Proof. Choose a finite Galois extension \(K/\mathbb{Q}\) containing the coefficients of \(t\). For \(a\in K\), the vector \[T_a=\mathop{\mathrm{Tr}}_{K/\mathbb{Q}}(at) =\sum_{\sigma\in\mathop{\mathrm{Gal}}(K/\mathbb{Q})}\sigma(a)\sigma(t)\] is rational. Each automorphism of \(K\) extends to \(\overline{\mathbb{Q}}\), so every summand belongs to \(P\); hence \(T_a\in T\cap P\). If \(b_1,\ldots,b_r\) is a rational basis of \(K\) and \(b_1^*,\ldots,b_r^*\) is its trace-dual basis, write \(t=\sum_jt_jb_j\) with \(t_j\in T\). Then \(T_{b_j^*}=t_j\), which proves the assertion. The condition that a complex-linear map have a fixed Hodge type is complex-linear, so the same argument applies in a Hom space. ◻

Proof. Dualizing a rational Hodge map gives a rational map on homology commuting with the complex structures. Multiplication by a positive integer makes it carry the integral lattice of \(A\) into that of \(B\). It consequently induces a holomorphic homomorphism of complex tori \(A^{\mathrm{an}}\to B^{\mathrm{an}}\), which is algebraic. Conversely, an algebraic homomorphism induces the indicated Hodge pullback. This proves the first assertion; equivalently it is the dual of the rational form of Riemann’s theorem (Milne 2005, Proposition 6.5, Theorems 6.7–6.8, and Corollary 6.9).

The Albanese map of \(S\) induces an isomorphism \(H^1(\mathop{\mathrm{Alb}}S,\mathbb{Q})\simeq H^1(S,\mathbb{Q})\). Apply the first assertion to the map with target \(H^1(\mathop{\mathrm{Alb}}S,\mathbb{Q})\), clear a denominator, and compose the resulting homomorphism \(\mathop{\mathrm{Alb}}S\to B\) with an Albanese map. ◻

Proof. The zero-dimensional case is immediate. Put \(H=H^1(A,\mathbb{Q})\), and first assume that \(J(H)\) is a torus. Its associative envelope \(E_0\subset\mathop{\mathrm{End}}_{\mathbb{Q}}(H)\) is a commutative semisimple algebra: after splitting the torus, its elements act diagonally by scalars on the weight spaces, so the envelope is a subalgebra of a product of fields and is reduced. This description descends to \(\mathbb{Q}\).

Let \(*\) be the polarization adjoint. It preserves \(E_0\), since it sends every symplectic torus element to its inverse. Every element of \(E_0\) commutes with the complex structure \(I\) supplied by the Hodge circle. Consequently \(*\) is also the adjoint for the positive definite real form obtained by inserting \(I\) in the polarization. In particular, \[\mathop{\mathrm{Tr}}_{H_{\mathbb{R}}}(a^*a)>0\qquad(0\ne a\in E_0\otimes_{\mathbb{Q}}\mathbb{R}).\] The involution cannot exchange two primitive factors of the real finite étale algebra \(E_0\otimes\mathbb{R}\): for the idempotent \(e\) of one factor, this would give \(e^*e=0\). On a complex factor it cannot be the identity, since for its idempotent \(e\) one would have \((ie)^*(ie)=-e\), of negative trace. Thus it acts as complex conjugation on each complex factor. There are no real factors. On the nonzero module of such a factor the circle would act by real scalars, and a continuous homomorphism from the circle to \(\mathbb{R}^\times\) is trivial; this contradicts the weight-one characters.

It follows that each rational field factor \(K\) of \(E_0\) is CM. Indeed, it has only complex archimedean factors, and its involution satisfies \(\tau(a^*)=\overline{\tau(a)}\) at every embedding. Its fixed field is therefore totally real and has index two. On the corresponding \(K\)-module the circle acts by \(K\otimes\mathbb{R}\)-scalars. Splitting that module into \(K\)-lines thus gives rational Hodge substructures. This proves (ii), without discarding multiplicities.

The separate scalar actions on the field lines in (ii) form a commutative semisimple Hodge-endomorphism algebra of dimension \(\dim_{\mathbb{Q}}H\). Lemma 3 realizes it in \(\mathop{\mathrm{End}}(A)\otimes\mathbb{Q}\), proving (iii). Conversely, write an algebra as in (iii) as \(\prod_jK_j\). Its faithful action on \(H\) has a positive rank \(m_j\) over each \(K_j\), and \[\dim_{\mathbb{Q}}H=\sum_jm_j[K_j:\mathbb{Q}]=\sum_j[K_j:\mathbb{Q}].\] Hence every \(m_j=1\). The centralizer of this action in \(\mathop{\mathrm{GL}}(H)\) is the torus \(\prod_j\mathop{\mathrm{Res}}_{K_j/\mathbb{Q}}\mathbf G_m\). It contains the Hodge circle because the algebra acts by Hodge endomorphisms, proving (i). ◻

Proof. Scalar conjugation transports the entire source and target decompositions. For each \(\sigma\), the only possibly nonzero component of \(\sigma T\) maps \(\sigma L\) to \(\sigma M\), whose types agree by hypothesis. All other components are zero. Thus \(\sigma T\) is a Hodge map on the whole source. Apply Lemma 2 in \(\mathop{\mathrm{Hom}}_{\mathbb{Q}}(H,H')\), followed, when applicable, by Lemma 3. ◻

Proof. Let \(A\) be a CM abelian variety and let \(h\in\mathop{\mathrm{Hdg}}^{2p}(A)\). The cases \(p=0\) and \(p=1\) follow from the unit class and the Lefschetz \((1,1)\) theorem, respectively. For higher codimension, we first replace a Hodge monomial by a tensor on auxiliary CM factors. We then connect its two lists of types by four-factor switches and compose the resulting algebraic tensors.

