid by taking the weight sufficiently close to zero. The resulting limiting polynomials have the uniform bound proved above. Substitution at the interior point respects their limits. Since \(b_w^0(z_0)=0\), the stage limits for the first period tend to zero there; the second limits remain bounded. Approximating each limit to any prescribed stage precision proves the assertion. ◻

Proof. Let \(a_{i,m}\) be any constant coefficient in an integral ordinary frame. It is the binary scalar times the local intertwiners applied to the corresponding projected tests. Square this expression and use the bilinear comparison just described. After cross-multiplication it has the form \[ A_{i,m}\,a_{i,m}^{2} =B_{i,m}\,P_{0,i}(m)^2P_{1,i}(m)^2. \tag{42}\] Here \(A_{i,m}\) and \(B_{i,m}\) have uniformly bounded group coefficients, and \(A_{i,m}(z_0)\) is nonzero with valuation bounded above, when \(m\) is sufficiently close to zero at stage \(i\). We verify that this bound is uniform in the cusp, rather than only for the reference binary test.

At a moving prime the projected test has integral support, depends on reduction modulo \(q\), and is determinant-equivariant under the full source compact. On a deficient-rank stratum the stabilizer determinant surjects onto the inertia group. Its group coefficient is therefore an integral multiple of the norm element: all coefficients are equal, without division by the group order. The remaining untwisted-unit integral is a sum over image lattices. Express an exact reduced line or zero image by subtraction from containment conditions. Each containment sum is a translate and a power of \(q\) times the full-lattice fake Euler series. Thus \(D_q^{\mathrm f}\) clears its denominator, while \(H_q^{\mathrm f}\) clears the raw toric factors. Their bounded powers have the bounded evaluated valuations just proved. This argument also applies to bounded integral sums of pivot translates; powers of \(q\) are \(p\)-units.

At \(p\) the canonical projection has integral matrix support, by the Fourier-support calculation in Proposition 55. Its determinant equivariance cancels the source twist on a fixed compact under which the bare vector is invariant. Expanding (40) by determinant valuation gives the factor \[p^{(m-1)v_p(\det h)}.\] Right-coset volume denominators are fixed, even though the target depth grows. The translates of the fixed compact automorphic function have bounded sup norm. For sufficiently large positive \(m\) this integral-matrix series converges with a uniform bound. Its finite determinant truncations are the Godement–Jacquet series and hence compute the same rational zeta operator. No projector onto a deep \(p\)-character space is used.

At a fixed split place the divided intertwiners above have Laurent polynomial bounds on the finite spaces of tests. Integral cusp representatives and their fixed variants give the same bound. Fixed nonsplit denominators have bounded inverse at \(z_0\); unmarked places have spherical normalization. Boundary approximation chooses representatives integral at the designated finite places, and at \(p\) preserving the deep canonical flag. Compact-model evaluations and frame changes therefore introduce only the fixed factors and unit character values already accounted for. These observations prove the claimed uniformity of (42).

The period identities first hold at actual finite characters. Those characters separate the characteristic-zero group algebra, so the cross-multiplied identity holds coefficientwise. The uniform bounds permit substitution at \(z_0\) to the increasing precision for which the finite-group relations hold. By Proposition 57, its right side tends to zero, and the valuation of the nonzero left multiplier stays bounded. Thus \(v_p(a_{i,m_i}(z_0))\to\infty\), uniformly in the coefficient. Taking a square root of this valuation inequality merely divides the required precision by two.

For clarity, the parameters are chosen in the following order. Fix the finite group stage and its required \(p\)-depth. At that stage take the positive weight sufficiently close to zero to pay every finite Fourier-inversion loss and approximate the two toric limits to the desired precision. Increase its ordinary size to meet the convergent-series and holomorphy requirements. Then let the group orders increase and retain a precision tending to infinity more slowly than their relation precision and the resulting constant-term precision. Fixed losses are absorbed in this retained precision. Additional finitely many weight congruences and lower bounds can be imposed at the first step. In particular the degree choices used to retain nonconstant content in Section 8 are compatible with this construction. ◻

Proof. The preceding argument proves the content assertion for the temporary tests. Restore the controls at each fixed split place by the spanning argument of [23]. At the auxiliary good places its finitely many Laurent denominators are nonzero at \(z_0\) by Lemma 51. At an additive place the same local rank filtration works at the present unramified scalar. A supercuspidal head has no proper Jacquet module. For a ramified principal or special head with elliptic determinant trivial on inertia, both inducing-position characters are ramified. Neither can match the unramified lower-rank orbit character. Thus the lower-rank coinvariants vanish for every unramified scalar, and there is no norm linkage between this head and the trailing unramified binary block. The open-orbit induction is the unlinked parabolic product, hence irreducible; a nontrivially paired pivot generates it. In a fixed finite-level model choose a spanning minor nonzero at \(z_0\). Cramer’s rule gives the required expression over a Laurent ring with that single minor inverted. This supplies the asserted finite denominators at the additive places as well.

