Proof. Negative odd fundamental discriminants in any admissible fixed congruence class have positive density. The splitting conditions at \(2Np\) are finitely many such conditions. We can exclude the finitely many unwanted quadratic fields and then apply Theorem 3 to choose \(D\) for which \(A_0^D\) has analytic rank at most one. Its rank is then minimal for its sign.

Keep \(D\) fixed. The conditions on \(D'\) of being coprime to \(D\), splitting \(2Np\), and satisfying the displayed Jacobi-symbol condition again contain admissible congruence classes of positive density. For example, impose \(D'\equiv-1\pmod{|D|}\) and \(D'\equiv1\) at a sufficiently divisible modulus supported on \(2Np\). The moduli are coprime, and \(|D'|\equiv1\pmod{|D|}\) gives the required Jacobi symbol. Exclude the finitely many remaining forbidden fields. Apply the density theorem to both \(A_0\) and \(A_0^D\). The union of their two exceptional sets has density zero, so some \(D'\) in the prescribed classes makes both \(A_0^{D'}\) and \((A_0^D)^{D'}=A_0^{DD'}\) have analytic rank at most one.

This use of density is unchanged if parameters are described by fundamental discriminants: for squarefree \(d\) the corresponding discriminant is \(d\) or \(4d\), so a zero-density exceptional set remains zero-density. In the odd classes used here the parameter itself is the discriminant.

The three quadratic subfields have discriminants \(D,D',DD'\). Consequently the biquadratic discriminant formula gives \[|\operatorname{disc}(L)|=D^2(D')^2 =\operatorname{disc}(F)^2.\] The discriminant formula for \(L/F\) therefore has relative discriminant of norm one, proving finite unramifiedness. The two imaginary subfields give the CM involution over the real field \(F\). The exclusions made above remove any extra roots of unity and the possible CM field of \(A_0\). ◻

Proof. For a fundamental discriminant prime to \(N\), the quadratic-twist root-number formula is \(w(A_0^d)=w(A_0)\chi_d(-N)\). The splitting conditions and the signs of \(D,D'\) therefore give \[w(A_0^D)=w(A_0^{D'})=-(-1)^r, \qquad w(A_0^{DD'})=(-1)^r.\] By Lemma 4, their analytic ranks are \(1-r,1-r,r\), respectively. Gross–Zagier–Kolyvagin gives the same Mordell–Weil and \(p\)-power Selmer ranks and finite Sha for these three twists. Restriction and corestriction for the quadratic extension \(K/\mathbb Q\) give \[H^1_f(K,V) =H^1_f(\mathbb Q,V)\oplus H^1_f(\mathbb Q,V^D),\] and the analogous decomposition for \(V^{DD'}\). Thus the two dimensions are \(r+(1-r)=1\) and \(r+(1-r)=1\).

For the second representation both summands come from twists with known analytic rank at most one. Its Selmer line is therefore spanned by a non-torsion global point on the corresponding twist over \(K\). Such a point has nonzero image under the local logarithm at either place above \(p\). Indeed, on a fixed \(p\)-adic completion, a finite-index open subgroup is a formal group on which the logarithm is injective after shrinking; a point with zero logarithm consequently has a torsion multiple in that subgroup, and is torsion. This argument uses no reduction hypothesis.

Every prime dividing \(N\) splits in \(K\), so the classical Heegner hypothesis holds. The Gross–Zagier formula identifies the nonzero height of \(y\) with simple vanishing of \[L(A_0/K,s)=L(A_0,s)L(A_0^D,s)\] at \(s=1\). Since the second factor has order \(1-r\), a non-torsion \(y\) implies \(a(A_0)=r\). The analytic rank-zero/one theorems then give \(\mathop{\mathrm{rank}}A_0(\mathbb Q)=r\) and finiteness of the whole \(\operatorname{Sha}(A_0/\mathbb Q)\). ◻

Proof. Let \(F_w=H^1_f(K_w,W)\) and let \(Q_w=H^1(K_w,W)/F_w\). Poitou–Tate duality identifies the image of \[H^1_{\mathrm{rel}}(K,W)\longrightarrow Q_v\oplus Q_{v^c}\] with the annihilator of the localization of the compact Selmer line in \(F_v\oplus F_{v^c}\). If that localization is nonzero, both of its coordinates are nonzero: the one-dimensional global space is stable under \(c\), which interchanges the two places. Its annihilator is therefore a line whose two coordinates are both nonzero for every nonzero vector. A class satisfying a zero condition at one place has zero singular image, hence belongs to the compact Selmer line; the zero condition then kills that line as well.

Suppose the compact Selmer line is everywhere locally zero. The same duality sequence makes the singular image all of \(Q_v\oplus Q_{v^c}\). The kernel of localization is precisely the compact Selmer line. Consequently the relaxed local image has dimension two and maps isomorphically to the singular plane. In each local hyperbolic plane choose the isotropic complement \(E_w\) of \(F_w\), choosing the two complements compatibly with \(c\). The relaxed local image is the graph of a map \[E_v\oplus E_{v^c}\longrightarrow F_v\oplus F_{v^c}.\] Global reciprocity makes the graph isotropic. In mutually dual bases its matrix is therefore skew-symmetric, hence of the form \(\left(\begin{smallmatrix}0&u\\-u&0\end{smallmatrix}\right)\). Conjugation interchanges both pairs of basis vectors and preserves the local pairings. Invariance of the graph therefore changes \(u\) to \(-u\), so \(u=0\). Lifting the first coordinate line gives \(h\); conjugating gives \(h^c\) and proves [eq:base-green-bases].

The auxiliary summand has a nontorsion global point and nonzero localization, by 5. If \(r=0\), the unique Selmer line of \(V/K\) likewise comes from a nontorsion point on the quadratic twist \(A_0^D/\mathbb Q\), so it cannot be strict. When \(r=1\), the twist by \(D\) has zero Selmer space and the line comes from \(H^1_f(\mathbb Q,V)\), giving \(a^c=a\). ◻

Proof. Choose representatives \(\mathfrak a_j\) for generators of the ideal class group, prime to \(S\). For their orders, their multiplication relations, and the expressions of the primes in \(S\) in these generators, choose principal generators. Adjoin generators of the finite-rank \(S\)-unit group and their conjugates to this finite list, denoted \(\alpha_1,\ldots,\alpha_d\). Fixed class-group orders and roots of unity will cost only a fixed index.

For a split prime \(q\), the kernel of the ring class group map to the ordinary ideal class group is the quotient of \((\mathcal O_K/q\mathcal O_K)^\times\) by diagonal residue units and global units. Its \(p\)-part is cyclic, of order \(p^{v_p(q-1)}\) up to a fixed factor. Require the \(\alpha_j\) to have zero residue logarithm modulo \(p^{n_i}\) at the pair over \(q_i\). This is ensured by splitting in their \(p^{n_i}\)-radical extensions, together with the corresponding roots of unity. The ring class presentation now allows us to set the logarithms of all \(\mathfrak a_j\) equal to zero and to take the surjective logarithm on the residue quotient. Each relation is respected, and the expressions chosen for the primes of \(S\) give \(\ell_i|_{G_{K_s}}=0\). Conjugation inverts the residue quotient and the ring class action, giving \(\ell_i^c=-\ell_i\).

We verify that imposing the full radical condition leaves room for \(\gamma\). Over the cyclotomic tower the Kummer Galois group is a closed subgroup of \(\mathbb Z_p(1)^d\). The joint image of \(V\) and \(\mathcal C\) is a \(p\)-adic Lie group. A common Lie quotient with this translation group is abelian, so it kills the derived Lie algebra of the image of \(V\). On the remaining torus Lie algebra, conjugation by an open cyclotomic subgroup is trivial; on the Kummer translations it is multiplication by the cyclotomic character. Any equivariant common quotient is consequently zero. This argument can be made after a fixed finite extension, which does not change Lie dimensions. Thus killing the radicals removes no Lie dimension from the joint image over the cyclotomic tower.

For a non-CM elliptic curve the determinant-one image has Lie algebra \(\mathfrak{sl}_2\), by the open image theorem (Serre 1972); for a CM curve its identity component contains the norm-one torus, by the CM description of the Tate-module character (Serre 1968). In either case the locus \(\det(V(g)-1)\ne0\) is nonempty and open in every sufficiently small identity neighborhood. The infinity type of \(\mathcal C\) makes its restriction over the cyclotomic tower nonconstant; its torsion locus is a proper closed subset in a sufficiently small identity neighborhood. The two complements intersect. We may also impose \(g\in G_L\) and any fixed open-kernel conditions. This gives \(\gamma\) fixing the full radicals and the cyclotomic tower with the required properties.

It remains to control \(v_p(q_i-1)\), rather than merely make it large. The cyclotomic–radical image is a closed subgroup of \[\mathbb Z_p(1)^d\rtimes\mathbb Z_p^\times\] with open cyclotomic projection. Choose an element in a pro-\(p\) subgroup whose cyclotomic projection is nontorsion. Its \(p^m\)-th powers tend to the identity; their translation coordinates are divisible by \(p^{m-C_0}\), and the valuation of their cyclotomic coordinate minus one is \(m+C_1\), for fixed \(C_0,C_1\). Multiplying \(\gamma\) by such a power with \(m=n_i+C_2\) therefore leaves all radicals trivial modulo the requested precision and makes the cyclotomic valuation exactly \(n_i+C\), for a fixed \(C\). Chebotarev applied to a growing sequence of finite quotients realizes these conditions and increasingly accurate approximations on all fixed data. Choosing a place above each prime selects the indicated Frobenius representative. The primes can be taken outside any fixed finite set.

Finally the \(p\)-part of the order of the full ring class group is \(p^{v_p(q_i-1)+O(1)}\). The kernel of its map to \(\Delta_i\) therefore has bounded \(p\)-part. The Frobenius of every \(s\in S\) belongs to this kernel. At \(p\) the extension is unramified, so the \(p\)-part of the local degree, which is the \(p\)-part of the Frobenius order, is bounded. This proves the proposition. ◻

Proof. Choose a stable lattice in \(V\). A bounded finite-stage cocycle modulo \(p^{A_i}\) has trivial wild inertia at \(q_i\). On tame \(p\)-inertia the Frobenius relation implies \[(V(\mathop{\mathrm{Fr}}_i)-q_i)z_i(\tau)=0\pmod {p^{A_i-C}}.\] The inverses of \(V(\mathop{\mathrm{Fr}}_i)-q_i\) have a fixed denominator: these matrices converge to the invertible matrix \(V(\gamma)-1\). The same is true at the conjugate prime and over \(L\). After one fixed loss of precision, the cocycle kills the normal subgroup generated by moving inertia, hence factors through \(G_{K,S}\). For each fixed finite coefficient module, there are only finitely many continuous cocycles on \(G_{K,S}\): cohomological finiteness follows from the finiteness of extensions with fixed ramification and bounded degree, and the group of coboundaries is finite. Taking the compatible ultralimits of these finite sets yields an actual continuous integral cocycle; then undo the fixed scaling. For cyclotomic coefficients the proof is the same once moving inertia has been required to vanish. Equality of classes is compatible with this procedure because bounded zero-cochains have compact ultralimits. It also shows that a representative of a given ordinary class can be replaced by a chosen continuous representative, up to a bounded coboundary and uniformly vanishing errors.

Exactness of [eq:coefficient-sequence] is coefficientwise: lift the finitely many coordinates continuously, and divide a zero constant coefficient by \(t\) by shifting coordinates. These operations do not enlarge a uniform denominator except for the fixed truncation loss in [eq:binomial-loss]. The character is exactly trivial at the fixed places in \(S\), so there \[H^1(K_s,M_b^{\eta})=H^1(K_s,V\otimes k)\otimes_k R_b, \qquad H^1_f(K_s,M_b^{\eta})=H^1_f(K_s,V\otimes k)\otimes_k R_b.\] These are fixed linear subspaces. A strictly zero local class can be made a zero local cocycle with bounded corrections: since \(H^0(K_s,V)=0\), finitely many elements \(g_1,\ldots,g_u\) make \[V\longrightarrow V^u,\qquad A\longmapsto((V(g_j)-1)A)_j\] injective. A fixed left inverse bounds the denominator of every such correction. Therefore division preserves the strict and Kummer conditions, both in rational cohomology and in their finite-precision representatives. ◻

Proof. We give the finite-precision argument, so that no continuity assertion about the varying Galois groups is implicit. For \(\eta=1\), write a first-order lift as \(x+t x_i'\) satisfying [eq:linear-lift]. Indeed, 9 permits subtraction of one bounded global coboundary and replacement of its constant coefficient by the chosen fixed cocycle \(x\). Equality on all sequences makes the error uniformly small. Its linear coefficient remains bounded. The cochain \[ x_i'\cup d-\ell_i\cup u \tag{13}\] is consequently a two-cocycle modulo the reduced precision. Its local invariants at \(S\) vanish to increasing precision. At \(p\) this follows from opposite strict conditions and \(\ell_i=0\); away from \(p\) it follows from the vanishing of rational \(H^1\) and the bounded exponents of the fixed integral local cohomology groups. At places outside \(S q_i\) it is unramified and has zero invariant.