By Lemma 4, choose a rational decomposition of \(H=H^1(A,\mathbb{Q})\) into rank-one CM-field modules, retaining separate copies of repeated summands. Choose a Galois CM field \(E\) containing all the fields and their conjugates. Over \(E\), \(H\) is the direct sum of its embedding lines. Since cup product gives \(H^*(A,\mathbb{Q})=\Lambda^*H\), the exterior monomials in these lines give a direct-sum decomposition of \(H^*(A,\overline{\mathbb{Q}})\) into one-dimensional lines (Milne 2005, Proposition 6.4). Every such line is contained in one Hodge bidegree.

Scalar conjugation permutes the monomial lines, with only the possible sign from reordering exterior factors. Because \(h\) is rational, a nonzero component on one monomial line gives a nonzero component on every line in its scalar-Galois orbit. All these lines must have type \((p,p)\). The decomposition is defined over \(E\), so the components of \(h\) have algebraic Betti coefficients.

Fix a supported monomial and list the complete type functions of its \(2p\) factors as \(z_1,\ldots,z_{2p}\). Each \(z_i\) is odd. In the conjugate indexed by \(g\), the number of \((1,0)\) factors minus the number of \((0,1)\) factors is \(\sum_i z_i(g)\). Its balance is therefore exactly \[\sum_{i=1}^{2p}z_i=0.\] For \(1\leq i\leq p\), set \(x_i=z_i\) and \(y_i=-z_{p+i}\). Then \(\sum_i x_i=\sum_i y_i\). The first \(p\) factors have the complete types of \(U(x_i)_1\), and the last \(p\) have those of \(U(y_i)_c\). For any ordered list \(a=(a_1,\ldots,a_p)\) of odd sign functions, put \(A_a=\prod_{i=1}^pB(a_i)\); for two such lists put \[\mathscr L_{a,b}= \bigotimes_{i=1}^pU(a_i)_1\ \otimes\ \bigotimes_{i=1}^pU(b_i)_c \subset H^{2p}(A_a\times A_b,\overline{\mathbb{Q}}).\] Repeated factors in these products are kept as separate slots. Lemma 5 reduces the chosen monomial to the algebraicity of \(\mathscr L_{x,y}\). We return to \(A\) after composing the transition tensors.

Regard each list as a matrix with \(p\) rows and one column for each representative of \(G_E/\langle c\rangle\). At any column, equality of the two sums is equality of the numbers of plus signs. Among the rows where the current \(x\)-entry differs from the desired \(y\)-entry, there are equally many mismatches of the forms \((+1,-1)\) and \((-1,+1)\). Pair these two sets of rows and swap the two current entries in every pair, making the opposite swap at the conjugate coordinate. This corrects the column, leaves the other columns unchanged, and preserves oddness of every row. Successively correcting the columns gives a finite sequence \[x=x^{(0)},\ x^{(1)},\ \ldots,\ x^{(N)}=y\] in which consecutive lists differ by one such two-row swap.

At a single swap the old rows \(a,b\) and new rows \(a',b'\) satisfy \(a+b=a'+b'\) at every coordinate. Theorem 6 supplies the pure algebraic tensor \[U(a)_1\otimes U(b)_1\otimes U(a')_c\otimes U(b')_c.\] Every unchanged row is joined to its new copy by \(U(v)_1\otimes U(v)_c\), an all-conjugate \((1,1)\) line and hence an algebraic two-factor tensor. Pull these classes to their distinct old and new slots and multiply. This gives a nonzero algebraic generator of \(\mathscr L_{x^{(j)},x^{(j+1)}}\). If the original lists already agree, the two-factor tensors alone generate \(\mathscr L_{x,x}\). It remains to compose consecutive transition tensors without losing their algebraicity or nonvanishing.

For three ordered lists \(a,b,e\), suppose that \(\mathscr L_{a,b}\) and \(\mathscr L_{b,e}\) have pure algebraic generators with algebraic Betti coefficients. Write them as \[\begin{split} t_{ab}&=\alpha_1\otimes\cdots\otimes\alpha_p\otimes \beta_1\otimes\cdots\otimes\beta_p,\\ t_{be}&=\gamma_1\otimes\cdots\otimes\gamma_p\otimes \delta_1\otimes\cdots\otimes\delta_p, \end{split}\] where \(\beta_i\in U(b_i)_c\) and \(\gamma_i\in U(b_i)_1\). Identify the middle copies and choose an ample divisor class \(\ell_i\) on each \(B(b_i)\). With \(r_{ab},r_{be},r_{ae}\) denoting the projections from \(A_a\times A_b\times A_e\), and with the \(\ell_i\) also denoting their pullbacks, cup product and pushforward give \[ \begin{split} (r_{ae})_*\left( r_{ab}^*t_{ab}\smile r_{be}^*t_{be} \smile\prod_{i=1}^p\ell_i^{\,s-1}\right) &= (-1)^{p(p-1)/2}\left(\prod_{i=1}^p q_i\right)t_{ae},\\ q_i&=\int_{B(b_i)}\beta_i\wedge\gamma_i\wedge\ell_i^{\,s-1}, \end{split} \tag{6}\] where \(t_{ae}=\alpha_1\otimes\cdots\otimes\alpha_p\otimes \delta_1\otimes\cdots\otimes\delta_p\). The sign is obtained by regrouping \(\beta_1\cdots\beta_p\gamma_1\cdots\gamma_p\) into \(\beta_1\gamma_1\cdots\beta_p\gamma_p\) in the indicated slot order.

Each \(q_i\) is nonzero. Complex conjugation exchanges the two middle embedding lines, so \(\gamma_i\) is a nonzero scalar multiple of \(\bar\beta_i\). Hodge–Riemann positivity for the ample class makes the pairing with \(\bar\beta_i\) nonzero, whether \(\beta_i\) is holomorphic or antiholomorphic. This requires no compatibility between \(\ell_i\) and \(E\). Moreover \(q_i\in\overline{\mathbb{Q}}^\times\), since its vectors have algebraic Betti coefficients and \(\ell_i\) is rational. The codimension on the left of Equation (6) is \[p+p+p(s-1)-ps=p.\] Thus it is an algebraic class of the required degree, and division by the nonzero algebraic coefficient shows that the endpoint line is algebraic. Iterating along the sequence of lists proves algebraicity of \(\mathscr L_{x,y}\).