Cross-multiply these expressions before evaluating the group algebras. Their clearing factors have nonzero limits at \(z_0\) and bounded inverse valuation. Prime-to-\(p\) target translations preserve ordinary integrality, since their differential maps are units and base change of the canonical level is integral. This includes the possibly long integral sums at the moving primes: the nonarchimedean bound for their coefficients is one, and there is no averaging. If every controlled section had content tending to zero, so would every such expression, contradicting [eq:source-20]. We may therefore select a controlled section with bounded content at each stage. Only a bounded number of kinds of fixed local tests is needed. At every upper \((2,2)\) control retain the two positive binary operators, the invertible center expansion and the full last spherical block. The pivot calculation is unaffected by a determinant twist on the head. Finally Proposition 58 applies to all these bounded test variants simultaneously, proving the constant-term assertion. ◻

Proof. At each depth normalize the good toroidal model in the generic level, after the finite extension needed for its cyclotomic frames. On its canonical ordinary branch the extra level is finite etale on the Cartier-dual character group of the multiplicative subgroup. At a boundary chart this adds a finite etale cover of the abelian extension-data scheme and leaves the center Fourier parameter unchanged. This description holds for each integer \(s_i\); it introduces no division by the degree of that cover.

A section whose constant terms vanish modulo \(p^M\) has positive Fourier–Jacobi indices. At such an index the Jacobi line bundle is ample on the abelian chart. The differential bundle has its integral column-degree filtration with constant graded pieces, so positive cohomology vanishes after tensoring with that ample line bundle. This remains true on each finite etale abelian cover. Cohomology and base change lift the positive coefficients modulo \(p^M\). Neatness removes the finite stabilizers of positive indices, permitting transport of lifts along the free index orbits without dividing by an orbit order. Formal functions then identifies these lifts with the cusp sheaf on the ordinary open. This proves the base-change assertion at each \(s_i\) with zero loss of precision, also over ramified finite coefficient extensions. Extend by zero on the other ordinary branches.

Multiply by a sufficiently high power of Hasse to extend across the complement of the ordinary open. Local lifts of Hasse differ by a multiple of \(p\); sufficiently divisible powers glue modulo \(p^M\) and equal one in multiplicative differential frames. Serre vanishing after a further ample twist lifts the resulting global section to characteristic zero. These exponents may depend on \(i,s_i,M\); the surjectivity statements themselves cost no precision. Choose the resulting shift \(H_i\) arbitrarily large and tending to zero on every weight branch in use. The classical weights are \[\lambda'_i=(m_i+H_i,m_i+H_i,1+H_i;-1-H_i).\]

The integral normalized operators remain uniform in depth. In Serre–Tate coordinates \(q_1,q_2,q_3\), each graph for \(U_j\) adjoins \(j\) roots \(u_a^p=q_a\). Its monomial basis gives \[\mathop{\mathrm{Tr}}\bigl(\mathcal O[u_1,\ldots,u_j]/(u_a^p-q_a)\bigr) \subset p^j\mathcal O.\] The transported multiplicative flags and frames depend on the graph, not on the chosen roots, even when marked modulo \(p^{s_i}\). They may therefore be taken outside this trace. The differential scalings have lower bounds \(\lambda'_3+\lambda'_4\) for \(j=2\) and \(\lambda'_4\) for \(j=3\), exactly as in [23]. Thus \[\widetilde U_2=p^{-(2+\lambda'_3+\lambda'_4)}U_2, \qquad \widetilde U_3=p^{-(3+\lambda'_4)}U_3\] preserve the integral lattice. The graph sum introduces no denominator. The Hasse shift changes these exponents by \(0,-H_i\); its determinant factor changes by exactly the same amounts. Take \(H_i\) divisible enough that all remaining unit frame factors are one modulo the chosen precision. The two comparison values are then one. Prime-to-\(p\) operators preserve the lattice and their comparison values for the same reason.

Use the intersection of the classical cusp space with the ordinary integral section module as its lattice. It is a full bounded lattice and is saturated for divisibility by \(p^M\): its definition as an intersection makes this immediate. The commuting integral operators generate an \(\mathcal O_k\)-algebra \(\mathbb T_i^0\) inside the endomorphisms of that lattice over a finite model extension of \(k\). This is a finite torsion-free \(\mathcal O_k\)-order: the endomorphism lattice is finite over \(\mathcal O_k\), so its submodule generated by the operators is finite. The scalar ring of the order is always \(\mathcal O_k\); only its rank and the field representing the lattice may vary with \(i\). Denote the lifted section by \(\mathfrak s_i\). If \(P\) is any integral polynomial relation among these operators, application to that section gives \[P(\hbox{comparison values})\,\mathfrak s_i \equiv0\pmod {p^{M_i}}.\] Every polynomial has this precision because all its operators are integral. Evaluate a positive coefficient of valuation at most \(C\) from Lemma 60. The scalar \(P(\hbox{comparison values})\) is divisible by \(p^{M_i-C}\). It lies in \(k\), so the divisibility descends from the evaluation field. This is the length-one version of content cancellation, performed at \(z_0\) itself. The initial clearing, \(C\), and the neat descent index are fixed; there is no other loss. Decrease and relabel \(M_i\), also slowing it so that every evaluated finite group relation holds. The scalar evaluation factors through the entire order, giving \(\lambda_i^0\). Its local target selects the residual local summand of that order, on which both \(\widetilde U_j\) are units. ◻