At \(\mathfrak q_i\), the fixed cocycles \(x,d\) are unramified and are coboundaries, say \(x=\delta A_i\), \(d=\delta B_i\), where \[A_i=(V(\mathop{\mathrm{Fr}}_i)-1)^{-1}x(\mathop{\mathrm{Fr}}_i),\qquad B_i=(V(\mathop{\mathrm{Fr}}_i)-1)^{-1}d(\mathop{\mathrm{Fr}}_i).\] These vectors have bounded denominators, because the inverse matrices converge to \((V(\gamma)-1)^{-1}\). By [eq:linear-lift], \(x_i'-\ell_i\cup A_i\) is a cocycle. Its cup product with \(d\) is a coboundary. The invariant of [eq:global-two-cocycle] therefore equals minus the invariant of \[ \ell_i\cup(u-A_i\cup d). \tag{14}\] The second factor is an unramified cyclotomic one-cocycle locally, since its differential is \(x\cup d-(\delta A_i)\cup d=0\). The tame local pairing evaluates this unramified factor at Frobenius and pairs it with the inertia logarithm. With compatible local invariant conventions, the multiplier at stage \(i\) is a unit \(\nu_i\) modulo the working precision, because the logarithm on tame inertia is surjective. At the conjugate prime the same calculation uses \(x^c,d^c,u^c\) and \(\ell_i^c=-\ell_i\); functoriality of the local invariant gives the same multiplier \(\nu_i\). Choose unit lifts to \(\mathbb Z_p\) and let \(\nu\in\mathbb Z_p^\times\) be their compact ultralimit. The global sum of invariants then gives \(\nu H_\gamma(x,d)=0\), hence [eq:obstruction-vanishing]. This calculation never divides by \(q_i-1\). We use arithmetic Frobenius in this calculation; replacing it throughout by geometric Frobenius changes the conventions compatibly and not the zero condition. For \(\eta=-1\), replace \(\ell_i\) by \(-\ell_i\).

Here is the precise bound on all reductions of precision. Let \(c_{\mathrm{fix}}\) bound the fixed cocycles and their finitely many primitives, and let \(c_{\mathrm{lift}}\) bound the particular admissible lift. Let \(c_\gamma\) clear both inverses \(V(\mathop{\mathrm{Fr}}_i)-1\) and \(V(\mathop{\mathrm{Fr}}_i)-q_i\) eventually. Let \(c_S\) bound the fixed left inverses detecting local \(H^0=0\) and the exponents of integral \(H^1,H^2\) at the fixed places away from \(p\). Finally use the binomial constant \(c_b\) of [eq:binomial-loss]. All displayed corrections use finitely many sums and products of these objects. A constant \(C=C(c_{\mathrm{fix}},c_{\mathrm{lift}},c_\gamma,c_S,c_b)\) therefore bounds every loss. If the original cochain identities hold to precision \(A_i\) and their replacement accuracies are \(B_i\), make the calculation to any precision \[ N_i\le\min\{n_i,A_i,B_i\}-C,\qquad N_i\longrightarrow\infty. \tag{15}\] The tame pairing above is defined at this precision since \(N_i\le n_i\). The constants may depend on a fixed lift and on \(b\), but never on \(i,q_i,n_i\). Only finitely many fixed primitives were chosen; no bounded contracting homotopy on all cochains is being asserted.

For independence of \(u\), the difference of two primitives is in \[H^1(G_{K,S},\mathbb Q_p(1)) =\mathcal O_{K,S}^{\times}\otimes_{\mathbb Z}\mathbb Q_p.\] The equality is the Kummer sequence and finiteness of the \(S\)-class group. The radical list in 7 spans this space, including its conjugate, up to a fixed index. Every such Kummer class evaluates to zero at \(\gamma\); a coboundary also evaluates to zero because \(\epsilon(\gamma)=1\). Changing \(u\) therefore changes neither term of [eq:obstruction-pairing]. Bilinearity follows from the fixed bilinear choices of primitives. ◻

Proof. First \(H_\gamma(x,d)=H_\gamma(d,x)\) whenever the pairs under consideration have locally zero cup product. To check the sign explicitly, put \(q(g)=e(x(g),d(g))\). The cocycle identities give \[\delta q=d\cup x-x\cup d,\] so \(u+q\) is a primitive for the switched cup product. At a place where \(x=\delta A\) and \(d=\delta B\), the difference between the two corrected evaluations is \((\epsilon(\gamma)-1)e(A,B)=0\). This proves the symmetry for the expression in [eq:corrected-primitive], and hence for \(H\). Its definition also gives \(H_\gamma(x^c,d^c)=-H_\gamma(x,d)\). Since \(a^c=a\) and \(p>2\), these identities give both diagonal zeros in [eq:obstruction-matrix] and the opposite signs of its off-diagonal entries.

It remains to make \(\kappa=H_\gamma(a,h^c)\) nonzero. Set \(d^-=h^c-h\). It is anti-invariant, and its cup product with \(a\) is everywhere locally zero. Average a primitive to be anti-invariant as well. Then \[H_\gamma(a,d^-)=2z_\gamma(a,d^-;u),\qquad H_\gamma(a,d^-)=2H_\gamma(a,h^c).\] It is enough to make \(z_\gamma(a,d^-;u)\) nonzero.

Let \(G=G_{K,S}\) and \(N_V=\ker(G\to\mathop{\mathrm{GL}}(V))\). The classes \(a,d^-\) are linearly independent: the former is nonzero and invariant under \(c\), whereas the latter is anti-invariant with nonzero local image. The image of \(G\) on \(V\) contains a central scalar different from one. The usual central-scalar argument gives \(H^1(\mathop{\mathrm{im}}(G\to\mathop{\mathrm{GL}}(V)),V)=0\): for a cocycle \(f\) and a central scalar \(z\ne1\), the cocycle identity is \((z-1)f(g)=(g-1)f(z)\), so \(f\) is a coboundary. Inflation–restriction thus makes restriction to \(N_V\) injective on \(H^1(G,V)\). On \(N_V\), both cocycles are additive homomorphisms. Their joint image spans \(V\oplus V\): its span is a \(G\)-submodule, and absolute irreducibility with \(\mathop{\mathrm{End}}_G(V)=\mathbb Q_p\) says that a proper submodule projecting onto each factor would impose a scalar linear relation between the two restrictions. That contradicts the preceding injectivity. The absolute irreducibility here follows from open image in the non-CM case; in the CM case it follows because neither \(K\) nor \(L\) contains the CM field. Passing to the finite-index subgroup over \(L\) leaves the span unchanged.

The Weil pairing implies \(\epsilon=1\) on \(N_V\). On this group, [eq:cup-primitive] reads \[u(gg')=u(g)+u(g')-e(a(g),d^-(g')).\] Consequently \[ u([g,g'])=-e(a(g),d^-(g'))+e(a(g'),d^-(g)). \tag{17}\] The alternating form on \(V\oplus V\) on the right is nonzero and nondegenerate. Since the joint translations span \(V\oplus V\), some \(g,g'\in N_V\cap G_L\) have a nonzero right-hand side. Write \(\sigma=[g,g']\). Both translations vanish on \(\sigma\), but \(u(\sigma)\ne0\).

Every Kummer character in our radical list is an additive homomorphism on \(N_V\), because \(\epsilon=1\) there. It therefore vanishes on \(\sigma\), so \(\sigma\) fixes all the radicals. A one-dimensional character such as \(\mathcal C\) also kills \(\sigma\). Replacing \(\gamma\) by \(\gamma\sigma^m\), for an integer \(m\), thus preserves its \(V\), \(\mathcal C\), cyclotomic, and radical values. The two translations entering the correction in [eq:corrected-primitive] do not change, whereas its primitive value changes by \(m u(\sigma)\). Some \(m\) makes the result nonzero. If further fixed open conditions are required, first replace \(\sigma\) by a suitable nonzero power; its nonzero primitive value survives. This constructs the required \(\gamma\). In particular the inverse matrix \(V(\gamma)-1\), and hence its denominator bound, is unchanged by the modification. Applying the prime construction to this element proves [eq:obstruction-matrix] for both signs of the tame twist. ◻

Proof. It suffices to work in each of the two summands over \(K\). Let \(\mathcal S_b^\bullet\) be the cone, shifted by \(-1\), of the map from the admissible global cochain complex to the local complex at the strict \(p\)-place. Its degree-one cohomology is exactly the indicated Greenberg group. Indeed global and local \(H^0\) vanish for \(V\) and, by the \(t\)-filtration, for every \(M_b\). The coefficient sequence in [eq:coefficient-sequence] gives a short exact sequence of these cone complexes. The resulting cohomology sequence therefore identifies \[ \ker\bigl(H^1(\mathcal S_b^\bullet) \longrightarrow H^1(\mathcal S_1^\bullet)\bigr) =\mathop{\mathrm{im}}\bigl(t:H^1(\mathcal S_{b-1}^\bullet) \longrightarrow H^1(\mathcal S_b^\bullet)\bigr). \tag{20}\] There is no flatness assumption on these cohomology groups.

If the base group is zero, induction in [eq:selmer-kernel] gives zero at every order. This deals with the auxiliary summand and with both summands outside the strict case. In the strict summand, a class at any order \(b\ge2\) reduces to an order-two class, whose base specialization is zero by [lem:obstruction-necessary,lem:nonzero-obstruction]. Equation (20), iterated, gives \[H^1(\mathcal S_b^\bullet) =\mathop{\mathrm{im}}\bigl(t^{b-1}:H^1(\mathcal S_1^\bullet) \longrightarrow H^1(\mathcal S_b^\bullet)\bigr).\] This image is killed by \(t\) and has \(k\)-dimension at most two. Its \(R_b\)-length is therefore at most two, as asserted. ◻

Proof. On a fixed sufficiently small formal subgroup, the formal logarithm and its inverse are integral, uniformly under unramified base extension. The intervening formal-group quotient has bounded \(p\)-exponent. In the Néron special fiber the geometric component group has fixed order, the additive part has bounded \(p\)-exponent, and a torus in characteristic \(p\) has no \(p\)-torsion. For the étale \(p\)-power torsion in the abelian part, let \(\alpha\) be a unit Frobenius eigenvalue. It is not a root of unity, by its complex absolute value. If \([E:\mathbb Q_p]=d\), the exponent is controlled by \(v_p(\alpha^d-1)\). Writing \(\alpha\) as its finite torsion part times a principal unit shows \[v_p(\alpha^d-1)\le C_\alpha+v_p(d)\] whenever the left side is positive. It is therefore bounded for these extensions. A fixed \(p\)-power, followed if necessary by an integer prime to \(p\), moves any point into the formal subgroup. Dividing its logarithm by that multiplier proves [eq:uniform-local-log], including at bad reduction.

For the precision assertion, the Kummer injection \[A_0(E)/p^n A_0(E)\hookrightarrow H^1(E,A_0[p^n])\] gives \(p^C P=p^n Q\) for some \(Q\in A_0(E)\). Consequently \[v_p(\log_{A_0,p}P)\ge n-C-c\] by [eq:uniform-local-log]. Thus the comparison loses only a fixed precision, without division by the local extension degree. ◻

Proof. Choose integral Kummer cocycles for \(y_i\) and corestrict with the tautological group-ring coefficients. Corestriction is a sum: its terms have integral coefficients regardless of the size of \(\mathcal G_i\). Truncate by the substitution in [eq:binomial-loss]. We obtain bounded admissible classes with coefficients \(M_b^{\pm}\), for every fixed \(b\), satisfying Kummer conditions at both \(p\)-places. The split Heegner norm relation gives augmentation equal to the Kummer class of \[(a_{q_i}-2)y.\] Indeed the \(q_i+1\) cyclic subgroups in the Hecke correspondence comprise two horizontal CM isogenies, corresponding to the two prime ideals over \(q_i\), and the \(q_i-1\) descending isogenies whose target order has conductor \(q_i\). The latter are the orbit over the conductor-one ring class field. Thus the relative trace is the Hecke action minus the two horizontal translates. Tracing further to \(K\) turns each horizontal translate into \(y\), and the Hecke eigenvalue on \(A_0\) is \(a_{q_i}\). This proves the displayed relation; see also the correspondence and CM action conventions in (Gross and Zagier 1986, I, Sections 2–3). Since \(y\) is a fixed torsion point, one fixed scalar kills all these augmentations in integral Kummer cohomology. Their rational augmentation is zero.

For \(b_*=1\) this proves the cohomological assertion. In the strict case divide the limiting Kummer trace once by \(t\), as in 9; its first specialization, say \(z_1\), is an ordinary compact Selmer class of \(V/K\). The cohomology sequence of [eq:coefficient-sequence] and \(H^0(K,V\otimes k)=0\) make multiplication by \(t\) injective on \(H^1(M_{b-1}^{\pm})\to H^1(M_b^{\pm})\), so this divided class is unique. All local Kummer conditions survive this division because the tame character is trivial at \(p\). The operation merely shifts bounded coordinates after subtraction of a fixed bounded coboundary; no denominator depending on \([H_i:K]\) is introduced.