Finally choose the line transfers in the direction from the auxiliary \(U\)’s to \(H^1(A,\mathbb{Q})\). Each is an algebraic-coefficient linear combination of rational pullbacks along maps from \(A\) to the corresponding \(B(v)\). Apply their exterior product on \(A^{2p}\) and pull back along the diagonal \(A\to A^{2p}\). The result is the cup product of the selected original factors. They occupy distinct embedding or multiplicity lines in the original exterior monomial, so this cup product is nonzero. We have proved that every supported monomial line is algebraic. Summing the components of \(h\) and using Equation (1) proves the rational assertion.

Finite products are covered because they are themselves CM, as observed after Lemma 4. ◻

Proof. If \(v_1\) and \(v_2\) are nowhere simultaneously positive, oddness also rules out simultaneous negativity. Hence \(v_2=-v_1\). The relation gives \(v_4=-v_3\), and the first two and the last two slots of Equation (4) are separately all-conjugate \((1,1)\) pairs.

Otherwise choose \(g\in G_E\) at which \(v_1(g)=v_2(g)=1\). The relation makes all four signs positive there. Choose a scalar automorphism extending \(g\). It sends the labels in Equation (4) to \(g,g,gc,gc\). The complete type function of the line labeled \(g\) is the right translate \[v_i^g(\tau)=v_i(\tau g).\] It is odd, satisfies \(v_i^g(1)=1\), and the four translated functions satisfy the same relation. Centrality of \(c\) in \(G_E\) makes the complete type of the \(gc\)-line equal to \(-v_i^g\). The transfers of Lemma 5 therefore carry the normalized switch for the \(v_i^g\)’s to the lines labeled \(g,g,gc,gc\) on the original factors. Applying the inverse scalar automorphism gives Equation (4).

When \(s=1\), there is only one odd sign function positive at the identity, so the normalized tensor is a product of ordinary pairs. ◻

Proof. Write \(\tau(\delta)=it_\tau\), where \(t_\tau\in\mathbb{R}^\times\). At the real place represented by \(\tau\in\Sigma\), write \(\psi_{P,\tau}\) for the complex bilinear extension of the real-place form. Our positivity convention is \[\psi_{P,\tau}(u,\overline u)/i>0 \qquad\bigl(0\ne u\in (\mathcal F_P)_\tau\oplus(\mathcal F_P)_{c\tau}\bigr).\] On the \(\tau\) summand its sign is the sign of \(t_\tau h_Vh_P\); on the \(c\tau\) summand the sign is the opposite. At \(\mu\), choose the coefficient of each line form to have sign \(-\operatorname{sgn}(t_\mu)\). The negative line \(L\) on the \(\mu\) side and the dual of the positive plane on the \(c\mu\) side then both have positive sign. At a definite place \(\tau\ne\mu\), choose the line coefficient to have sign \[-d_i(\tau)\operatorname{sgn}(t_\tau).\] The chosen full summand has positive sign in either case. These finitely many strict sign conditions on an element of \(F\) are simultaneously realizable by weak approximation.

At the active pair the two subspaces are annihilators under the complexified pairing; at every other pair one side is full and the other zero. Thus \(\mathcal F_P\) is isotropic of half the total dimension. The active annihilator description proves holomorphy, and the compact-place summands are constant over \(D\). Equivariance follows from the action on \(L\) and its annihilator at \(\mu\), and from preservation of the full embedding summands at the compact places.

Finally \[\det Q^*=L\otimes\det(V_\mu)^*.\] For general \(P\) the active determinant is therefore \[\det(L\otimes P_\mu)\otimes\det(Q^*\otimes P_\mu^*) \simeq L^{2n}\otimes\bigl(\det(V_\mu)^*\bigr)^{\otimes n} \otimes\det(P_\mu)^*.\] All factors other than \(L^{2n}\), including the determinants at the compact places, are constant lines. Their special-unitary actions are trivial, because they involve powers of determinants of \(V_\tau\) or \(V_\tau^*\), while \(G\) acts trivially on \(P\). This proves the first isomorphism in (17); choosing a square root of the constant line gives the second. ◻

Proof. We first check the infinite formula, keeping track of its signs. At one real place write \[f_{Z,z}(x)=\exp(\pi ix^tZx+2\pi ix^tz),\qquad K(g,Z)=J^{-1}C.\] The matrix \(K(g,Z)\) is symmetric; this follows from \(JC^t=CJ^t\), a symplectic identity. The formula to prove is \[ \omega_\infty(\widetilde g)f_{Z,z} =j(g,Z)^{-1}\exp(-\pi iz^tK(g,Z)z) f_{gZ,J^{-t}z}. \tag{24}\] For symmetric \(B\), the upper unipotent \(n(B)=\left(\begin{smallmatrix}I&B\\0&I\end{smallmatrix}\right)\) acts by multiplication by \(\exp(\pi ix^tBx)\), which gives (24) with \(j=1\). For \(m(A)=\mathop{\mathrm{diag}}(A,A^{-t})\), the chosen lift acts by \[\omega_\infty(\widetilde m(A))b(x) =j(m(A))^{-1}b(A^tx),\qquad j(m(A))^2=(\det A)^{-1}.\] For \(\det A>0\) the positive choice has \(j(m(A))^{-1}=|\det A|^{1/2}\). For negative determinant the metaplectic phase is retained in \(j\). Since \(m(A)Z=AZA^t\) and \(J^{-t}z=Az\), this is the required formula.