Proof. The chosen element has exactly two eigenvalues one, counted with multiplicity. Both are already supplied by the final unramified Jacquet block, so the complementary Weil positions cannot contain any inertia-trivial constituent. Thus the entire inertia-trivial Weil summand is this binary allocation. An unramified norm twist does not change inertia, so no segment can cross these allocations. The segment classification therefore identifies the allocated final binary representation with its unramified spherical block, whose predicted monodromy is zero. It is necessary to check actual monodromy as well: the Galois construction in the cited proposition is a direct sum of shifted copies of the Galois representations of generic unitary cuspidal factors. Inside one copy, the selected local segments have length one; a longer segment would cross the inertia separation or violate binary sphericity. Two such unramified length-one data in a generic unitary factor have real exponents in an interval of width strictly less than one, so their quotient cannot be \(Q_v\). Thus monodromy cannot link them. It cannot link different copies, which are direct summands, or link to the head, which is inertia-separated. The selected subspace consequently has trivial actual inertia. It has dimension two and is precisely the generalized \(1\)-primary space of the chosen element, proving the identity. The projection can be written explicitly. Suppress \(v\) and write \(P_{\iota_v}(Z)=(Z-1)^2C(Z)\) and \(\Delta_v=C(1)\). The polynomial \[E(Z)=C(Z)\bigl(\Delta_v^{-1} -C'(1)\Delta_v^{-2}(Z-1)\bigr)\] is one modulo \((Z-1)^2\) and zero modulo \(C(Z)\). Consequently \(P_v^{\mathrm{bin}}=E(\rho_\pi(\iota_v))\). This gives an algebraic identity with only powers of \(\Delta_v\) in its denominator, suitable for passage to the limiting corner algebra. The element \(\iota_v\) need not generate inertia.

For the order, apply the geometric-lemma filtration on each joint generalized controlled-operator space. There are at most \(4!\) allocation terms at each controlled place. Taking the required compact Levi invariants and a generalized eigenspace is exact in characteristic zero. At an inertia-separated place the two final unramified inducing positions are forced: no head position can occupy an unramified slot, because its inertia eigenvalue on \(\iota_v\) differs from one. At a Frobenius-separated place the resultants force the allocation. The full spherical binary invariants are one-dimensional, so its two operators are scalars on the surviving grade. At \(p\) the two contacts force the allocation to the \((2,1,1)\) Levi; both final center operators are scalars there at every depth. Permutations inside the head lie in the Levi Weyl group and do not create additional grades. Hence the separated generalized space has a single grade and no operator nilpotents.

On a component where \(\Delta\) has zero eigenvalue, it acts as zero on every allocation grade and lowers this filtration. The bounded number of control places therefore gives a fixed power, for example the product of their bounds \(24\), killing that component. The good spherical algebra is already semisimple; hence the kernel of eigensystem detection vanishes on the separated components and is killed by this power on the others. This is an operator identity in characteristic zero, and the integral order is torsion-free. It therefore holds in \(\mathbb T_i^0\) itself. ◻

Proof. Let \(Q=D_{xy}/(D_{xf}D_{fy}+\mathfrak mD_{xy})\), where \(\mathfrak m=\ker\lambda\). A functional \(Q\to k\) gives an extension of \(y\) by \(x\) with a separate \(f\) block; all discarded products vanish by (45) and the definition of \(Q\). If its Galois class splits, the splitting is preserved by the algebra generated by the group, hence by the corner idempotents, and the functional is zero. We thus obtain an injection into admissible \(H^1(L,k(1))\).

At \(\bar w\), stack the matrices \(\rho_*(g)-Y_{\bar w}(g)\). Their common kernel makes every four-row determinant zero. The vanishing of \(\mathop{\mathrm{Hom}}_{G_{L_{\bar w}}}(y,f)\) at \(z_0\) and \(x\ne y\) supplies three rows with an invertible minor on the competing columns \(f_1,f_2,x\). Matrix units allow either \(f\) coordinate to be used in either of the two output rows. Solve these three equations with \(y\) coordinate one. Each remaining equation follows from its vanishing four-row corner minor. This gives a local \(y\)-eigenline, hence a splitting. At \(w\) apply the transposed argument to the common quotient \(X_w\); the competing minor uses \(\mathop{\mathrm{Hom}}_{G_{L_w}}(f,x)=0\). These are exactly the local invariant conditions retained in Lemma 51. The extension is therefore strict at every place above \(p\).

At a binary place the projection of Lemma 62, or its Frobenius version, is the identity on the \(x,y\) extension and makes inertia trivial there. At a parahoric place its two-by-two minors give \((f(\iota)-1)_{ab}u(\iota)=0\) for every entry, where \(u\) is the cyclotomic extension entry. Some elliptic inertia difference is nonzero. Hence \(u(\iota)=0\) whenever that difference is nonzero; multiplying an element in its kernel by a fixed element outside the kernel and using the cocycle law gives zero there too. Elsewhere hyperspecial unramifiedness applies. In particular the class is unramified at the moving places and descends by the admissible comparison to the fixed field \(L\).