We verify the conjugation identity with its effect on this coefficient, using the CM coordinates of (Gross and Zagier 1986, II, Section 1). Write a CM point in fractional-ideal coordinates as \[x(\mathfrak a,\mathfrak n)= (\mathbb C/\mathfrak a\longrightarrow\mathbb C/\mathfrak n^{-1}\mathfrak a), \qquad \mathfrak n\overline{\mathfrak n}=(N).\] The dual cyclic isogeny gives \(w_Nx(\mathfrak a,\mathfrak n) =x(\mathfrak n^{-1}\mathfrak a,\overline{\mathfrak n})\). Consequently \[x(\mathfrak a,\mathfrak n)^c =w_Nx(\mathfrak n\overline{\mathfrak a},\mathfrak n).\] In the ring class group, \([\overline{\mathfrak a}]=[\mathfrak a]^{-1}\), so this is inversion followed by a fixed translation. The level ideal is invertible in every order in question because \((q_i,N)=1\). If \(\varphi\) is the modular parametrization, the differentials of \(\varphi\circ w_N\) and \(-\operatorname{sign}(A_0)\varphi\) agree. Their difference is therefore a constant \(Q\), the image of a cusp. The Manin–Drinfeld theorem (Manin 1972; Drinfel’d 1973) makes \(Q\) of fixed torsion order. In the strict case \(r=1\), so the Fricke scalar \(-\operatorname{sign}(A_0)\) is \(+1\). Conjugation also inverts the anticyclotomic weight, namely \[ t\longmapsto(1+t)^{-1}-1=-t+t^2-\cdots. \tag{23}\] The translation is multiplication by a group-ring unit of augmentation one. Because the constant Kummer coefficient has already vanished, that unit has no effect on the leading linear coefficient. Equation (23) thus gives \(z_1^c=-z_1\). But the compact Selmer line is generated by the invariant class \(a\). Therefore \(z_1=0\).

Divide once more by \(t\). Its specialization \(z_2\) again belongs to the compact Selmer line, by the same local Kummer and bounded division argument. That line is everywhere locally zero in the strict case. Thus the original trace has zero local Kummer class through degree two, which is the assertion required for \(b_*=3\).

It remains to convert the local Kummer vanishing into [eq:heegner-log]. At \(p\), Proposition 7 makes \(H_i/K\) unramified with local degree of bounded \(p\)-part, so 13 applies with a common constant \(c\). Restricting the local twisted Kummer trace to the unramified field of definition makes the coefficient action trivial. It becomes exactly the Kummer class of the raw weighted sum, modulo the precision allowed by [eq:binomial-loss]. The vanishing through degree \(b_*-1\) holds to precision tending to infinity as in [eq:precision-choice]. After one fixed factor \(p^C\) clears the cochain denominators, the local comparison gives the lower bound \(n-C-c\) at Kummer precision \(n\). Thus the corresponding logarithm coefficients have valuations tending to infinity. For any fixed higher truncation, the Kummer cocycles, logarithms, and binomial coefficients all have a fixed denominator, so those coefficients remain bounded as well. This proves the proposition. ◻

Proof. The upper shifts in the double coset are indexed by \(M_{j,4-j}(\mathbb F_p)\), of cardinality \(p^{j(4-j)}\). The expanding columns must become divisible by \(p\) before the multiplication by \(p^{-1}\). Since the first two columns have rank two modulo \(p\), these equations have \(p^{(j-2)(4-j)}\) solutions whenever the original test is supported, and none otherwise. The equations do not change the chosen head function. The determinant factor in (30) is \(p^{4-j}\). Hence the surviving factor is \[p^{4-j}p^{(j-2)(4-j)}=p^{(j-1)(4-j)}=p^2\] for both \(j=2\) and \(j=3\). The calculation is independent of the source representation and of its reduction at \(p\). ◻

Proof. This is the unramified theta correspondence for the matched lattices. At split places it can also be read directly from the open rank-two matrix orbit: its coinvariants are the normalized parabolic induction of the determinant-twisted source with the trivial binary representation. The latter has standard parameters \(|\cdot|^{-1/2},|\cdot|^{1/2}\). The open-orbit calculation, including the exclusion of the lower ranks near the comparison point, is given in Lemma 26. ◻

Proof. For a scalar position test \(\phi\), write \(J_{\mathbf d}\) for the normalized jet of multi-degree \(\mathbf d\) at the fixed CM object. For a polynomial \(P(X)=\sum_{\mathbf d}a_{\mathbf d}X^{\mathbf d}\) on the product position lattice \(\Lambda\), put \[L_\phi(P)=\sum_{\mathbf d}a_{\mathbf d}J_{\mathbf d}, \qquad \|P\|_\Lambda=\sup_{x\in\Lambda}|P(x)|_p.\] We will prove \(|L_\phi(P)|_p\leq C\|P\|_\Lambda\), where \(C\) depends only on the stated test bounds and \(p\)-levels. The proof constructs the square of this combination as an algebraic Igusa function, bounds its expansions, and removes its initial denominator.

The square argument in this proof is scalar. For a test with values in a tame group algebra, first expand it in the group basis and apply the argument separately to each scalar coefficient test. These tests have the same coefficient bound and the same position supports and levels at \(p\). Only after obtaining the scalar estimates do we reassemble the group-algebra-valued moments. In particular, no coefficientwise bound is inferred by taking a square root inside a growing group algebra.

Proof. Choose source integration representatives integral at \(p\), by weak approximation in the rational unitary group, or by changing the rational Hermitian basis. The source quotient then gives finite sums at a level of fixed volume denominator. This remains true at \(q_i\) because the determinant equivariance of (33) cancels the varying source character. The archimedean transformation factors cancel in the integral, and character evaluations on the remaining representatives are algebraic \(p\)-units.

A boundary representative may be chosen integral at \(p\) by Iwasawa decomposition, sending the specified multiplicative member to \(e+B\) modulo the fixed level. The translated \(B\)-tests remain integrally supported. The test in the column Fourier-dualized to produce \(x\) is invariant under integral translations, so \(x\) too is integral at \(p\). Rational Jacobi torsions can use unipotent parameters divisible by the fixed \(p\)-level; these torsions are still dense and preserve this structure. Representatives and unipotents can simultaneously be chosen integral at the moving and fixed exceptional places, using approximation in the cusp parabolic and the rational Cayley chart on its binary unitary factor.

After the source-equivariance cancellation just made, expand each remaining finite test in the tame group basis and decompose each scalar coefficient test into simple unary tensors, retaining integral support and the original coefficient bound. Apply the scalar unary estimate before reassembling the group-algebra coefficients. Additive Fourier transforms away from \(p\) cost only \(p\)-units; at \(p\) the levels are fixed. Each index coefficient at a prime-to-\(p\) Jacobi torsion is consequently a finite sum of the bounded moments in Lemma 19, multiplied by integral powers of the coordinates of \(x\) and by the unit scales already separated in Convention 18. The nonarchimedean bound is unaffected by the number of summands.

At each fixed weight and Fourier–Jacobi index, these algebraic evaluations first give an algebraic coefficient section. Indeed the coefficient bundle on the abelian chart is algebraic. By proper algebraic–analytic comparison, the holomorphic coefficient lies in the complexification of its finite-dimensional algebraic space of sections. Choose finitely many algebraic prime-to-\(p\) torsion evaluations that separate this space, in the algebraic split differential frames just used. In an algebraic basis their evaluation matrix has full rank. Algebraicity of the values therefore gives algebraicity of the section, with an initial denominator that may depend on the weight, index, and tame level.

The full family of these torsion evaluations detects vertical integrality. The prime-to-\(p\) torsions, and those satisfying the fixed congruences just imposed, are dense on the special fiber. Their split differential frames are obtained by prime-to-\(p\) isogenies. After clearing an individual denominator, a section whose normalized torsion evaluations vanish modulo the uniformizer is therefore zero modulo it. The same least-denominator argument as above proves the uniform bound for the coefficient section. These algebraic coefficients first give algebraicity of the form by the characteristic-zero expansion principle. The ordinary Fourier–Jacobi expansion principle then transfers their uniform bound to the form itself. The canonical cusp charts with their finite étale level are those of Lemma 28. Their construction uses only ordinary PEL geometry and the flag and frame data of \(J\), so it is independent of the theta estimates and the cusp-lifting conclusion.

Finally the substitution defining \(R_b\) uses only the finitely many binomial coefficients through degree \(b-1\). Its factorial loss is a fixed constant depending on \(b\). This proves the last assertion. ◻

Proof of Proposition 17. The Fock calculation gives the weight and holomorphy, the mixed expansion gives its extension at cusps and its algebraic integral interpretation, Lemma 15 gives the exact operator values, and Lemma 16 gives the spherical parameters. Lemma 20 gives the uniform bounds. ◻

Proof of Lemma 22. The equal-rank Rallis inner product formula and the Waldspurger toric period formula give \[ S_i(m)^2=C_i(m)\,P_{0,i}(m)^2P_{1,i}(m)^2. \tag{43}\] We use these formulas with bilinear dual data, so the squares are algebraic squares, rather than absolute squares. For the theta identity, the precise Siegel–Weil input is the anisotropic boundary case of (Yamana 2011, Theorems 2.1(i) and 2.2(i)), also stated in (Gan et al. 2014, Theorem 7.1). In the notation of those results, \(m=n=2\), \(r=0\), and \(s_0=(m-n)/2=0\). On the doubled group \(U(2,2)\) the Siegel Eisenstein series is holomorphic at \(s_0\) and \(E(0,\Phi)=2I(\Phi)\), where \(I\) is the ordinary convergent theta integral over the compact binary group with quotient volume one. The conversion to our global measure convention contributes only the fixed scalar permitted in 2.3. Restrict this identity to the two copies of \(U(A)\) and integrate against the source cusp forms. The theta seesaw gives their bilinear theta pairing on one side; unfolding the Eisenstein series gives the doubling zeta integral on the other. Its Euler factorization is the central standard \(L\)-value times the normalized local integrals used below. This proves the equal-rank Rallis identity needed here directly from that boundary Siegel–Weil identity. The doubling and local factorization conventions are those of (Gan et al. 2014, secs. 11.3, 11.5–11.6). The source groups are compact, the standard base-change parameter is cuspidal and tempered, and its zero-space theta lift vanishes, since that lift could only be one-dimensional. The Waldspurger identities (Waldspurger 1985, sec. II.7 and Section III.3, Proposition 7) apply to cuspidal transfers with compatible central character and opposite torus characters in the two slots. Thus the identities involve central values, not residues or derivatives.

For the theta pairing, pull back the same data by \((h,g)\mapsto(h^c,g^c)\). Field conjugation is anti-symplectic on the tensor product and identifies the oscillator with its dual by Poisson summation, without a global conductor factor. Pair fixed compact vectors nontrivially; if necessary two different vectors can be used, since the scalar is the same. On the toric side, translate the vector by a rational torus normalizer inducing conjugation. This turns the second period into the first, up to a fixed character value from a base-point translation. At \(p\) and \(q\) the normalizer exchanges the distinguished coordinates with unit scales.

The global central \(L\)-values in the formulas match by quadratic base change over \(K\): \[ L\bigl(\tfrac12,\pi_L\otimes\eta_L\bigr) =L\bigl(\tfrac12,\pi_{0,K}\otimes\eta_K\bigr) L\bigl(\tfrac12,\pi_{1,K}\otimes\eta_K\bigr). \tag{44}\] Here \(\eta_L\) is induced by norm from \(\eta_K\), and simultaneous inversion of the character gives the same comparison. One may retain raw local integrals at every removed place and use partial \(L\)-values. Fixed adjoint, norm, and global zeta factors then contribute a single fixed scalar. We compare the remaining local factors explicitly.

Proof. At a fixed split place away from \(p\), use small support transverse to the orbit of \(e_1\) in the mixed polarization, leaving arbitrary desired unary tests on the remaining coordinates. Unfolding the rational norm-one orbit then cuts the finite integration to its stabilizer \(T\). At a nonsplit place the single compact orbit of primitive norm-one vectors gives the same cut, with all remaining functions spherical.

At \(q=q_i\), integral vector–covector pairs pairing to \(1\) are one \(\mathop{\mathrm{GL}}_2(\mathcal O_q)\)-orbit. The pivot test (33) therefore effects the same cut. At augmentation its determinant character is trivial. Setting the position vector equal to \(e_1\) makes invertibility of the head precisely the condition that the second entry of \(Y_{B_1}\) is a unit; the other remaining entries are integral. The orbit volume is the unmodified ratio \(\mathop{\mathrm{vol}}(K_{A,q})/\mathop{\mathrm{vol}}(T(\mathcal O_q))\), so it does not introduce the residue unit-group order.

At \(p\), take \(Y_e\) in a small fixed neighborhood of \(e_1\), \(Y_{B_1}\) integral with any prescribed locally constant test on its second entry supported on units, and \(Y_{B_2},Y_f\) integral. The dual coordinate of \(x\) is integral and pairs to \(1\) with \(Y_e\). Such pairs form one orbit under a sufficiently deep source congruence subgroup enlarged by the integral lower unipotent. The latter fixes \(\varphi_p\) by its choice above, and its transpose on the covector test leaves the relevant second entry unchanged. Thus the row-two \(B_1\) test can have arbitrary fixed locally constant unit level while the row-two \(B_2\) test is the integer test. The depth of \(Y_e\) and all other \(p\)-levels can be fixed once and for all. These tests still have the form (31).

At infinity, rotating \(x\) introduces exactly the determinant multiplier already present in the source type. The minor coefficient therefore becomes the specified second-row powers. The Heisenberg translation acts only on the first row; the second row gives (54). On a line, the two torus factors act by their product, so translating the binary argument by \(b_j\) is the same as \(s\mapsto sb_j\), up to its explicitly removable splitting character. This proves the factorization. ◻

Proof. Begin with large weights satisfying the indicated congruences. The local theta Hom for each individual line is nonzero: at a nonsplit finite place this is the unramified correspondence for the matched self-dual lattice, at infinity it is the chosen Fock degree, and at a split place there is no local obstruction.