For \(w=\left(\begin{smallmatrix}0&-I\\I&0\end{smallmatrix}\right)\), choose the lift \[b\longmapsto e^{-\pi ia/4} \int_{\mathbb{R}^a}b(y)e^{-2\pi ix^ty}\,dy.\] The Gaussian integral is \[\int_{\mathbb{R}^a}f_{Z,z}(y)e^{-2\pi ix^ty}\,dy =\det(-iZ)^{-1/2} \exp\!\bigl(-\pi i(x-z)^tZ^{-1}(x-z)\bigr),\] where the square root of \(\det(-iZ)\) equals \(1\) at \(Z=iI\). Taking \[j(w,Z)=e^{\pi ia/4}\det(-iZ)^{1/2}\] gives \(j(w,Z)^2=\det Z\) and proves (24) for \(w\).

These generators give the formula for the whole real symplectic group. To check that the displayed exponent also composes correctly, let \(J_2=J(g_2,Z)\) and \(J_1=J(g_1,g_2Z)\). The symplectic identities give \[J(g_1g_2,Z)=J_1J_2,\qquad K(g_1g_2,Z)=K(g_2,Z)+J_2^{-1}K(g_1,g_2Z)J_2^{-t}.\] For the second identity, use \(A_2-(g_2Z)C_2=J_2^{-t}\) to write \(C_{12}=C_1J_2^{-t}+J_1C_2\), and multiply by \((J_1J_2)^{-1}\). Together with the metaplectic group law for \(j\), this is exactly the composition law in (24). Multiplying the formula over the real places gives its infinite adelic version.

Apply rational invariance of \(\lambda\) to \(\phi\otimes\prod_\tau f_{Z_\tau,z_\tau}\). By (24), the transformed sum is \[j(g,Z)^{-1} \exp\!\left(-\pi i\sum_\tau z_\tau^tJ_\tau^{-1}C_\tau z_\tau\right) \vartheta_{\omega_f(\widetilde g_f)\phi}(gZ,J^{-t}z).\] Equating this with the original sum gives (22). In particular the finite action on the left is \(\omega_f(\widetilde g_f)\), with no inverse. If one instead keeps the finite input on the left fixed, the input on the right is acted on by \(\omega_f(\widetilde g_f)^{-1}\).

For the completion, put \(Y=\operatorname{Im}Z\) and \(Y'=\operatorname{Im}(gZ)\) at one place. From \(Y'=\bar J^{-t}YJ^{-1}\), and then \(\bar J^t=J^t-2iYC^t\), we obtain \[ J^{-1}(Y')^{-1}J^{-t}=Y^{-1}-2iJ^{-1}C. \tag{25}\] Here \(C^tJ^{-t}=J^{-1}C\) by symmetry. Thus the completion factor at \((gZ,J^{-t}z)\) differs from that at \((Z,z)\) by \(\exp(-\pi iz^tJ^{-1}Cz)\). This cancels the exponential in (22), so \[\widehat\vartheta_{\omega_f(\widetilde g_f)\phi}(gZ,J^{-t}z) =j(g,Z)\widehat\vartheta_\phi(Z,z).\] Comparison of homogeneous terms yields (23). Finally, \[g e_Z=e_{gZ}J.\] Hence \(J\) is the transformation matrix for vectors in the positive Lagrangian; this proves the asserted tensor target and its orientation. The completion has no linear term in \(z\), so \(T_{1,\phi}\) is the ordinary holomorphic theta gradient. For comparison, the first-order law agrees with the classical theta-gradient formula in (Möller and Zagier 2016, sec. 9.1). ◻

Proof. First we check that the required splitting characters exist. For the idele class groups \(C_F\) and \(C_E\), the natural map \(C_F\to C_E\) is injective: if an \(F\)-idele becomes a principal \(E\)-idele, its generator is fixed by \(c\) and hence belongs to \(F\). This map is also proper. The absolute idele norm on \(C_E\) restricts to the square of the norm on \(C_F\), whose norm-one kernel is compact. A compact subset of \(C_E\) has norms in a compact subinterval of \(\mathbb{R}_{>0}\); its preimage is therefore a closed subset of a compact norm interval times the compact norm-one kernel of \(C_F\). Thus \(C_F\) is a closed locally compact subgroup of \(C_E\). The character-extension theorem for a closed subgroup of a locally compact abelian group extends each unitary character \(\epsilon_{E/F}^{\,r}\) from \(C_F\) to \(C_E\) (Folland 2015, sec. 4.3, Corollary 4.42). Apply this with \(r=m,n\).

The splitting theorem applies to our form by replacing \(h_P\) with the skew-Hermitian form \(\delta h_P\). With both forms linear in the first variable, the trace product in (Borade et al. 2025, sec. 2.1 and 3.1) is exactly (15). We take the same global additive character as in Lemma 12. The determinant twists from changing the characters vanish on \(G=\mathop{\mathrm{SU}}(V)\). The product of the finite local splittings is therefore an honest action \(\omega_{P,f}\) on \(G(\mathbb{A}_{F,f})\), independent of those characters.

We compare its infinite circle lift with the double lift needed for the differential bundles. At the active maximal compact subgroup \(S(\mathop{\mathrm{U}}(2)\times\mathop{\mathrm{U}}(1))\), let \(\ell\) be the character on the negative line. The determinant of the positive plane is \(\ell^{-1}\), so the determinant character on \(\mathcal F_P\) is \(\ell^{2n}\). At every other real place the determinant character is trivial on \(\mathop{\mathrm{SU}}(3)\). On the maximal compact \(\mathop{\mathrm{U}}(a)\) of each real symplectic factor, the metaplectic double cover is the determinant-square-root cover. For the product of the real factors its square is the product of those determinant characters. Its pullback to the maximal compact subgroup of \(G(F\otimes_\mathbb{Q}\mathbb{R})\) therefore splits with square-root character \(\ell^n\).