Kummer theory identifies the resulting group with \(p\)-units tensored with \(k\), with zero valuations and logarithms at all \(p\)-places. Zero valuations leave the global units. The CM biquadratic field \(L\) has unit rank one. A nontrivial fundamental unit has nonzero \(p\)-adic logarithm at a split \(p\)-place: a unit of \(\mathbb Q_p\) with zero logarithm is a root of unity, and the field embedding is injective. Thus this one-dimensional space has no element with all local logarithms zero. The extension vanishes, so \(Q=0\). Nakayama for the finite module \(D_{xy}/D_{xf}D_{fy}\) proves the assertion. ◻

Proof. If \(D_{fy}\) vanished at such a field, then Lemma 63 would make \(D_{xy}\) vanish there as well. The last idempotent line would be globally stable. The competing-column minor used at \(\bar w\) is a unit in \(\mathscr T\). In a matrix with this last line stable its upper-right column is zero. The common \(Y_{\bar w}\) kernel, with this competing minor invertible, must therefore be the last line, so its character is \(Y_{\bar w}\).

These finite matrix relations pass to a filter-large set of actual systems, as in [23]. More explicitly, stability uses only the bounded algebra generators; the idempotent, matrix-unit, and competing-minor relations use finitely many further matrices. The algebraically closed residue ultraproduct realizes the henselian choices, so their finite equalities and inequalities hold on actual systems. Thus each such system has a global one-dimensional constituent \(\xi\) restricting to \(Y_{\bar w}\).

The character \(\xi\) is de Rham and hence an algebraic Hecke character. Its conjugate weight sums are parallel, by the relation on the units of the real subfield. Its weight at \(\bar w\) is \(3-h_4\), the isolated lowest weight. In the construction of the Galois package it cannot belong to a factor of Arthur length greater than one: successive copies would also supply an adjacent integer weight, whereas the next weight is separated by \(h_4-h_3\). Thus its factor has length one. The strict unitary generic bounds on the unramified cuspidal roots place its absolute Frobenius size strictly between \(Q\) and \(Q^2\) after the cohomological shift. A character of parallel conjugate weight sum \(j\in\mathbb Z\) has size \(Q^{j/2}\), so \(j=3\).

In particular the weight of \(\xi\) at \(w\) is \(h_4\). Let \(F_3\rho_\pi\) be the rank-three local subrepresentation at the second Hodge–Newton contact. Its weights are \(h_1,h_2,h_3\), so the \(\xi\) line cannot be contained in it: the de Rham functor is exact and a subrepresentation’s weights occur in this multiset. Its map to \(\rho_\pi/F_3\rho_\pi=X_w\) is therefore nonzero, and is an isomorphism of characters. This identifies the entire local character of \(\xi\) at \(w\), including its smooth factor, without a bound on conductor. For every local \(g\) we now have the exact identity \[\xi(g)\xi(g^c) =X_w(g)Y_{\bar w}(g^c)=\epsilon(g)^{-3}.\]

Choose one fixed inertia element \(g\) with \(\epsilon(g)\) nontorsion and include \(g,g^c\) in the finite matrix list. The preceding identity makes \[(e_y\rho_*(g)e_y)(e_y\rho_*(g^c)e_y) -\epsilon(g)^{-3}e_y\] zero in the field evaluation. It is, however, a unit in \(\mathscr T e_y\): its residue under \(\lambda\) is \((\epsilon(g)^{-4}-\epsilon(g)^{-3})e_y\ne0\). This contradiction proves the nonvanishing of \(D_{fy}\). An element annihilating \(D_{fy}\) consequently vanishes in every ambient field, hence is zero by reducedness and faithfulness. The zeroth Fitting ideal of a finite module is contained in its annihilator by the adjugate identity. This proves its asserted vanishing. ◻

Proof. The local invariant tests can be checked before replacing the fixed conditions. At a \(p\)-place, an inertia element with nonzero true exponent gives an equation in \(t\) for any invariant vector. A Frobenius lift gives a second equation with a nonzero exponent vector in the artificial variables. No height-one prime can contain both equations. This applies to the coefficient and its dual, so local \(H^0\) and \(H^2\) vanish. At a moving prime, inertia makes the complex acyclic unless its own variable is zero. On that divisor the Frobenius determinant is nonzero: setting all parameters to zero gives the chosen Tate action without eigenvalue one. Hence the moving local complexes are acyclic on every horizontal height-one test. Global invariants vanish because the untwisted planes are absolutely simple and a scalar twist cannot create a stable line.