We check the global root number separately for each line. If \(\alpha\) is its \(U(1)\)-character and \(\chi_W\) the corresponding splitting character, the standard unitary Hecke character entering its central \(L\)-value is \[\chi_W^{-1}\alpha_L,\qquad \alpha_L(z)=\alpha(z/z^c).\] Since \(\alpha_L\) is trivial on \(\mathbb A_F^\times\) and \(\chi_W|_{\mathbb A_F^\times}=\omega_{L/F}\), this character restricts to \(\omega_{L/F}\). The dimension-one epsilon dichotomy identifies its local epsilon factor, for the chosen trace-zero element and additive character, with the product of the two local line invariants (Borade et al. 2025, Theorem 3.5). The product formula for these invariants is \(1\) for each coherent global pair of lines. Thus each of the two global root numbers is \(+1\); it is not only their product that is known to be positive.

At infinity the pullback of \(\xi_j\) has exponent \(2(d_j+c_j)\), modified by \(\pm1\) by the splitting. After the arithmetic norm-half shift, and if necessary conjugating the CM orientation using self-duality, its critical algebraic character has type \[ (a_j+1)\Sigma-a_j\Sigma^c, \qquad a_j=d_j+O(1)\geq0, \tag{57}\] and restriction \(\omega_{L/F}|\cdot|_{\mathbb A_F}\). Its conductor has only split prime factors.

For the simultaneous nonvanishing, use the characteristic-zero theorem of Burungale–He–Tian–Ye (Burungale et al. 2026, Theorem 1.1 and Corollary 4.20). Write \(u_j=\chi_{W,j}^{-1}(\xi_j)_L\) for the unitary standard character of the \(j\)th line and put \[\lambda_j=u_j|\cdot|_{\mathbb A_L}^{-1/2}.\] This is the opposite norm-half shift from the arithmetic character in (57). Since the idelic norm on \(L\) restricts to the square of that on \(F\), \[\lambda_j|_{\mathbb A_F^\times} =\omega_{L/F}|\cdot|_{\mathbb A_F}^{-1},\qquad L(1,\lambda_j\nu)=L(1/2,u_j\nu).\] Thus \(\lambda_j\) has the self-duality convention of that theorem. For the large weights already chosen, its infinity type has the required form after choosing the corresponding CM orientation; the theorem allows an arbitrary CM type, including an induced one.

Choose a degree-one prime \(\mathfrak l\) of \(F\) above an odd rational prime \(\ell\), split in \(L\) and outside all fixed conductors and \(p\). Let \(\Gamma_{\mathfrak l}\) be the Galois group of the maximal \(\mathbb Z_\ell\)-free anticyclotomic extension of \(L\) unramified outside \(\mathfrak l\); it is isomorphic to \(\mathbb Z_\ell\). The two base signs are separately \(+1\) by the preceding local calculation. At this split prime they remain \(+1\) for all sufficiently ramified finite-order characters of \(\Gamma_{\mathfrak l}\) (Burungale et al. 2026, Lemma 4.12 and Section 4.4.3). The nonvanishing theorem therefore gives finite exceptional sets for each of \(L(1/2,u_1\nu)\) and \(L(1/2,u_2\nu^{-1})\). Choose \(\nu\) outside their union.

Because \(\ell\) is odd, \(\nu\) has a unique square root among the \(\ell\)-power-order characters. Restrict \(\nu^{1/2}\) to \(\mathop{\mathrm{U}}(1)(\mathbb A_F)\) and call the resulting character \(\tau\). Anticyclotomicity gives \[\tau(z/z^c)=\nu^{1/2}(z/z^c)=\nu(z).\] Replacing \(\xi_1,\xi_2\) by \(\xi_1\tau,\xi_2\tau^{-1}\) preserves Equation 55 and gives the two nonzero central values just selected. This adds conductor only at the split prime \(\mathfrak l\), so all prescribed local conditions are preserved. Only characteristic-zero nonvanishing is used here.

The equal-rank line theta norm formula, equivalently the global line-lift criterion (Borade et al. 2025, Theorem 4.2 and Proposition 4.3), now supplies nonzero unary periods with suitable local tests. At \(p\) use the appropriate unit-character test for row-two \(B_1\) and the unit cut of the unramified test for row-two \(B_2\); their split local Mellin integrals are nonzero. Keep the spherical tests at nonsplit places. Compact line groups and the nontrivial infinity powers give the cuspidal situation with no occurrence from dimension zero. The nonzero output on the other line is a character, so it is already nonzero at identity: translating that argument factors through the input torus, up to the known splitting scalar. This proves the assertion at the chosen cut weights. Fix this auxiliary finite-order choice permanently. Its conductor and every additional local denominator are now fixed; enlarge the finite set in Proposition 7 before the final moving-prime sequence is chosen. ◻

Proof. Apply Lemma 19 to the two unary periods. For the actual \(B_2\) integer test, positive degrees tending to infinity pick out its unit cut. Thus the analytic limiting function specializes at \(d_{20}\) to the nonzero cut value obtained in Lemma 24; the same is true for \(B_1\). Hence neither \(\mathcal T_j\) is identically zero on its chosen disc. This is the reason for testing nonvanishing with unit cuts, while retaining the integer test in the actual third column of (31).

By the congruence imposed on \(d_{10}+d_{20}+1\), the discs can be arranged so that \(s\) in the first disc implies \(-1-s\) in the second. The analytic functions \(\mathcal T_1(s)\) and \(\mathcal T_2(-1-s)\) are nonzero functions. Choose a point \(a\) avoiding their zero sets. We may take positive integer sequences \(d_{1i}\to a\) and \(d_{2i}\to-1-a\), both tending to infinity, on the required torsion branches. Their sum plus one then tends to zero. The two period values stay bounded away from zero.

Restore the unit cut above \(q_i\). Subtracting the uniformizer translate of the relevant unary Schwartz function from its integer test and using torus equivariance expresses its effect by finitely many factors \[ 1-\alpha_{iv}\mathcal C(\mathop{\mathrm{Fr}}_{iv})^{\pm d_1}. \tag{58}\] Here \(\alpha_{iv}\) consists of residual-cardinality powers, fixed finite characters, splitting characters, and the fixed weight shifts. Its limit is a nonzero \(p\)-unit; the dependence through \(d_2=m-1-d_1\) is incorporated in the sign and these shifts. Pass to sublimits for the bounded-degree roots occurring in these normalizations. Proposition 7 gives \(\mathcal C(\mathop{\mathrm{Fr}}_{iv})\to\mathcal C(\gamma)\), which is a \(p\)-unit because \(\mathcal C\) is a continuous \(p\)-adic Galois character, and is not a root of unity. On a fixed weight branch its unit logarithm is consequently nonzero. None of the limiting analytic factors (58) is identically zero. Refine the choice of \(a\) to avoid their finitely many zero sets as well. The product of the two unary periods, all these local factors, the fixed period (52), and the chosen row-one evaluations is then bounded away from zero. The remaining \(m\)-dependent normalization factors in (53) are only the unit powers already treated in the mixed expansion.

There is no growing averaging denominator in turning this product back into a coefficient. The auxiliary finite-order choices and the torus levels have been fixed; the \(q_i\) unit tests are invariant under local torus units. Thus the averages over the \(b_j\) in (56) use a fixed finite set of representatives, chosen away from the varying \(q_i\). If every one of the corresponding coefficient evaluations had valuation tending to infinity, so would their fixed finite weighted sum, contrary to the product just obtained. Hence one of finitely many evaluations of the positive-index coefficient has valuation bounded above. Passing to the filter chooses one option if needed.

Finally choose the positive integers \(d_{1i},d_{2i}\) sufficiently close to their limits at each stage, while keeping them arbitrarily large. Their sum plus one, \(m_i\), can be as close to zero as required by the fixed-\(i\) constant-term estimate. Simultaneous congruences and lower bounds on the positive integers cause no conflict. This proves the assertion with the required constant-term precision. ◻

Proof. Work in the split local theta coinvariants over the Laurent ring of the unramified determinant deformation, localized near its central value. Filter locally constant Schwartz functions on \(M_{2,4}\) by rank. The rank-zero contribution would require a character quotient of the generic source representation, which is impossible. For a rank-one image fix its first covector. The stabilizer Levi \(\mathop{\mathrm{diag}}(1,d)\) would require exponent \(|d|^{-1}\) on the raw lower Jacquet module of \(\pi^\vee\): the Weil factor is \(|d|^2\) and the vector-orbit Jacobian is \(|d|\). In normalized Jacquet conventions the required exponent is \(|d|^{-3/2}\), which is excluded by temperedness of the elliptic source representation. For a supercuspidal source the relevant Jacquet module is zero; for a tempered principal or special source its exponents have the usual tempered bounds and do not equal \(-3/2\).

These are calculations over the deformation ring, not merely at isolated characters. Use restriction of locally constant functions for the rank filtration and integration along the vector orbit, or smooth compact-induction coinvariants, on compact open charts of the row/image parameters. The difference between the required Levi character and each of the finitely many actual Jacquet characters is invertible after localization at the comparison point. Thus the lower-rank coinvariants vanish in that localization, and the open rank-two part surjects onto all coinvariants.

On the open rank-two orbit, direct orbit induction identifies the coinvariants with normalized parabolic induction of the determinant-twisted source and the trivial binary representation. At weight zero this induction is irreducible by irreducibility of unitary parabolic products for general linear groups (Tadić 1986). The pivot test is nonzero there, since its head can be paired with a nonzero local matrix coefficient. Its translates consequently generate. In the Steinberg case there are nonzero \((3,1)\)-parahoric invariants: in the induced model place the two Steinberg positions on different sides of the single-step flag. Their finite-Weyl sign condition then imposes no forbidden merging of those positions. Project an open test nontrivially to these invariants; its translates also generate the central specialization.

Only finitely many original test vectors at one fixed level must be spanned. In that finite-dimensional induced model, select a finite collection of the permitted translates whose spanning matrix has a minor \(f\) with \(f(1)\ne0\). Cramer’s rule expresses the needed vectors after inverting \(f\). The coefficients are rational functions of the unramified parameter. Since \(\eta_m\to1\) and \(\nu_i\) is locally trivial at this fixed place, \(f(\eta_m)\to f(1)\ne0\). All these coefficients therefore have bounded denominator. There are only finitely many fixed places, so tensoring the finite expressions gives one bound. No inverse involving the varying place \(q_i\) has entered this argument. ◻

Proof. Lemma 25 chooses the degrees and obtains the content bound simultaneously with Proposition 21. The bounded fixed-place spanning and the preceding content argument impose all additional controls without losing that bound. Every selected test still satisfies the hypotheses of Proposition 17 and the constant-term proof, which allowed arbitrary bounded fixed split tests. The explicit positive-operator calculation gives the last binary allocations. If finitely many choices remain, make the constant-term approximation sufficiently accurate for all of them and then choose the one with bounded content. This gives all the properties at once. ◻

Proof. The ordinary group has a canonical multiplicative subgroup at every fixed level \(p^s\) (Katz 1981, proof of Theorem 2.1, pp. 149–150). Once the rank-three flag is this subgroup, its rank-two subflag and frame are equivalent by Cartier duality to quotients and markings of its finite etale character group. The full upper radical of \(J\) is essential: specifying a splitting of the etale quotient would impose different data. The canonical subgroup separates the corresponding generic level choices. Their idempotents extend in the finite normalization over the ordinary completion; equivalently, one may normalize after completion, since the base is excellent. This proves the asserted branch decomposition.

At a rank-one boundary component, the Raynaud semiabelian group is an extension of the binary abelian group by a rank-one torus. The preimage of the binary multiplicative torsion is an extension of multiplicative-type finite flat groups, hence is of multiplicative type: its Cartier dual is an extension of finite etale groups. This is the entire distinguished rank-three multiplicative subgroup. Its construction precedes quotienting by the Raynaud period lattice and does not involve the center Fourier parameter. Its character-group level therefore produces only the asserted finite etale chart covers.

After passage to a completed strict boundary base and choice of an origin, a connected finite etale cover of an abelian chart is an abelian scheme and the covering map is an isogeny. On a geometric fiber this follows by lifting addition to the pointed cover: the subgroup defining the cover in the abelian fundamental group is preserved by addition, and uniqueness of pointed lifts gives the group laws. The same construction extends over the strict base by the equivalence for finite etale covers under henselian thickening. Thus the new charts have the same abelian geometry needed for Fourier–Jacobi coefficients; compare the ordinary chart description in Lan (2018, Proposition 4.2.1.34 and Corollary 4.2.1.36). ◻

Proof. For a finite locally free module \(\mathcal W\), its divided-power module is \[\Gamma^d(\mathcal W) =\bigl(\operatorname{Sym}^d(\mathcal W^\vee)\bigr)^\vee.\] Use this integral definition with \(\mathcal W=\bigwedge^2\mathcal H\). A minor monomial and its dual basis vector have pairing one, without a factorial. All the modules in this definition are finite locally free, so it commutes with base change. Functoriality for linear maps proves preservation under integral frame changes and gives the displayed diagonal entries, including when some \(c_j\) is not a unit.