A connected real reductive group retracts onto its maximal compact subgroup. The preceding determinant computation says that every fundamental-group loop of \(G(F\otimes_\mathbb{Q}\mathbb{R})\) has even determinant winding in the symplectic group. The lift from the universal covering group consequently descends to a group lift of \(G(F\otimes_\mathbb{Q}\mathbb{R})\) into the double cover. Its half-determinant action is the one on \(L^n\) in (17). The original circle lift and this double lift, pushed out to the circle extension, differ by a continuous character of \(G(F\otimes_\mathbb{Q}\mathbb{R})\). This group is connected and semisimple. The differential of such a character is zero, and connectedness makes the character trivial. The two lifts therefore agree. The rational compatibility of the global splitting now pairs \(\omega_{P,f}(\gamma)\) with this infinite lift for every \(\gamma\in G(F)\), proving the claimed rational covariance.

At a split finite place the two embedding summands are dual stable Lagrangians. If \(h\) is the action on the position summand, the symplectic matrix is \(\mathop{\mathrm{diag}}(h^{-t},h)\). The Schrödinger linear formula gives (26), up to the scalar of the chosen bare lift. On \(\mathop{\mathrm{SL}}_3(F_v)\), the normalized substitution lift and the unitary lift have a character as their ratio. This character is trivial because \(\mathop{\mathrm{SL}}_3(F_v)\) is perfect: elementary matrices generate it, and for distinct \(i,j,k\), \[e_{ij}(u)=[e_{ik}(u),e_{kj}(1)].\] The determinant factor is also \(1\) there. This proves the split-place assertion without removing scalars from lifts outside \(\mathop{\mathrm{SL}}_3\).

Finally the finite Weil action is smooth, so finitely many Schwartz functions have a common open stabilizer. Intersect it with the fixed lattice stabilizer in \(G(\mathbb{A}_{F,f})\). Principal lattice congruence neighborhoods form a basis at the identity in that stabilizer, so a sufficiently deep principal one lies in the intersection. This proves the last assertion. ◻

Proof. Holomorphy follows from that of \(T_{1,\phi}\), the holomorphic embedding projection, and (14). Descent follows from Lemma 14.

Fix a point \(x_*\in D\) and its period point \(Z\). Here \(a=3\). For \(0\ne x_0\in\mathcal O_F^a\) and positive integers \(k\), take \[\phi_k=\mathbf1_{\,x_0+k\widehat{\mathcal O}_F^{\,a}}.\] Its rational support is exactly \(x_0+k\mathcal O_F^a\). The first Taylor tensor is the ordinary gradient, so in the chosen graph frame \[T_{1,\phi_k}(Z) =2\pi i\sum_{x\in x_0+k\mathcal O_F^a} (x_\tau)_{\tau\in\Sigma} \exp\!\left(\pi i\sum_\tau x_\tau^tZ_\tau x_\tau\right).\] Under the Minkowski embedding \(\mathcal O_F^a\) is a full lattice. If \(\rho>0\) is the minimum norm of a nonzero lattice vector, then, for \(m\ne0\) in this lattice and \(k\ge2\|x_0\|/\rho\), \[\|x_0+km\|\ge k\|m\|/2.\] The Gaussian bound used above, with any polynomial factor absorbed into a weaker Gaussian, bounds the tail by a constant times a convergent lattice sum of the form \(\sum_{m\ne0}\exp(-c k^2\|m\|^2)\). That sum tends to zero. The same argument is uniform for \(Z\) in a compact neighborhood. We have proved \[ T_{1,\phi_k}(Z)\longrightarrow 2\pi i\,(x_{0,\tau})_{\tau\in\Sigma} \exp\!\left(\pi i\sum_\tau x_{0,\tau}^tZ_\tau x_{0,\tau}\right). \tag{27}\]

The scalar exponential is nonzero. As \(x_0\) varies, the vectors \((x_{0,\tau})_\tau\) span the real position space, and their complex span is the entire first-tensor fiber in this frame. The complex span of the actual values \(T_{1,\phi}(Z)\) is a subspace of a finite-dimensional space and hence is closed. It contains all the limits in (27), so it is the full fiber. Projection onto \((\det\mathcal F_i)^{1/2}\otimes(\mathcal F_i)_{c\mu}\) is surjective; therefore the projected values span \(\Omega_D^1|_{x_*}\). Choose finitely many actual inputs whose values contain a basis, and then apply the stabilizer assertion of Lemma 14.

The inputs \(\phi_k\) in this limit need not have a common congruence stabilizer as \(k\) varies. The limit takes place in one fiber over \(D\); a common level is chosen only after finitely many actual inputs have been selected. ◻

Proof. For (i), the rational orthogonal decomposition gives a rational symplectic direct sum \(\mathbb V_W=\mathbb V_{W_1}\perp\mathbb V_{W_2}\). Choose rational coordinates adapted to it. By Remark 13, the total reference input becomes another finite Schwartz input in the adapted coordinates. Such an input is a finite sum \[\sum_r\phi_{1,r}\otimes\phi_{2,r}.\] Indeed, a locally constant compactly supported function on the product of two finite-adelic vector spaces is invariant under a sufficiently small product compact open subgroup. Only finitely many of its cosets meet the support, and each coset is a product coset.

For a product input the completed theta series multiply, since both the sum and its completion split in the adapted coordinates. If \(T_j^{(i)}\) denotes the degree-\(j\) coefficient for the \(i\)-th line, the coefficient of total degree two is \[ T_2^{(1)}T_0^{(2)} +T_1^{(1)}T_1^{(2)} +T_0^{(1)}T_2^{(2)}. \tag{28}\] The coefficient of the mixed product is \(1\), by the \(1/k!\) normalization in (21). The two unmixed terms have both multiplicity factors in the same one-dimensional \((W_i)_\mu^*\); they vanish under \(\operatorname{alt}_2\). The mixed term becomes the wedge of the two projected first tensors. With compatible constant-line trivializations it is exactly \(\Theta_1(\phi_{1,r})\wedge\Theta_2(\phi_{2,r})\); other trivializations multiply it by a fixed nonzero scalar. Rational changes of coordinates are already accounted for by (23). Moreover, two identifications with the differential bundle used here have a ratio invariant under the transitive infinite special-unitary action on \(D\), so that ratio is constant. This proves the wedge expansion and its holomorphy. We have proved holomorphy only after the double-alternating projection: the whole completed second Taylor tensor need not be holomorphic. The finitely many individual inputs descend on a further common principal level by Lemma 14.