The unramified complexes at the remaining places include \(H^0\) and are exact complements under duality. The same holds at the companion’s nonsplit ramified places, where the inertia invariants are zero. The invariant vanishings, Poitou–Tate duality, and Euler characteristic calculation therefore give the asserted amplitude and equal ranks. The localization triangles identify the quotient of a full fixed-place condition by its unramified condition with residue cochains on inertia coinvariants, twisted by \((-1)\) and shifted to degrees \(1,2\). Its determinant is the corresponding Euler factor. Multiplication of determinant lines gives the two displayed identities. In particular \(v_{\mathfrak q}(M_{a,w})\geq0\), and \(M_{a,w}\) vanishes on the field fiber exactly when that one-sided complex jumps.

By [eq:source-11], at this DVR \[U_w=\text{unit}\cdot (b_w^0)^2/M_{0,w}.\] Consequently \(v_{\mathfrak q}(U_w)>0\) implies \(v_{\mathfrak q}(b_w^0)>0\), even when \(D_0\) also vanishes. Proposition 59 then gives a one-sided jump for one of the two representations. Thus \(\mathfrak q\) divides one of \(L_{0,\mathfrak p},L_{0,\bar{\mathfrak p}}, L_{1,\mathfrak p},L_{1,\bar{\mathfrak p}}\). Each of these determinants has nonzero reduction modulo \((p,t)\) by [eq:source-1]. A factor of such a determinant also has nonzero reduction modulo \((p,t)\), since \(R/(p,t)\) is a domain. Finally \(p\nmid B_w\) by [eq:source-14]; integrality of \(U_w\) and \(B_w=L_wU_w\) imply \(p\nmid U_w\). ◻

Proof. The preceding lemma and factoriality show that \(U_w\) has nonzero reduction modulo \((p,t)\): it is a unit times a finite product of factors with that property. The same is true for \(U_{\bar w}\). Proposition 46 says that their specializations at every sufficiently high finite \(p\)-power character of \(t\) are associates after inverting \(p\) and extending the constants.

Both quotients are bounded integral series, so the regularity just verified is precisely the hypothesis of Lemma 18. Apply it first with \((L,B)=(U_w,U_{\bar w})\) and then with the two interchanged. The characterwise association supplies divisibility, with all multiplicities, in each application. The constant extension is faithfully flat; equivalently the integral Weierstrass remainder in that lemma vanishes after extension and hence before it. We obtain \(U_w\mid U_{\bar w}\) and \(U_{\bar w}\mid U_w\) in \(R\). Mutual divisibility in this domain proves [eq:source-17]. ◻

Proof. For the first assertion introduce \([v]\in\mathbf P^{s-1}\) and impose \(v^{\mathsf t}A=0\) and \(v^{\mathsf t}B=0\). For fixed \([v]\) these are \((s-1)+2\) independent linear conditions: after a change of basis they are the nontrivial entries of the first row of \(A\) and the two entries of the first row of \(B\). If \(N=\dim W_s\), the incidence variety has dimension \(N-2\). Projection from its projective factor is proper and its image is exactly the rank-defect locus, which proves the assertion.

For a positive even size the determinant of an alternating matrix is a nonzero nonconstant polynomial: it vanishes at zero and is nonzero on a block diagonal matrix with blocks \(\left(\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}\right)\). Its zero locus is therefore a hypersurface. The two loci in the second assertion involve disjoint coordinate sets; their product has codimension two. ◻

Proof. In the first case the rank-defect locus on the slice is contained in the locus of Lemma 67. The full tangent rank-defect locus is a homogeneous closed cone. A codimension-one component, after algebraic closure of \(k\), would have a nonconstant homogeneous defining polynomial \(h\). Its restriction to the slice is either zero, in which case the entire slice lies in the rank-defect locus, or is homogeneous of positive degree, in which case a hypersurface of the slice lies there. Both conclusions contradict the codimension-two bound on the slice. The tangent rank-defect locus thus has codimension at least two.

Each nonzero maximal minor of \(D_1\) is the degree-\(s\) initial form of the corresponding minor of \(D\). Hence \[I_s(D_1)\subseteq\operatorname{in}_{\mathfrak m} I_s(D).\] The associated graded ring of \(A\) is a polynomial ring over \(k\), and passage to the initial ideal preserves the dimension of the quotient. The asserted height bound follows.

For the simultaneous assertion use the closed cone defined by the sum of the two tangent minor ideals. Its intersection with the slice is contained in the simultaneous alternating-defect locus, which has codimension two by the second part of Lemma 67. The same homogeneous restriction argument applies. Initial forms of the minors of each size lie in the initial ideal of the sum, so the same dimension argument proves the simultaneous assertion. ◻

Proof. There are finitely many old tests, the tuple satisfies [eq:source-1](b), and at least two independent old inertia variables remain. These are the hypotheses of Lemma 29. Apply that lemma to both representations and all old tests. It makes each new prime Frobenius-acyclic at every test, retains a nonzero old exponent at that prime, and preserves the Tate, weight, radical, and fixed theta prescriptions. Its construction also shows that at least one old inertia scalar has infinite order at each test; we use that scalar in the evaluation argument below.

The new logarithms are zero at the old support, including the old moving primes. Unramified inflation and the marked comparisons of Lemma 10 identify the specialized complexes with the old ones. The same comparisons hold after localization at \((\mathfrak q,\mathbf y)\) and at all its generizations. All local conditions remain exact complements; in particular the unramified fixed-place comparisons retain their cup nullhomotopies. Their perfect complexes have the amplitudes used below.