To check the filtration assertion, split the filtration locally. Exterior and symmetric powers have their degree filtrations, with graded pieces given by tensor products of the corresponding powers of the original graded pieces. Dualizing the symmetric-power filtration gives the divided-power filtration. Block-triangular changes between local splittings preserve these filtrations, so they descend. Their graded pieces are constant whenever those of \(\mathcal H\) are constant. No rational splitting or rational identification with ordinary symmetric powers is used. ◻

Proof. The boundary has rank one over \(L\), and the residual binary Shimura datum is zero dimensional. In its formal charts, a cusp tuple is indexed by strictly positive integral \(\alpha\in F\). Here positivity means positivity at both real embeddings. There are no nonzero rational indices on the boundary of the positive cone: an element of \(F\) with one embedding zero is zero. Vanishing of the constant term therefore imposes positive order on every relevant cone ray.

For \(\alpha>0\), the coefficient is a section on the abelian extension-data chart \(C\) of \[\mathcal L_\alpha\otimes\mathcal E_\lambda|_C,\] where the Jacobi line bundle \(\mathcal L_\alpha\) is ample. This description persists on the finite etale abelian covers in 28. The semiabelian differential sequence has constant locally free graded pieces over the fixed boundary base. Apply 29, and tensor with the determinant and opposite differential line factors. This gives \(\mathcal E_\lambda|_C\) a finite filtration with constant graded pieces, in precisely the integral lattice used by the theta section. This construction needs no rational splitting of the semiabelian differential sequence.

We recall why the required vanishing is valid in every residue characteristic. If \(L\) is ample on an abelian variety, choose multiplication by an integer \(n\) prime to the characteristic. Its trace splits \(\mathcal O\) inside \([n]_*\mathcal O\), so \(H^r(L)\) is a summand of \(H^r([n]^*L)\). For sufficiently large such \(n\), the latter vanishes for \(r>0\): \([n]^*L\) has arbitrarily large ample degree, and Serre vanishing is uniform under the algebraically trivial twists occurring in the theorem of the square, by properness of the dual abelian variety. The filtration just described gives the same vanishing after tensoring with \(\mathcal E_\lambda|_C\). Cohomology and base change, or induction through the exact sequences for successive uniformizer powers, then lifts every positive coefficient modulo \(p^M\) integrally.

It remains to lift the arithmetic identifications between coefficients. Modulo translations, the stabilizer acts on positive indices by unit norms. Fixing a positive index forces this norm to be one; the resulting CM norm-one unit group is finite. The remaining binary arithmetic stabilizer is finite by definiteness. Neatness removes these finite stabilizers. We may therefore choose one lift in each free index orbit and transport it around the orbit, without averaging or dividing by an orbit order.

The resulting tuples converge on every formal cone. In fact, an interior rational ray gives a linear functional strictly positive on the nonnegative index cone. A bound on its value bounds both real coordinates of a positive index, and hence leaves only finitely many points of the index lattice. This is exactly the formal convergence condition on the completed monoid algebra. The formal-chart description and formal functions now identify these tuples with the completed pushforward sections; see Lan (2013, sec. 7.1.2) and Hartshorne (1977, III, Theorem 11.1). Thus the required reduction map is surjective at a cusp. Its injectivity follows from applying the left-exact pushforward to multiplication by a uniformizer on the torsion-free source sheaf. Off the boundary the assertion is immediate from the finite level normalization. Faithfulness of completion for coherent modules gives the assertion on the ordinary open.

The branch decomposition allows the zero lift on every other branch. Finally, \(\mathcal O[t]/(t^b)\) is a finite free \(\mathcal O\)-module, so the entire construction applies coefficientwise and gives the stated Artin-ring assertion. This proves the required special case directly; the general ordinary subcanonical vanishing theorem is Lan (2018, Theorem 8.2.1.3, with the setup of Section 8.1). ◻

Proof. It suffices to work at one distinguished \(p\)-adic factor. The split PEL idempotents select the rank-three multiplicative character lattice and the rank-one etale lattice; polarization determines the conjugate factor. Choose Serre–Tate coordinates \(q_1,q_2,q_3\) for their bilinear pairing (Katz 1981, Theorem 2.1(1),(2)). Locally their deformation ring is \[R=\mathcal O[[q_1-1,q_2-1,q_3-1]].\] For \(U_j\), expand the lattice transverse to a multiplicative \(j\)-summand by \(p^{-1}\). The expanded multiplicative directions are selected by \(p^{j(3-j)}\) integral graphs. Fix one graph. Functoriality of the Serre–Tate pairing (Katz 1981, Theorem 2.1(4)) for the lattice inclusion and \(p\) times the reverse inclusion shows that the remaining extension choices replace the \(j\) retained coordinates by their \(p\)th roots; the other \(3-j\) coordinates are unchanged. Indeed, expanding the etale generator requires a root when paired with a retained multiplicative character, whereas the simultaneous expansion of a transverse multiplicative direction cancels this change for the other coordinates.

The corresponding finite free extension is the appropriate completion of \[R'=R[u_1,\ldots,u_j]/(u_a^p-q_a:1\leq a\leq j).\] Its basis consists of \(\prod_a u_a^{e_a}\) with \(0\leq e_a<p\). The trace of every such basis element except \(1\) is zero, and \(\mathop{\mathrm{Tr}}_{R'/R}(1)=p^j\). Hence \[ \mathop{\mathrm{Tr}}_{R'/R}(R')\subseteq p^jR. \tag{61}\] The same monomial calculation applies on a boundary cone whenever the Hecke correspondence takes a \(p\)th root of a torus coordinate. For \(j=2\) there are \(p^2\) graphs and a degree-\(p^2\) trace, and for \(j=3\) there is one graph and a degree-\(p^3\) trace. Thus the total counts are \(p^{j(4-j)}\), as required by the double cosets.

The transported multiplicative flags and differential comparison maps depend only on the graph, not on these root choices. They can therefore be put outside the trace. In an integral graph basis, pullback scales \(3-j\) positive differential directions by \(p\). The opposite differential scales by \(p^{-1}\), by duality to the distinguished etale direction.

For the weights in use, the positive integral coefficient lattice is \[(\det\omega^+)^{\lambda'_3} \otimes \mathcal D_{m-1}(\omega^+),\] and the opposite line occurs with exponent \(-\lambda'_4\). By 29, the integral graph changes preserve this lattice. For \(j=2\), choose a graph basis in which the positive differential map is \(\mathop{\mathrm{diag}}(1,1,p)\). It acts on the three minor directions with entries \(1,p,p\); hence on their monomial-dual coefficient of degrees \((a_{12},a_{13},a_{23})\) it acts by \(p^{a_{13}+a_{23}}\). This is integral for every \(m\), without a factorial or a change of lattice. The determinant factor supplies \(p^{\lambda'_3}\). For \(j=3\), the positive differential map is the identity in a graph basis. The opposite line supplies \(p^{\lambda'_4}\) in both cases. Before taking trace, the lower bounds are consequently \[\lambda'_3+\lambda'_4=N_2-2 \quad(j=2),\qquad \lambda'_4=N_3-3 \quad(j=3).\] Combining these with [eq:root-trace-divisibility] gives \(p^{N_j}\) divisibility. The graph sum introduces no denominator. This proves integrality of \(p^{-N_j}U_j\) on the stated lattice, including when \(N_j\) is negative. The other \(p\)-adic factor is a spectator, so the argument applies independently at both places.

In the original weight \((m,m,1;-1)\), both \(N_j\) are \(2\); the theta eigenvalue \(p^2\) thus gives normalized value \(1\). The shift \((H,H,H;-H)\) changes \(N_2\) by zero and \(N_3\) by \(-H\). Its differential determinant contributes \(p^{(2-j)H}\) on the same graph, precisely this change of normalization. The remaining unit frame factors, raised to the chosen divisible \(H\), are \(1\) modulo \(p^M\); in canonical multiplicative frames the lifted Hasse section itself is \(1\). Thus multiplication by the Hasse lift intertwines the normalized operators modulo \(p^M\). Explicitly, in the shifted weight \[N_2=2,\qquad N_3=2-H.\] Prime-to-\(p\) Hecke correspondences preserve the canonical branch and its integral lattice as well. Their Hodge determinant changes are \(p\)-adic units; the same divisibility of \(H\) makes their additional weight factors trivial modulo precision. Hence all prescribed comparison eigenvalues survive the lift. ◻

Proof. Write \(a=a_0(1+u)\), with \(u\in t\mathcal O_{K'}[1/p][t]/(t^b)\). Then \[a^{-1}=a_0^{-1}\sum_{j=0}^{b-1}(-u)^j \in p^{-bC}\mathcal O_{K'}[t]/(t^b).\] Multiplication by this inverse proves the assertion. The use of the constant coefficient is essential, since it makes \(a\) invertible over \(K'[t]/(t^b)\). ◻

Proof. The comparison eigenvalues lie in the fixed coefficient field \(k\) and are integral in the specified normalizations. This follows from the theta parameter, the normalized value \(1\) at \(p\), and the rational auxiliary block parameters. Away from \(p\), the Frobenius-scaled polynomial coefficients are integral; the character and residual-cardinality factors in these normalizations are \(p\)-adic units. The coefficients of \((1+t)^\ell\) are integral for \(\ell\in\mathbb Z_p\), and the preceding choice \(M_i\leq n_i-c_b\) makes the finite tame relations valid. Consider the polynomial \(\mathcal O\)-algebra on all the chosen Hecke generators and its evaluation at these comparison eigenvalues in the target of [eq:hecke-congruence]. We must show that every relation in its action on \(S_i\) belongs to the kernel of this evaluation.

Let \(s_i\) be the lifted cusp section, viewed with its \(t\) coefficients. On the canonical ordinary locus it agrees modulo the working precision with Hasse times the integral theta section. Every generator acts there with the prescribed comparison value modulo that precision, by 31. The same is true for any integral polynomial in the generators: multiplication and addition of integral operators consume no precision, regardless of the degree or number of its monomials. If \(P\) is a relation on the classical cusp space, it follows that \[P(\text{comparison values})\,s_i\equiv0 \pmod{p^{M_i}}.\] Evaluate the nonconstant coefficient furnished by 27, using the same Fock monomial and its literal dual coefficient basis. The definition of \(\mathcal E_{\lambda'_i}\) retains this integral pairing through the lift. Hasse is \(1\) in that frame modulo precision, so its value \(a_i(t)\) has integral coefficients and \(v_p(a_i(0))\leq C\) for one fixed \(C\). The preceding display and 32 imply \[P(\text{comparison values})\equiv0 \pmod{p^{M_i-bC},t^b}.\] The scalar polynomial on the left has coefficients in \(k\); integrality and divisibility can be descended from the finite evaluation field because \(p^n\mathcal O_{K'}\cap\mathcal O=p^n\mathcal O\). The fixed loss \(bC\), together with the earlier fixed clearing and neat-descent losses, is independent of \(P\) and \(i\). Reducing and relabeling \(M_i\) gives the factorization through the entire operator order \(T_i^0\).

Finally, a finite algebra over the complete local ring \(\mathcal O\) decomposes as a product of its factors at the residual maximal ideals. The target ring is local, with residue field obtained by setting \(t=0\) and reducing modulo the uniformizer. Hence \(\lambda_i^0\) factors through the summand selected by this residual augmentation. The images of \(\widetilde U_2,\widetilde U_3\) are \(1\), so these operators lie outside that summand’s maximal ideal and are units there. This construction keeps every coefficient through degree \(b_*-1\) and permits nilpotents in \(T_i^0\), as required for the next section. ◻

Proof. First consider one real factor \(\mathop{\mathrm{U}}(3,1)\), and omit the subscript \(i\). The Harish–Chandra coordinates of the selected holomorphic discrete series \(D\) are \[k=(m+H+\tfrac12,\ m+H-\tfrac12,\ H-\tfrac12,\ \tfrac12-H).\] Their successive gaps are \(1,m,2H-1\). They are regular integral, and the unique negative-sign coordinate of \(D\) is \(k_4\). Choose its individual pseudocoefficient \(f_D\), with trace one on \(D\) and trace zero on every other basic standard module (Clozel and Delorme 1990, Proposition 4 and its corollary, pp. 212–213). In particular it annihilates all properly parabolically induced standard modules. Standard-module expansions preserve infinitesimal character, so an irreducible representation with nonzero trace against \(f_D\) must have infinitesimal character \(k\).

Every unitary representation with this regular integral character is an admissible cohomologically induced module \(A_{\mathfrak q}(\mu)\) (Salamanca Riba 1988, Theorem 1.2). That theorem applies to \(\operatorname{SU}(3,1)\); adjoining the compact central character gives the same assertion for \(\mathop{\mathrm{U}}(3,1)\). Write the theta-stable Levi of \(\mathfrak q\) as ordered signature blocks \(\prod_j\mathop{\mathrm{U}}(p_j,q_j)\), where \(\sum p_j=3\) and \(\sum q_j=1\). The inducing character is constant on each Levi block and dominant across the nilradical roots. Hence the infinitesimal coordinates in each block form a consecutive string, with successive difference one, and the blocks occur in their decreasing order in \(k\). There is exactly one block with \(q_j=1\).