For (ii), fix a point of \(D\). Proposition 15 provides an input for \(W_1\) with a nonzero cotangent value and an input for \(W_2\) whose value is outside the line it spans. Their wedge is nonzero at that point. The product of these inputs in adapted coordinates has a nonzero double-alternating second tensor by the calculation above. Transporting this product back to the reference coordinates gives a total finite input with \(\Xi_W(\phi)\ne0\).

For (iii), no rational coordinate change is made as \(t\) varies. The rational symplectic space, its rational \(G\)-action, the finite representation, and the total input all remain fixed. The changed compact-place Lagrangians remain invariant under \(\mathop{\mathrm{SU}}(3)\), since all changes are in the multiplicity factor. Their determinant characters on \(\mathop{\mathrm{SU}}(3)\) are trivial. The active pieces have not changed, so their determinant character remains \(\ell^4\). The infinite comparison in Lemma 14 therefore gives the same half-determinant action \(L^2\) at every \(t\). Consequently the same subgroup fixing \(\phi\) gives equivariance and descent at every parameter. There is no finite translation, Hecke choice, or change of level in this variation.

It remains to check uniformity on the quotient. The constant determinant factors in (17) vary smoothly with the compact-place subspaces. On a small parameter neighborhood choose smooth nonzero frames and their square roots, agreeing with the chosen frames at the center. Cover the fixed compact \(X\) by finitely many charts with precompact lifts to \(D\). On the closure of each lifted chart times a sufficiently small compact parameter neighborhood, the period map is smooth, its image is a compact subset of the Siegel domain, and every fixed order of its derivatives is bounded. The imaginary parts thus have a common positive lower bound. The rational support of the fixed \(\phi\) continues to lie in one fixed finite union of lattice cosets. The Gaussian estimates following (21) now give uniform convergence of every fixed order of derivatives in the chart coordinates and the parameters. Projection to (19) and the chosen smooth trivializations preserve these bounds. The unchanged equivariance glues these convergent local expressions on \(X\). This is continuity in the \(C^\infty\) topology. Compactness of \(X\) is used here; no estimate at a cusp or boundary is needed. ◻

Proof. At a compact place \(\tau\ne\mu\), equality of the two sign sums says exactly that the multisets \(\{d_1(\tau),d_2(\tau)\}\) and \(\{d_3(\tau),d_4(\tau)\}\) coincide. Reorder the two original local lines so that the first has the sign required by \(d_3(\tau)\), and choose a nonzero vector on it. After fixing an \(E\)-basis of \(W\), weak approximation in \(E\), coordinate by coordinate at these finitely many complex places, gives a single \(w\in W(E)\) arbitrarily close to all the chosen local vectors. Nondegeneracy and the signs of their Hermitian norms are open conditions. For a sufficiently close approximation, \[W_3=Ew,\qquad W_4=W_3^\perp\] is therefore a rational orthogonal decomposition. Its first line has the prescribed compact-place signs for \(d_3\); orthogonal inertia gives the complementary signs for \(d_4\). At \(\mu\) the form on \(W_\mu\) is definite, so both new lines automatically have the same required sign there.

The total compact-place Lagrangians converge to the original ones. Reordering the original lines only relabels their total positive summands. For a nondegenerate local vector the orthogonal projection onto its line is \[z\longmapsto h_W(z,w)h_W(w,w)^{-1}w.\] It varies continuously with \(w\), so both the line and its orthogonal complement converge to the chosen original ones. The selected signed subspaces, and hence the total positive Lagrangians, converge. At the active pair the total pieces are always \(L\otimes W_\mu\) and \(Q^*\otimes W_\mu^*\), so they are unchanged, not merely close. The approximating rational lines may have small active norms or large arithmetic denominators. No bound on these is needed: the active total period data are fixed, and the continuity assertion concerns the total tensor in the original reference coordinates.

All the hypotheses of Proposition 16(iii) now hold with the same rational \(W\), the same \(G\), the same total finite \(\phi\), and the same compact \(X\). Thus \[\Xi_W(\phi;t')\longrightarrow \Xi_W(\phi;t_0)\] smoothly on \(X\). The limit \(\xi\) is a nonzero holomorphic top form. In local complex coordinates \(\xi=f\,dz_1\wedge dz_2\), the form \(\xi\wedge\bar\xi\) is a positive multiple of \(|f|^2\,dx_1\wedge dy_1\wedge dx_2\wedge dy_2\). Consequently \(\int_X\xi\wedge\bar\xi>0\). Continuity of the integral gives (29) for all sufficiently close rational choices. At each such endpoint holomorphy follows from Proposition 16(i), not from a holomorphy assertion along the intermediate parameter path.

Factor the two endpoints only now. There are finitely many line inputs in their two wedge expansions. Choose a principal subgroup contained both in the level of \(X\) and in their finite stabilizers. It gives a finite unramified cover \(\pi:X'\to X\) on which all individual factors descend. The covering formula is \[ \int_{X'}\pi^*\xi\wedge\overline{\pi^*\xi'} =\deg(\pi)\int_X\xi\wedge\overline{\xi'}\ne0. \tag{30}\] Both quotients are compact and torsion-free, so there is no cusp term or ramification correction. This proves the last assertion. ◻

Proof. The Lie-rank condition need not identify a single rational Hermitian isomorphism class. We therefore construct the whole finite-type action locus, identify its complex components as ball quotients, and then prove projectivity and identify the component of \(\Lambda\).