For one representation, the conjugate self-duality and Lemma 25 make a minimal one-sided differential alternating. Its generic fiber is acyclic by [eq:source-1]. It follows that every one-sided jump has even dimension. If the both-strict degree-one dimension on a field fiber is \(s\), duality and Euler characteristic make the both-full dimension \(s+2\). The two one-sided spaces embed in the both-full space and have intersection the both-strict space. In particular, if \(s=0\) they cannot both be nonzero: each would have dimension at least two inside a two-dimensional space, and their intersection would then be nonzero. Thus the locus in assertion (1) is contained in the both-strict jump locus.

At the old field \(k_{\mathfrak q}\) let \(x_1,\ldots,x_s\) be a basis of the both-strict space and let \(d_1,\ldots,d_{s+2}\) be a basis of the conjugate both-full space, with the first \(s\) vectors corresponding to the conjugates of the \(x_i\). Scalar localization in degree two is injective by the scalar models of Section 4. We can therefore form the cup primitives used in Lemma 26. For a new direction \(\lambda\), a first-order correction \(x^{(1)}\) to \(x\) must satisfy \(dx^{(1)}=-\lambda\cup x\). Pairing with \(d\) and applying the transferred invariant sum gives the Frobenius test of that lemma. At old places \(\lambda=0\). At another new prime \(\lambda\) is unramified; after the local acyclic trivialization both factors are unramified, and their invariant cup is zero. Thus these tests add linearly when several new directions are used.

For completeness, their span has precisely the form needed in Lemma 68. On the termwise joint kernel of the Tate and old character actions, the evaluations of the \(x_i,d_j\) are additive and detect their classes by Lemma 12. The absolutely simple coefficient planes with opposite scalar twists are nonisomorphic, as detected on the infinite-order old inertia scalar. Joint evaluations therefore span the full tuples. Multiplying a Frobenius address by a commutator \([v,v']\) in this kernel preserves its Tate and old character values and all radical prescriptions, since its action on the radical extension is abelian. The cup-primitive value changes by \[-e(x(v),d(v'))+e(x(v'),d(v)),\] with the conjugate expression subtracted at the other prime. Elementary evaluation pairs consequently span all skew entries in the first \(s\) columns and all entries in the last two columns. This is the space \([A\mid B]\) of Lemma 67. Finitely many addresses suffice; finite-precision Chebotarev realizes them with the old markings retained, as proved in Lemma 27.

Put \(A=A_{\mathfrak q}=S_{(\mathfrak q,\mathbf y)}\); this is regular local with residue field \(k_{\mathfrak q}\) of characteristic zero. Cancel the invertible differential blocks there. The both-strict model becomes \(D:A^s\to A^{s+2}\) with zero residue. A vector in the kernel of its first-order differential is a liftable class and hence is annihilated by the cup tests. Choosing a linear section of their span gives the injected tangent slice in Lemma 68. The identities are identities of finite matrices, so this kernel inclusion persists over every extension of \(k_{\mathfrak q}\). That lemma proves codimension at least two. If \(s=0\), the minimal degree-one term is zero and there is no jump in this local spectrum. This proves assertion (1).

For assertion (2) use the two indicated one-sided spaces, of dimensions \(s_0,s_1\). If either is zero there is again no joint jump in the local spectrum. Otherwise both dimensions are even. The four coefficient planes, consisting of each representation with each of the two opposite scalar twists, are pairwise nonisomorphic. Opposite twists are distinguished by old inertia. For the remaining comparison, \(V_0|_{G_L}\) is absolutely simple. An isomorphism \(V_0\simeq V_0\otimes\chi_{L/K}\) would restrict to a scalar on \(G_L\) by Schur’s lemma, and its intertwining equation on the other coset would say that a nonzero scalar equals its negative. Such an isomorphism is impossible. This also applies to the twisted planes.

Joint evaluation spanning and the same commutator calculation therefore provide an independent slice \(\operatorname{Alt}_{s_0}\oplus\operatorname{Alt}_{s_1}\) for the two one-sided differentials. Apply the simultaneous part of Lemma 68. It gives codimension at least two for the joint jump locus, proving assertion (2).

Finally take the union of the finite direction lists for every old test and every allocation of sides. Every new prime was made acyclic at every test. Hence adjoining another list changes none of the old field spaces, and the vanishing of the cross-prime invariants proved above preserves each list’s tangent slice. This proves the simultaneous assertion. The nonzero old exponent also preserves [eq:source-1] after adjoining a prime: on \(\mathbf y=0\) its singular determinant is nonzero modulo \((p,t)\). The remaining tuple prescriptions, including the fixed theta tests, were retained in the finite tables. The exact norm and localization identities of Lemma 43 therefore apply to the enlarged tuple. ◻

Proof. Choose the enlargement in Proposition 69 and put tildes on its quotients. By Corollary 49, \[\widetilde U_w(t,\mathbf u,0) =\varepsilon_w(t,\mathbf u)U_w(t,\mathbf u), \qquad \varepsilon_w\in R^\times.\] Indeed both numerator and denominator acquire the same square of each factor in [eq:source-13]. Those factors are nonzero; cancellation in the fraction field proves the identity, and the marked integral determinant triangles make \(\varepsilon_w\) an integral unit.