If this Levi is noncompact, its block containing the negative coordinate has size at least two. In the standard-module resolution of \(A_{\mathfrak q}(\mu)\), the discrete-series terms arise from positive systems extending the nilradical roots (Adams and Johnson 1987, Theorem 8.2 and Lemma 8.8). They may reorder coordinates within a Levi block, but not between blocks. Their negative coordinate must therefore belong to that same block. For \(D\) to occur, the block would have to contain \(k_4\). A consecutive string of size at least two containing \(k_4\) also contains \(k_3\) and forces \(k_3-k_4=1\), contrary to \(k_3-k_4=2H-1\geq5\). Thus the coefficient of \(D\) in this resolution is zero. All other terms are annihilated by \(f_D\). If the Levi is compact, the induced module is itself discrete series, and its trace is one precisely when it is \(D\). We have proved that \(\operatorname{tr}X(f_D)\) is zero or one for every irreducible unitary \(X\), with value one only for \(X=D\). This argument treats each real factor separately. Their product pseudocoefficient therefore satisfies the sign property \((*)\) of Labesse (2011, sec. 5.1), including competitors with different types at the two real places. No lower bound on the first gap beyond its actual value one is used.

Apply Labesse (2011, Corollary 5.3) to \(\pi\). It gives one automorphic transfer induced from conjugate-dual discrete general-linear representations. The discrete-spectrum classification of Mœglin and Waldspurger (1989) expresses its factors as Arthur blocks and gives the formal parameter \[\Psi=\boxplus_\beta\tau_\beta[a_\beta], \qquad \sum_\beta d_\beta a_\beta=4,\] with unitary cuspidal factors. The same corollary gives local base change at every split finite place and at every unramified place. It also retains the full infinitesimal character at both real places: its proof uses the coefficient systems for the complex group and compatibility with infinitesimal-character multipliers, not just the Casimir eigenvalue. We do not invoke its additional conclusion requiring cuspidality of the transfer.

Choose \(S\) to contain the infinite places and the finite ramification of \(\pi\) and the chosen standard embedding. All its finite places can be chosen split in \(L/F\). At a nonsplit finite place, the transferred representation is spherical, so its Weil–Deligne parameter has trivial inertia and zero monodromy. Formula (63) and compatibility of local Langlands with the normalized product express this parameter as the direct sum of the shifted parameters of \(\tau_{\beta,w}\). Each summand therefore has trivial inertia and zero monodromy, so each \(\tau_{\beta,w}\) is unramified. The corollary’s stronger matching at every split place concerns this same \(\Psi\) and will also be used below. ◻

Proof. At a distinguished complex embedding the first infinitesimal coordinates are \[ (k_1,k_2,k_3,k_4) =\bigl(\lambda'_1+\tfrac12,\lambda'_2-\tfrac12, \lambda'_3-\tfrac32,\lambda'_4+\tfrac32\bigr). \tag{64}\] Indeed the half-sums of positive compact and noncompact roots are \(\rho_c=(1,0,-1;0)\) and \(\rho_n=(\tfrac12,\tfrac12,\tfrac12;-\tfrac32)\), and the holomorphic discrete-series formula is \(\lambda'+\rho_c-\rho_n\). The coordinates are distinct and belong to \(\mathbb Z+\tfrac32\); the coordinates at the conjugate embedding form the multiset \(\{-k_1,-k_2,-k_3,-k_4\}\).

Consider an archimedean character \(z^u\bar z^v\) in a cuspidal factor \(\tau\) before inserting a shift \(s\in S_a\). The two coordinates \(u+s\) and \(v+s\) occur in the two prescribed infinitesimal characters. Hence \(u,v\) are real and \(u+v\in\mathbb Z\). A unitary generic representation of \(\mathop{\mathrm{GL}}_d(\mathbb C)\) has \(|\operatorname{Re}(u+v)|<1\) for every inducing character. Cuspidality supplies genericity, so \(u+v=0\). This proves temperedness at infinity.

The archimedean parameters of \(\tau\) and \(\tau^{c,\vee}\) are now identical. A pair of distinct conjugate-dual cuspidal factors in \(\Psi\) would consequently repeat the first infinitesimal coordinates, contrary to regularity. Thus all factors are conjugate-self-dual. Repeated copies of an identical Arthur block are excluded for the same reason. In particular the resulting unitary parameter is elliptic.

Choose a unitary Hecke character \(\chi\) of \(L\) with \(\chi^c=\chi^{-1}\), with infinity type \(z/|z|\) at both distinguished embeddings, and with conductor supported on split auxiliary primes. Its existence is the character construction of 2.3, with the following parity prescription. Require \(\chi|_{\mathbb A_F^\times}=\omega_{L/F}\), the quadratic character attached to \(L/F\). At infinity this agrees with the restriction of \(z/|z|\) to \(\mathbb R^\times\). At nonsplit finite places use unramified extensions of \(\omega_{L/F,v}\), and use a sufficiently deep conductor at split auxiliary primes to remove the remaining ray-class unit obstruction. The resulting Hecke character satisfies \[\chi(x)\chi(x^c)=\chi(\mathop{\mathrm{N}}_{L/F}x) =\omega_{L/F}(\mathop{\mathrm{N}}_{L/F}x)=1,\] which gives the required global identity \(\chi^c=\chi^{-1}\). For a factor \(\tau\) of rank \(d\) and block length \(a\), choose \(\delta\in\{0,1\}\) with \[\delta\equiv a+d-1\pmod2.\] If \(u\) is one of its first infinitesimal coordinates, then \(u\in\mathbb Z+(4-a)/2\), and therefore \(u+\delta/2\in\mathbb Z+(d-1)/2\). Thus \(\Pi=\tau\otimes\chi^\delta\) is regular algebraic in the cohomological normalization for \(\mathop{\mathrm{GL}}_d\). Regularity follows because its shifted coordinates form a subset of the distinct coordinates in [eq:transfer-infinitesimal]. The equality \(\Pi^c\simeq\Pi^\vee\) follows from those for \(\tau\) and \(\chi\). ◻

Proof. Take the single weak parameter supplied by 34. Its archimedean character is the required one, and its unramified cuspidal factors at nonsplit places give assertion (iii) by [eq:arthur-block-wd]. The application of Labesse’s corollary in that lemma already gives local base change at every split finite place. Under the chosen identification \(G(F_v)=\mathop{\mathrm{GL}}_4(F_v)\) this is exactly assertion (ii), including the places above \(p\), the bad places, and the moving levels. All three assertions thus concern the same parameter. This transfer argument places no restriction on the coefficient prime \(p\). ◻

Proof. Apply 37 to the representations \(\Pi_\beta=\tau_\beta\chi^{\delta_\beta}\) supplied by 35. For a factor of rank \(d\), length \(a\), and \(s\in S_a\), put \[t_s=s+\frac{d-4}{2},\qquad \eta_s=\chi^{-\delta}|\cdot|_L^{t_s}.\] At a distinguished complex embedding its exponents are \(t_s-\delta/2\) and \(t_s+\delta/2\). Both are integers: \(2t_s\equiv a+d-5\equiv\delta\pmod2\). Thus \(\eta_s\) is an algebraic Hecke character even when its two displayed factors are not separately algebraic. Let \(r_p(\eta_s)\) be its Galois character and define \[ \rho_\pi =\bigoplus_\beta\ \bigoplus_{s\in S_{a_\beta}} r_p(\Pi_\beta)\otimes r_p(\eta_{\beta,s}). \tag{68}\] Its dimension is \(\sum_\beta d_\beta a_\beta=4\). On the automorphic side, the normalization in each summand is \[(\tau\chi^\delta)|\det|^{(1-d)/2}\eta_s =\tau|\det|^{s-3/2}.\] Equations (63) and (65) now give [eq:four-dimensional-compatibility]. The same construction at the unramified standard parameters in 36(iii) gives unramifiedness: the bound forces the monodromy to vanish, and the semisimplified Weil inertia is trivial. Twisting and direct sums preserve the de Rham property.

For completeness, the polarization follows directly with the stated normalization. Since \(\chi^c=\chi^{-1}\), \[\eta_s^c\eta_{-s}=|\cdot|_L^{d-4}.\] Geometric reciprocity sends \(|\cdot|_L\) to \(\epsilon\). Consequently the summand \(R_s=r_p(\Pi)r_p(\eta_s)\) satisfies \[R_s^c \simeq R_{-s}^{\vee}\epsilon^{1-d+d-4} =R_{-s}^{\vee}\epsilon^{-3}.\] The set \(S_a\) is invariant under \(s\mapsto-s\), proving the polarization of [eq:four-dimensional-construction].

If \(u\) is a first infinitesimal coordinate of \(\tau\) and \(k=u+s\) is the corresponding coordinate of the full parameter, the associated summand has de Rham weight \[\frac{d-1}{2}-u-\frac\delta2 -\left(t_s-\frac\delta2\right) =\frac32-k.\] Substitution from [eq:transfer-infinitesimal] proves [eq:hodge-weights]. In particular, for \(\lambda'=(m+H,m+H,1+H;-1-H)\) these weights are \[1-m-H,\quad 2-m-H,\quad 2-H,\quad 1+H,\] with the asserted gaps \(m\) and \(2H-1\). ◻

Proof. Positive finite-slope Jacquet restriction at \(w\) (Casselman 1995, sec. 4.1) allocates two unramified characters to the final two slots of the Levi \(\mathop{\mathrm{GL}}_2\times\mathop{\mathrm{GL}}_1\times\mathop{\mathrm{GL}}_1\). Write \(\alpha_3,\alpha_4\) for their geometric Frobenius eigenvalues with the cohomological norm shift of 38. The half-modulus in normalized Jacquet restriction and the integral normalization of the two operators give \[ \widetilde U_2=\frac{p^{h_3+h_4}}{\alpha_3\alpha_4}, \qquad \widetilde U_3=\frac{p^{h_4}}{\alpha_4}. \tag{71}\] Thus \(v_p(\alpha_3)=h_3\) and \(v_p(\alpha_4)=h_4\). The sum of the other two slopes is \(h_1+h_2\), since the whole potentially semistable module is weakly admissible. Its Newton polygon lies above its Hodge polygon. The sum of the lowest two slopes is at most the sum of the other two slopes and at least \(h_1+h_2\); hence equality holds. The strict Hodge gap at position two forces a strict Newton gap there. Similarly the lowest three slopes sum to \(h_1+h_2+h_3\), with a strict gap at position three. In particular the two final roots are uniquely selected by their individual slopes.

After a finite semistable extension, the low-slope spaces are stable under inertia and monodromy. On either such space its induced Hodge number is at least the sum of the corresponding lowest Hodge weights, whereas weak admissibility bounds it above by its Newton number. Both inequalities are equalities. The subspaces are therefore weakly admissible. The Colmez–Fontaine equivalence and its subobject correspondence (Colmez and Fontaine 2000, Theorem A, Proposition 4.2 and the corollary following Theorem 4.3) give semistable subrepresentations over this finite extension. The argument also applies to the descended module: the slope spaces are canonical and hence stable under descent. All embeddings above the fixed embedding have the same Hodge multiset, so the same inequalities hold after this extension.

The last quotient is unramified in its Weil part, has de Rham weight \(h_4\), and its Frobenius value is \(\alpha_4\). This proves [eq:last-character]; the polarization proves [eq:first-character]. Their values belong to the integral operator ring. To check uniform convergence on local elements, recall that \(h_4=1+H_i\) and that \(H_i\) is divisible by \((p-1)p^{M_i+c}\). Hence \[u^{H_i}\equiv1\pmod{p^{M_i}}\qquad(u\in\mathbb Z_p^\times)\] uniformly in \(u\), after the fixed choice of \(c\). This controls the cyclotomic factor on the whole local Galois group. For the unramified factor, continuous powers of the unit \(\widetilde U_3^{-1}\) define the character on the procyclic unramified quotient. Since \(\lambda_i^0(\widetilde U_3)=1\) in the full truncated ring, every such power also has image \(1\) there. Formula (69) therefore gives \(X_w\equiv\epsilon^{-1}\) at the working precision, uniformly on local elements and coefficientwise in \(t\); the polarization gives \(Y_{\bar w}\equiv\epsilon^{-2}\). These congruences hold on every sequence of local elements. ◻

Proof. At the moving place \(Q_v=q_i\) tends to \(1\) and the head Frobenius matrix tends to \(V(\gamma)\), while the binary roots tend to \(1\). The invertibility of \(V(\gamma)-1\) in 7 gives the required bound on its nonzero valuation. At a fixed potentially good place the head roots have the elliptic purity size, namely \(Q_v^{3/2}\) after the cohomological twist, whereas the final roots are \(Q_v,Q_v^2\) up to the specified fixed unitary character. The half-integral difference in weights prevents equality after an integral power of \(Q_v\). At a ramified potentially multiplicative place choose the Frobenius lift whose quadratic sign is negative; its head roots cannot equal the positive final roots after such a power. These are finitely many fixed nonzero comparisons. The weight and tame specializations converge to them, so their valuations are uniformly bounded.

The nonzero resultants exclude both coincidences and norm linkages between the two allocations. The segment classification of Bernstein and Zelevinsky (1977) and Zelevinsky (1980, Theorem 6.1 and Proposition 8.5) then factors the local representation into the unlinked head and binary representations. A segment crossing the two allocations would yield a forbidden ratio \(Q_v^d\) with \(|d|\leq4\). The binary Jacquet allocation is therefore the whole binary representation; its prescribed hyperspecial invariants make it spherical, with zero predicted monodromy.