The finite-type action locus. The fine-moduli theorem for polarized abelian varieties gives a quasi-projective scheme with universal polarization and full level at sufficiently large invertible level (Mumford and Fogarty 1982, chap. 7, Definitions 7.1–7.2, Proposition 7.6, and Theorem 7.9). In that theorem the polarization degree is fixed and the level is an ordered torsion basis, with no prescribed pairing matrix. After inverting the polarization degree, the polarization kernel is finite étale, so its geometric elementary-divisor type is locally constant. We may therefore take the open-and-closed part of the required type. If necessary, first apply the theorem to a sufficiently large divisor of \(N\) prime to the polarization degree and then add the remaining full \(N\)-basis as a finite étale level cover. These constructions work over a ring of integers with finitely many primes inverted, as well as in characteristic zero.

We next impose the action by a finite-type construction. Choose finitely many ring generators \(b\) of \(\mathcal O_E\) and parameterize their graphs in relative Hilbert schemes of the square of the universal abelian scheme. The degrees of the graphs in question are bounded. Indeed, put \(\alpha=1+\bar b b\). It is a nonzero totally positive element of \(F\). Choose a positive integer \(m\) with \(m\alpha^{-1}\in\mathcal O_E\). In every such action \(\alpha\) is an isogeny, and \[\deg(\alpha)\deg(m\alpha^{-1})=m^{6s}.\] The second factor is a positive integer, so \(\deg(\alpha)\) is bounded independently of the point.

To translate this into a Hilbert bound, the normalized Poincaré bundle on the universal abelian scheme times its dual, pulled back by \((\mathop{\mathrm{id}},\lambda)\), is a relatively ample bundle inducing \(2\lambda\). The tensor product of its two pullbacks to the square restricts on the graph of \(b\) to a bundle inducing \(2\lambda(1+\bar b b)\). Its polarization degree is bounded by the preceding identity. Riemann–Roch on an abelian variety of dimension \(3s\) gives the Hilbert polynomial \(\chi(\mathcal L)n^{3s}\), with \(\chi(\mathcal L)^2=\deg(\phi_{\mathcal L})\). Only finitely many polynomials can therefore occur. On their graph loci impose the homomorphism, ring, Rosati, and Lie-rank identities. This gives a finite-type action locus.

Over the characteristic-zero polarized base this locus is unramified. One form of rigidity suffices: an infinitesimal difference of two homomorphisms between fixed abelian schemes would give a homomorphism from a proper connected abelian fiber to an additive tangent group, and must vanish. The locus is also proper over that base. Given a discrete valuation ring mapping to the polarized base and a lift of its generic point to the action locus, each generator homomorphism extends uniquely between the fixed abelian schemes by the Néron mapping property (Bosch et al. 1990, sec. 1.2, Definition 1 and Proposition 8). Its graph remains a graph of the same Hilbert polynomial. The ring and Rosati identities extend by uniqueness. Since the base contains \(E\), the action algebra splits into idempotents in characteristic zero, and their Lie summands have locally constant ranks. Thus the extended point lies in the same action locus. The valuative criterion gives properness. A proper finite-type unramified morphism is quasi-finite and hence finite, so the action locus is finite over the polarized base.

The complex components. We now identify the reduced complex structure of this algebraic locus. In a marked period chart, the integral matrices of the action are locally constant. Such a matrix acts on the complex torus exactly when it preserves the homology filtration; the resulting holomorphic endomorphism of a polarized complex torus is algebraic. Hence the reduced local locus is precisely the locus of positive filtrations stable under the fixed matrices. Trace duality and the Rosati identity express the alternating form as an imaginary trace multiple of an \(E/F\)-Hermitian form on the rational homology, of \(E\)-dimension three. Isotropy makes the two filtrations in a conjugate pair annihilators. Positivity and (32) leave one negative line at the active place and constant data at every other place. The local locus is therefore a smooth complex two-ball.

This description gives whole ball quotients, not only local ball charts. Fix a marked polarized \(E\)-lattice occurring in \(S\). Every positive filtration in its compatible ball gives a complex torus with the fixed integral Riemann form, and hence an abelian variety; the preserving integral matrices give its algebraic \(\mathcal O_E\)-action. Conversely every tuple with these lattice data gives such a filtration. Its connected component is thus the quotient of the whole ball by the marked lattice stabilizer.

Projectivity and the principal level. For each such rational Hermitian space, a vector isotropic over \(E\) would remain isotropic at a definite real place. Since \(s\geq2\), there is no such vector, and its special-unitary group is \(F\)-anisotropic. The arithmetic lattice and compactness theorems give compact ball quotients (Borel and Harish-Chandra 1962); see (Milne 2005, Theorem 3.3 and Proposition 3.5). The level removes stabilizers. There are finitely many components because \(S\) is of finite type. The finite action locus is quasi-projective, and its reduced complex space is smooth with compact components. It is therefore proper, and a proper quasi-projective scheme is projective. Smoothness and properness descend from the complex base change to \(K_0\).

Finally the initial lattice \(\Lambda\) is among these data. Even if it is not free, its determinant module is a fractional \(\mathcal O_E\)-ideal. The determinant of a unitary lattice automorphism is consequently a unit of \(\mathcal O_E\) of norm one. All its complex conjugates have modulus one, so it is a root of unity. A sufficiently divisible principal level kills every nontrivial such determinant. Its stabilizer is then precisely the principal special-unitary lattice level, as claimed. ◻

Proof. Models and Albanese generation. Spread the finitely many schemes, morphisms, and universal data over an open part of the ring of integers of \(K\). One can obtain the proper model of \(S\) by taking its closure in a projective compactification of the full parameter scheme used in the proof of Lemma 19. Its intersection with the boundary has proper closed image on the integer base. That image misses the generic point because \(S\) is proper and lies in the parameter scheme. Removing the finitely many primes in the image leaves the closure inside the full parameter scheme. Thus its geometric points are still distinguished by their polarized, acted-on, marked tuples, including the graph coordinates. Further shrinking gives smooth proper models with geometrically connected component fibers, good abelian models, and the extended Albanese maps.