Suppose an irreducible \(F\) divided one enlarged quotient. It is horizontal by Lemma 65. Its specialization \(F_0=F(t,\mathbf u,0)\) is nonzero, since it divides the nonzero specialized quotient. It is a nonunit: a unit specialization would give a unit constant term of \(F\) in the local ring \(S\). Choose an irreducible factor \(f\) of \(F_0\) in \(R\). The displayed identity makes \(f\) an old quotient divisor; in particular \(f\ne p\). Moreover \(F\in(f,\mathbf y)\), so \((F)\) survives as a height-one prime in \(S_{(f,\mathbf y)}\).

Equation [eq:source-17] for the enlarged tuple shows that \(F\) divides both enlarged quotients. The calculation in Lemma 65 makes both undepleted analytic series vanish on this prime. Applying the two conjugate theta implications, either one representation jumps on both sides, or a one-sided complex for each representation jumps. Each possibility is excluded at a height-one prime of \(S_{(f,\mathbf y)}\) by Proposition 69. This contradiction proves that the enlarged quotients are units. Their specialization, and the displayed identity, imply that the original quotients are units too. If there were no old divisors, the original quotients were already units and the argument requires no enlargement.

Removing all artificial variables uses the same integral specialization identity. The restored factors [eq:source-13] are nonzero even if they are not units at the center, and occur with the same multiplicity on both sides. Their cancellation leaves a unit in \(\mathbb Z_p[[t]]\). No loss of \(p\)-adic precision occurs in this passage. ◻

Proof. After removal of the artificial variables, the marked arithmetic models compute the fixed-support complex over \(K\) itself by Lemma 10. Central specialization is derived specialization of its finite free model. In particular it retains the finite local Kummer modules and their invariant torsion. The strict/full complex is rationally acyclic: the global compact Selmer space is the line generated by \(P_K\), and its localization at each \(p\)-place is nonzero, since the kernel of the elliptic logarithm on local points is torsion. Exact complementary duality, or Lemma 15, then gives the acyclicity. The inverse-determinant convention consequently gives the first line of [eq:source-22].

Begin with the all-Kummer complex. By Lemma 17, its class-functional tensor has valuation \[d(C_{\rm Kum},(P_K,P_K^\vee))=2n_P+2\tau_g-s_K.\] Here the torsion lengths in degrees one, two, three are \(\tau_g,s_K,\tau_g\), and both occurrences of the point contribute its index \(n_P\). Thus this is a calculation on the full lattice.

First compare the paired \(p\)-adic Kummer lines with strict at \(w\) and full at \(\bar w\), keeping the Kummer conditions at every non-\(p\) place. Denote this intermediate complex by \(C_w^{\rm Kum-away}\); it differs from \(C_w^{s=1}(0)\) only at the non-\(p\) support places. The unchanged conditions are thus still exact complements, as required by Lemma 15. Choose a free local Kummer generator \(e\) with logarithm generating \(p^l\mathbb Z_p\). Localization of \(P_K\) has free coordinate of valuation \(v_p(\log_{\omega_E}P_K)-l\). The local Kummer complex also contains degree-one torsion of length \(\tau_p\); its volume in this rational basis is therefore \(\tau_p\). The two paired localization triangles in (15) subtract twice the sum of these two quantities, giving \[d(C_w^{\rm Kum-away},1)=2n_P+2\tau_g-s_K -2\bigl(v_p(\log_{\omega_E}P_K)-l+\tau_p\bigr).\]

Now relax the conditions at the non-\(p\) support places. At such a place compact \(H^1\) is already the completed point module, while the quotient of full cochains by the Kummer condition has degree-two torsion of length \(\tau_q\), by local duality. Its inverse-determinant valuation is \(-\tau_q\). Each rational \(q\in S^\circ\) gives two places of \(K\), hence a total change \(-2\sum_{q\in S^\circ}\tau_q\). These last comparisons use determinant multiplication in localization triangles and do not require full local conditions to be self-complementary. Combining the two changes proves [eq:source-22]. In particular the torsion term \(\tau_p\) cannot be discarded by replacing the point module by its free quotient.

By Theorem 70 the analytic series and strict determinant on the true line differ by an integral unit. Their values at \(t=0\) are therefore of the same valuation. Equation [eq:source-12] gives \[v_p(L_w^{s=1}(0)) =-2v_p(c_E)+2v_p(\log_{\omega_E}P_K) +2v_p(P_p(1))+2\sum_{q\in S^\circ}v_p(P_q(1)).\] Every argument of \(v_p\) in this display is nonzero. In particular \(P_q(1)\) is a nonzero inverse local Euler factor, and the point logarithm is nonzero as observed above.