Separation from the head still allows norm linkages within the binary pair. Since [eq:four-dimensional-compatibility] gives only monodromy domination, we use the direct-sum construction of \(\rho_\pi\) to exclude those links as well. In each individual shifted copy of a cuspidal factor \(\tau\) in [eq:four-dimensional-construction], a selected root has a length-one local segment. A longer segment either crosses the forbidden allocation boundary or lies wholly in the binary part, contradicting sphericity. Two length-one unramified data inside one unitary generic \(\tau_v\) cannot differ by \(Q_v\): their real exponents lie in an interval of width strictly less than one (Tadić 1986, Theorem 7.5) and (Zelevinsky 1980, Theorem 9.7). Thus the actual monodromy cannot link two selected roots within that copy. It cannot link different shifted copies, since the Galois construction is their direct sum; it cannot link to the complement by the resultants. This proves that actual monodromy vanishes on the selected part even though [eq:four-dimensional-compatibility] only gives an upper bound.

The semisimple Weil inertia is finite. Its fixed subspace is stable under Frobenius, and its selected generalized Frobenius primary spaces have inertia one and monodromy zero. Monic coprime factorization gives the projection onto their sum, including any Frobenius Jordan blocks. This proves [eq:binary-inertia-identity]. ◻

Proof. Iwahori invariants make semisimple Weil inertia trivial. Write the local Langlands parameter as unramified Steinberg blocks of lengths \(s_j\). The standard induction of these blocks surjects onto the local representation. Compact invariants are exact in characteristic zero, so the standard induction must have \((3,1)\) parahoric invariants. Restriction to the maximal compact and its double-coset decomposition reduces this to invariants of each Steinberg factor under the induced partial flag. The finite Weyl sign representation of a Steinberg factor has such invariants only when all its positions are separated. A single-step flag can therefore accommodate at most one block of length two, all others having length one. The predicted monodromy has rank at most one and square zero. The monodromy bound in 38 gives the same for the actual representation; its inertia matrices are \(\exp(t_p(\iota)N)\). ◻

Proof. Work first on one irreducible local representation over an algebraically closed field of characteristic zero. Realize it as a subquotient of induction from its full supercuspidal support. The geometric lemma filters the relevant block Jacquet module by at most \(4!=24\) allocation terms (Bernstein and Zelevinsky 1977, Lemma 2.12). Taking the required compact Levi invariants is exact. On the resulting finite-dimensional space, taking a joint generalized eigenspace for the commuting controlled operators is also exact. These operations therefore select a filtration of the actual operator module.

At \(p\), the unit summand was chosen before taking any reduced quotient. On every eigensystem its final slopes are \(h_3,h_4\) by 39, and the other slopes are at most \(h_2\). The two gaps and [eq:unit-root-formulas] select a unique allocation to the \((2,1,1)\) Levi. On its surviving grade the two final \(\mathop{\mathrm{GL}}_1\) center operators are scalars. At a binary control place where \(\Delta_v\ne0\), the determinant and the degree-one spherical operator determine the full quadratic \(A_v\). Separation from the head similarly selects one allocation. The hyperspecial invariants of the full unramified binary induction are one-dimensional, so all the binary operators are scalars on that grade, including when the two binary inducing characters coincide.

Repeated factors within the head create no second allocation: permutations inside that block already belong to the Levi Weyl group in the geometric-lemma double-coset indexing. No supercuspidal factor splits internally across an allocation. Thus there is exactly one nonzero grade in the selected generalized eigenspace. It equals that grade, and the controlled operators act by actual scalars there. In particular there is no residual Jordan extension on the separated locus. The good spherical operators are already scalar on each automorphic representation.

For comparison, on a joint generalized component where \(\Delta\) has eigenvalue zero, it acts by zero on every associated allocation grade and therefore lowers the filtration. A fixed power kills that component. Tensoring the filtrations at the bounded number \(r\) of control places gives, for example, the coarse bound \(24^r\). On the complementary components the kernel of eigensystem detection is zero by the preceding scalar-action argument. This proves [eq:nilpotent-annihilation] as a characteristic-zero operator identity. The integral order \(T_i^0\) is torsion-free, so the same identity holds integrally.

Finally the constant coefficient of \(\lambda_i^0(\Delta)\) has uniformly bounded nonzero valuation by 40, and all of its other coefficients are integral. Successive coefficient comparison in the truncated polynomial ring shows that multiplication by its fixed power can lose only a bounded amount of \(p\)-adic precision. More explicitly, inversion over \(k[t]/(t^{b_*})\) uses at most \(b_*\) powers of the inverse constant coefficient; the resulting denominator bound depends only on \(C,b_*\) and that valuation bound. Applying this to [eq:nilpotent-annihilation] kills the kernel modulo \(p^{M_i-O(1)}\). Renaming these still increasing precisions proves [eq:reduced-hecke-congruence]. ◻

Proof. The residual characteristic polynomials are fixed by the comparison system. After a fixed finite extension of \(L\) killing their semisimplification, the residual image in every factor is unipotent, and the full integral image is pro-\(p\). All factors are unramified outside the fixed set \(S\) and the bounded number of places over \(q_i\). The maximal pro-\(p\) Galois group of this extension with such ramification has a bounded number of topological generators: its minimal number is the dimension of \(H^1(-,\mathbb F_p)\), which the class field sequence bounds by a fixed contribution from \(S\), the units and class group, plus one bounded contribution per new place. This is the usual global Galois finiteness theorem (Neukirch et al. 2008, X). Add fixed coset representatives for the finite extension. It follows that uniformly many elements generate the matrix image topologically.

Their algebraic span is already closed, because any \(\mathcal O\)-submodule of the finite integral matrix lattice is finite and closed. Inverses are polynomial expressions with integral coefficients by the characteristic equation and the unit determinant. Hence the same bounded list generates \(B_i\) as an algebra.

Reduce modulo the maximal ideal of \(T_i\). Matrix algebras of size four satisfy the degree-eight standard polynomial identity (Amitsur and Levitzki 1950), and this identity passes to the quotient. Shirshov’s height theorem bounds the number of power factors needed to span the quotient, with base words of length less than eight; the bound depends only on the identity and the number of generators (Shirshov 1957, sec. 2, Theorem 1 and its proof). Each base word before reduction is a group image and satisfies its degree-four characteristic polynomial over \(T_i\). Reducing its powers to exponents at most three leaves a uniformly bounded list of monomials. These span \(B_i\) modulo the maximal ideal of \(T_i\). Since \(B_i\) is finite over \(T_i\), Nakayama’s lemma proves the assertion. ◻

Proof. Let \(b_{ij}\), \(1\leq j\leq N\), be the monomial generators from 43, and \(b_j\) their images in \(B\). Only the fixed finite list of matrices \(C_j=\Phi(b_j)\) is needed. One denominator clears all coefficients of all \(C_j\) in the fixed basis of \(R_b\).

For any increasing precision \(s_i\), the compact surjection \(T_i^N\to B_i\) induces a surjection on the finite quotients modulo \(p^{s_i}\). Choose a section on that finite set. Composing with \(\rho_i\) produces locally constant coefficient functions \(a_{ij}(g)\in T_i\) such that \[\rho_i(g)\equiv\sum_{j=1}^N a_{ij}(g)b_{ij} \pmod {p^{s_i}B_i}.\] Apply [eq:reduced-hecke-congruence] and multiply the coefficients by successively accurate approximations to the fixed \(C_j\). The resulting continuous matrix functions have the one denominator already chosen, and their limit is \(\Phi\rho_*\) by \(T\)-linearity. The precision may be slowed to the minimum of \(s_i\) and \(M_i\) minus this fixed loss; the bound \(M_i\leq n_i-c_b\) already respects the tame relation. The variable element \((p^{s_i})_i\) has image zero under \(\lambda\) in every coefficient of \(t\), so the remainders vanish. Although the constant element \(p\) is a unit in \(T\), this does not change the assertion about the variable sequence.

If a matrix identity holds for all sequences but fails uniformly to increasing precision, choose a violating local element at each of a filter-large set of stages. That sequence contradicts the identity in the limit. The same argument applies to local splittings after choosing the finitely many splitting coefficients. In particular no coordinatewise lifting of the henselian idempotents is required. ◻

Proof. Each signed determinant monomial has entries directed from its column index to its row index. At every vertex the incoming and outgoing counts agree except for a path from \(y\) to \(u\). Decompose the directed graph into that path and cycles. Compose the path in \(D_{uy}\); each cycle is a scalar in its rank-one diagonal corner and hence in \(T\). The inverse matrix units identify the two \(f\) positions whenever needed. This gives the claimed element. In a rank-one matrix realization it is exactly the usual product of entries, independent of the chosen bases, since the intermediate basis changes cancel. Faithfulness in \(M_4(\mathcal R)\) and reducedness of \(\mathcal R\) prove the vanishing assertion. The construction uses sums and corner products, so is preserved by algebra homomorphisms. ◻

Proof. Let \[Q_{xy}=D_{xy}/(D_{xf}D_{fy}+\mathfrak mD_{xy}).\] A \(k\)-linear functional \(Q_{xy}\to k\) defines a triangular representation, retaining the separate \(f\) block and an extension of \(y\) by \(x\). Products through \(f\) have been killed, and other crossing products vanish by [eq:crossing-cycles]; thus the triangular assignment is an algebra map out of \(B\). It yields an injection \[ \mathop{\mathrm{Hom}}_k(Q_{xy},k)\hookrightarrow H^1_{\mathrm{adm}}(\Gamma,k(1)). \tag{81}\] Indeed, a split Galois extension has a splitting preserved by the algebra generated by the group, hence by \(B\). The idempotents then force this splitting to be the diagonal \(y\) line. Its invariance annihilates the whole functional. Admissibility follows from 44.

We verify the strict local conditions on every class in [eq:xy-cohomology]. At a conjugate place \(\bar w\mid p\), set \(S_g=\rho_*(g)-Y_{\bar w}(g)\) for local sequences \(g\). In every characteristic-zero system these matrices have a common kernel by 39. Take four rows of their stack with types \(f,f,x,x\). The four-row determinant is an element of \(D_{xy}\) by 45, and it is zero by the common kernel. Its identity therefore holds in the triangular quotient.

On the three competing columns \(f_1,f_2,x\), one may choose two \(f\) rows and one \(x\) row with an invertible \(3\times3\) minor. Here the inverse matrix units of the \(f\) block transport the selected coordinate rows to the two required output positions. Writing these units as \(E_{ab}\), the rows can be represented by \(e_1E_{1a}S_g\) and \(e_2E_{2b}S_h\), with \(a,b\in\{1,2\}\) allowed to coincide. These operations preserve the common kernel and the corner identities of 45. For the \(f\) part this follows from \[\mathop{\mathrm{Hom}}_{G_{L_{\bar w}}}(y,f) =H^0(L_{\bar w},V)=0;\] there is no common kernel, so finitely many rows span the two-column dual. The tame character is trivial locally at these fixed places. For the \(x\) row use \(x\ne y\) on the local group. Normalize the \(y\) coordinate to one and solve these three equations for the other coordinates. Each additional \(x\) row is consistent with this solution because its four-row determinant is zero. The \(f\) coordinates are zero, all the remaining \(f\) equations already hold, and the \(y\) rows are zero. Thus a common \(y\)-eigenline exists in the triangular extension: its restriction at \(\bar w\) splits.

At \(w\in\Sigma_p\), use the common quotient of character \(X_w\) in 39 and apply the transposed minor argument. The competing-column condition becomes \[\mathop{\mathrm{Hom}}_{G_{L_w}}(f,x)=H^0(L_w,V)=0,\] using \(V\simeq V^*(1)\), together with \(x\ne y\). The resulting quotient of character \(x\) splits the same extension of \(y\) by \(x\). [eq:last-character,eq:first-character] supply the required characters on all local elements. This proves strictness at every place above \(p\).

Away from \(p\) the extension is unramified, as follows. Outside the controlled places this holds for the original representations. At a binary control place, the image of \(\Delta_v\) is a unit in \(T\). The coprime factorization of the full Frobenius polynomial therefore defines \(P_v^{\mathrm{bin}}\) over \(B\), and the identity [eq:binary-inertia-identity] holds there by faithful field testing. In the triangular representation this projection is the identity on the \(x,y\) block: both of its Frobenius roots are in \(A_v\) and are separated from the \(f\) roots. Hence inertia acts trivially on that extension. This includes the moving places \(q_i\).

At a Steinberg control place, the rank bound of 41 gives two-by-two minors of \(\rho_*(\iota)-1\) with rows \(f,x\) and columns \(f,y\), which are zero in \(D_{xy}\). In the triangular representation they say \[(f(\iota)-1)_{ab}\,u(\iota)=0\] for every matrix entry and every inertia element, where \(u\) is the extension entry. The comparison elliptic Steinberg representation has a nonzero actual inertia difference for some element \(h\). Thus \(u(\iota)=0\) whenever \(f(\iota)-1\ne0\), in particular for \(h\). If \(f(\iota)=1\), apply the same assertion to \(\iota h\) and use the cocycle equation and the unramifiedness of \(x,y\) to obtain \(u(\iota)=u(\iota h)-u(h)=0\).