Set \(h=v_p(\log_{\omega_E}P_K)\), \(e_q=v_p(P_q(1))\), and \(c=v_p(c_E)\). Equating this last expression with the negative of [eq:source-22], then cancelling \(2h\), gives \[2n_P+2\tau_g =s_K+2c+2(\tau_p-l-e_p) +2\sum_{q\in S^\circ}(\tau_q-e_q).\] The Haar identities of Lemma 16 identify the parentheses with \(v_p(c_p(E))\) and \(v_p(c_q(E))\), respectively. All bad primes are in the fixed support, and every omitted good prime has component number one. This proves [eq:source-23] with the sum over all finite primes. ◻

Proof. The full group index in Proposition 9 has valuation \[v_p([E(K):\mathbb ZP_K])=n_P+\tau_g.\] Substitute [eq:source-23] into the resulting valuation identity (11). Its right-hand side is zero, proving [eq:source-24]. That arithmetic-volume identity already uses the total real periods, full-lattice regulator, and height diagonal \(H\). The field-degree factor over \(K\) is included there; none is appended here. ◻

Proof. Let \(M\) be the CM field. Theorem 1.1 and Corollary 1 of Burungale–Flach [5] give exact BSD for an elliptic curve \(A/F\) with CM by the maximal order of \(M\), provided \(F(A_{\rm tors})/M\) is abelian and the associated Hecke \(L\)-value at one is nonzero. We verify these hypotheses after an isogeny over \(M\).

All CM endomorphisms of \(E_M\) are defined over \(M\). If the endomorphism order is not maximal, at each prime \(\ell\) replace \(T_\ell E\) by its span under \(\mathcal O_M\otimes\mathbb Z_\ell\) in \(V_\ell E\). These are Galois-stable lattices, equal to the original lattices except at the finitely many primes dividing the conductor of the order. Their finite quotients define a finite Galois-stable subgroup of \(E\), and quotienting gives an elliptic curve \(A/M\) isogenous to \(E_M\) with maximal-order CM. Its Tate modules have rank one over the corresponding local CM orders. The Galois actions commute with those orders, so their images, and hence the full torsion extension over \(M\), are abelian.

Choose a nonzero trace-zero CM endomorphism after multiplying by an integer to make it integral. Galois conjugation outside \(M\) sends it to its negative. The twisting descent criterion therefore gives a \(\mathbb Q\)-isogeny from the quadratic twist \(E^M\) to \(E\). In particular \[L(E/M,s)=L(E/\mathbb Q,s)L(E^M/\mathbb Q,s)=L(E/\mathbb Q,s)^2.\] This has nonzero value at one. Isogeny does not change that \(L\)-function, and its factorization into the Hecke character and conjugate Hecke character \(L\)-functions shows that the nonvanishing hypothesis of Burungale–Flach holds for \(A\). Their theorem gives exact BSD over \(M\), and isogeny invariance transfers it to \(E_M\).

Finally restriction of scalars is compatible with the BSD arithmetic volume, and \[\operatorname{Res}_{M/\mathbb Q}E_M\ \sim_{\mathbb Q}\ E\times E^M \ \sim_{\mathbb Q}\ E^2.\] Thus the normalized BSD discrepancy for \(E\) has square one. It is positive: analytic rank zero gives rank zero and finite \(\mathop{\mathrm{Sha}}\), so the two-primary result supplies positivity and rationality. It is therefore one. This is the descent argument of Burungale–Flach, Corollary 2, using the restriction-of-scalars and isogeny compatibilities of [6, 19]; the preceding lattice argument supplies the additional nonmaximal-order case. ◻

Proof. For a non-CM curve, the single-curve inequality [eq:source-8] gives \(X(E)\geq0\). Its twist \(E^D\) is also non-CM and has the complementary analytic rank, so the same inequality gives \(X(E^D)\geq0\). Their sum is zero by [eq:source-24]; hence \(X(E)=0\).

For a CM curve of analytic rank zero use Lemma 73. For a CM curve of analytic rank one, the companion \(E^D\) is CM of analytic rank zero. That lemma and [eq:source-24] again give \(X(E)=0\). The construction was made for an arbitrary odd \(p\), including \(p=3\), so the displayed equality holds at every odd prime. ◻

Proof of Theorem 1. Apply the low-corank converse (Theorem 2) at the prime \(q\) in the hypothesis. It gives \(a(E)=r(E)=s_q(E)\in\{0,1\}\) and finiteness of the whole \(\mathop{\mathrm{Sha}}(E/\mathbb Q)\). The Kummer exact sequence then gives \(s_p(E)=r(E)\) for every prime \(p\), in particular \(s_2(E)\leq1\). The two-primary formula (Theorem 3) gives positivity and rationality of \(Q_E\) and its correct \(2\)-adic valuation. Proposition 74 supplies its correct valuation at every odd prime, independently of \(q\). Consequently the positive rational number \(Q_E/\#\mathop{\mathrm{Sha}}(E/\mathbb Q)\) has zero valuation at every rational prime. Its prime factorization is empty, so it equals one. Substituting the definition of \(Q_E\) is exactly the asserted real formula, with the total real period and full Mordell–Weil regulator. ◻