By admissibility and unramifiedness at the moving places, the class now descends to ordinary continuous cohomology over \(L\), by 9. Its coefficients are \(k(1)\), it is unramified away from \(p\), and it is zero locally at all \(p\)-places. Kummer theory identifies such classes with \(p\)-units tensored with \(k\), subject first to zero valuations at the \(p\)-places and then to zero local logarithms. The valuation condition leaves global units. The CM biquadratic field \(L\) has unit rank one, and a nontrivial fundamental unit has nonzero \(p\)-adic logarithm at a split \(p\)-place: if its image in \(\mathbb Q_p^\times\) had logarithm zero, being a unit it would be a root of unity, and injectivity of the field embedding would make the global unit torsion. Thus the local logarithm is injective on this one-dimensional space, and the class vanishes. We have shown \(Q_{xy}=0\). Nakayama’s lemma for the finite module \(D_{xy}/D_{xf}D_{fy}\) proves [eq:xy-factorization]. ◻

Proof. Suppose its image vanished at such a field. By [eq:xy-factorization], \(D_{xy}\) also vanishes there, so the one-dimensional \(e_y\)-space is globally stable. We will obtain a rank-one geometric character with conjugate weight sum three. The residual corner character \(y=\epsilon^{-2}\) has weight sum four. Evaluating on a fixed pair of conjugate local units will turn this discrepancy into a scalar unit that would have to vanish in the ambient field.

We first identify the stable \(e_y\)-space at a conjugate place \(\bar w\mid p\) with the first line in 39. Choose the two \(f\) rows and one \(x\) row used above from the stack \(\rho_*(g)-Y_{\bar w}(g)\), and project their columns to the complement of \(e_y\). Their \(3\times3\) determinant is a scalar in \(T\) and is a unit: its reduction is the invertible minor in \(f\oplus x\oplus y\). On a filter-large set of actual systems, the common kernel is consequently a unique line with nonzero \(e_y\) coordinate. Once \(e_y\) is stable, all competing entries in its column are zero. The invertible complementary minor therefore forces the existing common kernel of the matrices \(\rho_*(g)-Y_{\bar w}(g)\) to be the \(e_y\) line. Its character is thus \(Y_{\bar w}\).

Here these assertions really do refer to arrays of actual characteristic-zero systems. Stability need only be checked on the uniformly bounded list of module generators in 43. The rank, inverse matrix-unit relations and the competing-column condition also use finitely many matrices. Include also \(\rho_*(g)\) and \(\rho_*(g^c)\) for the fixed unit element selected below; this only enlarges that finite list. Elements introduced by henselization can be represented in the algebraically closed residue ultraproduct of [eq:ambient-ring], with their finite relations. Vanishing or nonvanishing of each of these finitely many matrix expressions then holds on a filter-large set of actual systems. Thus at almost every such system its semisimple global representation has a one-dimensional constituent whose local character is the prescribed \(Y_{\bar w}\). This passage uses finite matrix equations, not an assertion about infinitely many coordinatewise projectors.

Denote this actual character by \(\xi\). It is de Rham above \(p\), hence Hodge–Tate with semisimple rank-one inertia, and therefore locally algebraic (Serre 1968, III, Section 1.2 and Appendix A7, Theorem 3). Global class field theory and the Serre-group construction then identify it with the \(p\)-adic realization of an algebraic Hecke character; see Serre (1968, III, Sections 2.2–2.3, including the proof, and Chapter II, Section 2.7). Here the construction is over the coefficient field. The sums of its weights at conjugate embeddings are parallel: apply its infinity type to the global units of the real subfield. Write \(w(\xi)\in\mathbb Z\) for this common sum.

The character occurs in a shifted copy of a cuspidal factor \(\tau[a]\) in [eq:four-dimensional-construction]. If \(a>1\), the same rank-one constituent occurs in all its consecutive shifts. Its weight \(3-h_4\) at \(\bar w\) would then have an adjacent integer weight in the full representation. This is impossible: it is the lowest weight there, separated from the next by \(h_4-h_3\), which was chosen arbitrarily large. Hence \(a=1\). At every unramified place, the roots of a unitary generic cuspidal local representation have absolute values strictly between \(Q^{-1/2}\) and \(Q^{1/2}\) (Tadić 1986, Theorem 7.5) and (Zelevinsky 1980, Theorem 9.7). After the cohomological shift in [eq:four-dimensional-compatibility], the value of \(\xi\) has absolute size strictly between \(Q\) and \(Q^2\). On the other hand, for an algebraic Hecke character with parallel conjugate weight sum \(w(\xi)\) its absolute value is \(Q^{w(\xi)/2}\): dividing out this norm power leaves a unitary character. Consequently \[ w(\xi)=3. \tag{82}\]

There is a uniform open subgroup of the local units above \(p\) on which the smooth factors of all these characters are trivial. Indeed the compact open level \(J\) at \(p\) is fixed. Bernstein’s decomposition theorem gives finitely many inertial cuspidal-support classes among representations with \(J\)-invariants (Bernstein et al. 1986, sec. 2.3); local Langlands therefore gives finitely many semisimple inertia types. The smooth factor of a rank-one constituent belongs to this finite list. This bounds its conductor independently of the growing algebraic weights. Choose in this fixed open subgroup an element \(g\) with \(\epsilon(g)\ne1\), deep enough that \(\epsilon(g)\) is not torsion, and use its conjugate \(g^c\) at the opposite place. [eq:rank-one-weight] implies at every actual constituent under consideration \[ \xi(g)\xi(g^c)=\epsilon(g)^{-3}. \tag{83}\] Thus it implies the same identity in the evaluated ambient field.

But the scalar corner element \[(e_y\rho_*(g)e_y)(e_y\rho_*(g^c)e_y) -\epsilon(g)^{-3}e_y\] is a unit in \(T e_y\). Its reduction in the residual diagonal representation is \((\epsilon(g)^{-4}-\epsilon(g)^{-3})e_y\ne0\), since the residual \(y\) is \(\epsilon^{-2}\) and cyclotomic is invariant under conjugation. A unit cannot vanish in a field evaluation, contradicting [eq:weight-three-unit]. The assumed vanishing of \(D_{fy}\) is impossible. ◻

Proof. If \(a\in T\) annihilates \(D_{fy}\), then its image at every ambient field is zero by 47. Since \(T\) embeds in the reduced ring \(\mathcal R\), this gives \(a=0\). For a finite module over an arbitrary commutative ring its zeroth Fitting ideal is contained in its annihilator: choose finitely many generators and apply the adjugate identity to each square matrix of relations. This does not require a finite presentation. It proves [eq:fitting-vanishing]. ◻

Proof. Put \(M=D_{fy}\otimes_{T,\lambda}R_b\). A functional \(M\to R_b\) gives a triangular extension of \(y\) by \(f\) while retaining the separate \(x\) block. We check the only additional product relation. The path from \(y\) through \(x\) to \(f\) has image in \[D_{fx}D_{xy}=D_{fx}D_{xf}D_{fy}\] by [eq:xy-factorization]. Its first two factors form a crossing diagonal cycle, killed by [eq:crossing-cycles] after applying \(\lambda\). All other discarded products are killed by the same cycle relation or by the diagonal representation. Hence the triangular assignment is an algebra homomorphism out of \(B\). As in [eq:xy-cohomology], a split extension must have its splitting preserved by \(B\) and then by its idempotents; it forces the functional to be zero. This proves injectivity into extension cohomology. Moreover \[\mathop{\mathrm{Hom}}(y,f)=V\otimes\psi^{\pm1},\] and 44 proves admissibility with coefficients in the full ring \(R_b\).

At a conjugate \(p\)-place, use the same common-kernel stack \(S_g=\rho_*(g)-Y_{\bar w}(g)\). Choose four rows of types \(f,f,x,f\), with either of the two \(f\) positions as the final row. By 45 the vanishing four-row determinants are identities in the corresponding entry of \(D_{fy}\), and they survive in the triangular extension. The first two \(f\) rows and one \(x\) row give a \(3\times3\) minor on the competing columns whose residue is invertible by \(H^0(L_{\bar w},V)=0\) and \(x\ne y\). It is therefore a unit over \(R_b\). Solve these equations with \(y\) coordinate one. Varying the final \(f\) row proves that every other \(f\) equation is satisfied. The \(x\) coordinate is zero and the \(y\) rows vanish. This produces a common \(y\)-eigenline and splits the local extension over the full Artin ring, including its nilpotent coefficients.

The Greenberg condition is full at \(\Sigma_p\), so no splitting is required there; it is relaxed at the remaining fixed bad places. Outside those places and the moving places the representations are unramified, as required for admissible classes. Thus the preceding local splitting is precisely the one-sided condition of 12, and gives [eq:green-injection].

Fitting ideals commute with base change for finite modules, so [eq:fitting-vanishing] gives \(\mathop{\mathrm{Fitt}}_{0,R_b}(M)=0\). The finite module \(M\) over the principal Artin ring \(R_b\) has an elementary-divisor decomposition \[M\simeq\bigoplus_j R_b/(t^{a_j}), \qquad 1\leq a_j\leq b.\] Its zeroth Fitting ideal is \((t^{\sum_j a_j})\), with the evident interpretation for free summands \(a_j=b\). Vanishing is equivalent to \(\sum_j a_j\geq b\). Each summand and its \(R_b\)-dual have length \(a_j\), since \(\mathop{\mathrm{Hom}}_{R_b}(R_b/(t^{a_j}),R_b)=(t^{b-a_j})\). Thus the source in [eq:green-injection] has length at least \(b\). ◻

Proof. We will express each normalized factor as the same scalar times \(I(T_i)\). First, the exponential sequence for the complex dual torus is \[0\longrightarrow Y\longrightarrow Y\otimes_{\mathbb Z}\mathbb C \longrightarrow\widehat T(\mathbb C)\longrightarrow 1.\] The Galois action factors through a finite group. Taking its invariants, and averaging on the complex vector space, identifies the component group of \(\widehat T^{\Gamma_K}\) with \(H^1(K,Y)\). Indeed, the image of the invariant vector space is precisely the identity component of the fixed torus. Thus \[ b(T)=|H^1(K,Y)|. \tag{87}\]

We specify the measure comparison. Choose invariant top differential forms \(\omega_i\) over \(K\) with \(\ell^*\omega_2=\omega_1\). Compatibility of the local Haar measures means \[\eta_{i,v}=a_v|\omega_i|_v\] with the same positive constant \(a_v\) for \(i=1,2\). Put \(d=\dim T_i\). Let \(P_v(T,t)\) be the local Artin polynomial and \(\rho(T)=\lim_{s\to1}(s-1)^rL(T,s)\), in the conventions of Geisser and Suzuki (2022, sec. 6). The rational Galois-module isomorphism gives the same \(d,r,P_v\), and \(\rho\) for both tori. In these conventions the Tamagawa measure is \[ \mu_i=\frac{1}{\rho\,q^{(g-1)d}} \prod_v P_v(q_v^{-1})^{-1}|\omega_i|_v, \qquad q_v=|k(v)|. \tag{88}\] Consequently \(\eta_i=\alpha\mu_i\) with the same scalar \[\alpha=\rho\,q^{(g-1)d}\prod_v a_vP_v(q_v^{-1}).\] At almost all places the forms are integral generators and the tori have good reduction; there the volume of the integral compact under \(|\omega_i|_v\) is \(P_v(q_v^{-1})\). The displayed product therefore has only finitely many factors different from \(1\). Replacing a global form by a \(K^\times\)-multiple does not change the global measure, by the product formula. This also explains why clearing denominators in a rational torus correspondence adds no unrecorded global scalar.

For clarity, the function-field Tamagawa number includes a degree correction. The definition in Geisser and Suzuki (2022, sec. 6) and their Proposition 6.3 give \[\operatorname{vol}_{\mu_i} \bigl(T_i(K)\backslash T_i(\mathbb A_K)^1\bigr) =(\log q)^r I(T_i)\tau(T_i), \qquad \tau(T_i)=\frac{|H^1(K,X^*(T_i))|}{|\operatorname{Sha}^1(K,T_i)|}.\] Here their discriminant \(\operatorname{Disc}(h_{T_i})\) is exactly \(I(T_i)\): the degree pairing identifies the torsion-free adelic class group with the image lattice of \(\deg_{T_i}\) in \(\operatorname{Hom}(X^*(T_i)^{\Gamma_K},\mathbb Z)\). Together with (87), this proves the more explicit formula \[ \frac{c_{\eta_i}(T_i)}{b(T_i)} =\alpha(\log q)^r I(T_i), \tag{89}\] and hence (86).

Finally suppose \(T\) splits at a degree-one place \(v_0\). The invariant sublattice \(Y^{\Gamma_K}\) is saturated in \(Y\): if \(ny\) is invariant for \(n\ne0\), torsion-freeness of \(Y\) makes \(y\) invariant. Therefore restriction gives a surjection \[X_*(T_{K_{v_0}})=\operatorname{Hom}(Y,\mathbb Z) \longrightarrow\operatorname{Hom}(Y^{\Gamma_K},\mathbb Z).\] For a local cocharacter \(\lambda\) and uniformizer \(\varpi_{v_0}\), the adele equal to \(\lambda(\varpi_{v_0})\) at \(v_0\) and to \(1\) elsewhere has degree \(\chi\mapsto\langle\chi,\lambda\rangle\). Thus the degree map is onto and \(I(T)=1\). ◻

Proof. We apply the general transfer principle for parameterized integrals (Cluckers et al. 2011, Theorems 2.8.1–2.8.3). To do so, we construct a common parameter space for the paired integrals, express their measures by differential forms, and remove the volume and discriminant normalizations. The resulting identity will be the vanishing of one constructible function.