For a component \(S_j\), let \(\Delta_j=a_j(S_j)-a_j(S_j)\subset A_j\). It is closed, irreducible, symmetric, and contains zero. The successive sum images \(n\Delta_j\) are nested irreducible closed sets. Their dimensions eventually stabilize; then equality of two successive sets implies stability under addition by \(\Delta_j\). Symmetry shows that the stable image is the abelian subvariety generated by \(\Delta_j\), which is \(A_j\) by the Albanese property. Thus for some fixed \(n\) the proper sum map \[(S_j\times S_j)^n\longrightarrow A_j\] is surjective. Spread this map together with the other data. Its closed image contains the generic fiber of the flat abelian scheme; that fiber is dense, so the image is the whole abelian scheme. Consequently the sum map is surjective on every geometric fiber.

Ordinary points. For each characteristic-zero component, we now choose a point whose reduction at the split primes retained below is ordinary. In its rational Hermitian three-space choose, by weak approximation, an \(E\)-line negative at \(\mu\), and diagonalize its orthogonal complement. At the period whose negative line is this coordinate line, the three rational orthogonal projectors split the rational homology into three \(E\)-lines and preserve the filtration. Integral multiples of them are endomorphisms of the corresponding abelian variety.

These points and endomorphisms are defined over \(\overline{\mathbb{Q}}\). Indeed, add their graphs as extra parameters with the Hilbert polynomials at the selected point. This is a finite-type parameter scheme over a number field. Locally the projector matrices are fixed by rigidity. A negative line preserved by all three projectors must be the unique negative coordinate line, so the selected enhanced point is isolated. An isolated complex point of a finite-type scheme over a number field has algebraic coordinates. Adjoin the finitely many fields of definition to \(K\), spread these points and endomorphisms, and exclude their projector and isogeny denominators.

Now let \(q\) be a remaining completely split prime. At each selected point the integral projectors and the idempotents of \(\mathcal O_E\otimes\mathbb{Z}_q\) split the \(q\)-divisible group into pieces of height one. Their heights are the corresponding characteristic-zero Tate ranks and stay constant under specialization. A height-one \(q\)-divisible group over an algebraically closed field has dimension zero or one. In the first case it is étale; in the second its dual has dimension zero, so it is multiplicative. Thus each selected reduction is ordinary. Ordinariness is open, and a smooth geometrically connected fiber is geometrically irreducible. This nonempty open is dense on each component. Chebotarev’s theorem supplies infinitely many completely split rational primes after the finite exclusions. ◻

Proof. Let \(C\to S\) be one of the finite étale source correspondences. On a source component, compose each target branch with the appropriate Albanese map and sum in \(A_S\). This sum descends from a cover trivializing the branches, because it is invariant under their permutations. It is a morphism \(n_j:S_j\to A_S\). Subtracting \(n_j\) at the basepoint and applying the Albanese universal property gives an integral homomorphism \(A_j\to A_S\). On differences it has the exact formula \[ T\bigl(a_j(x)-a_j(y)\bigr) =\sum_{z\in C_x}a_{j(z)}(t(z)) -\sum_{z\in C_y}a_{j(z)}(t(z)), \tag{37}\] where \(t\) is the target map, \(j(z)\) indexes the component containing \(t(z)\), fibers are counted with their branches, and \(a_{j(z)}\) is included in \(A_S\). These homomorphisms on the factors define the operator on \(A_S\). Their pullback on one-forms is the branch sum, since a constant translation contributes no differential.

Composition has the raw spherical convolution multiplicities. For a selected lattice \(\Lambda_j\), a branch is an overlattice \(g_j^{-1}\Lambda_j\) with a representative modulo the integral subgroup on the left. If its cumulative multiplier is \(q^h\), the opposite target lattice is forced to be \(q^{-h}(\Lambda_j')^\vee\). Consequently a chain of quotient choices is precisely a chain of selected overlattices, as in spherical convolution. The opposite lattices, polarizations, and transported markings agree for the same composite. The numbers of branches landing in each target component are locally constant on a connected source. Thus the component basepoint translations cancel when (37) is composed.

Consider the central position with all selected exponents equal to one and multiplier exponent two. It is quotient by \(B[q]\); under the multiplication-by-\(q\) identification its target has the same polarization and action and marking \(q\eta_N\). It is therefore a finite-order marking automorphism, since \(q\) is invertible modulo \(N\). Adding a sufficiently large power of this position to any integer exponent position puts it in (35). Define its operator by clearing that power with the inverse marking automorphism. The positive convolution identity shows that this definition is independent of the clearing power. This gives the full rational spherical action. The spherical algebra is commutative, as follows also from the Satake isomorphism (Gross 1998, sec. 3). ◻

Proof. Let \(k=\overline{\mathbb{F}_q}\), and let \(x\) be an ordinary geometric point of the special fiber of \(S\). Its active \(q\)-divisible group has the canonical splitting \[G_w=M_0\oplus E_0,\] where \(M_0\) is multiplicative of height one and \(E_0\) is étale of height two. Over the perfect field \(k\), the reduced subgroup at each finite torsion level is the canonical étale part; it is preserved by the endomorphism idempotents. Each nonactive selected block is wholly multiplicative or wholly étale according to its Lie rank.

Three quotient constructions will account for the neighbors. Write \(X_0\) for the quotient with active kernel \(M_0[q]\) and multiplier \(q\), \(Y_0\) for the sum of quotients with active kernel a line \(\ell\subset E_0[q]\) and multiplier \(q\), and \(Z_0\) for the quotient with active kernel \(E_0[q]\) and multiplier \(q^2\). For multiplier \(q^h\), prescribe the connected \(q^h\)-torsion on every nonactive selected block and the forced annihilator on every opposite block. We will prove that these quotients define full tuples of \(S\), that \(X_0\) is Frobenius, and that the specialized neighbor sums are \[D_1:\ X_0+qY_0,\qquad D_2:\ X_0Y_0+q^2Z_0,\qquad D_3:\ X_0Z_0.\] The displayed formulas are the identities to establish; at this point the quotient constructions have not yet been identified with operations on \(S\). Once they are, their cancellation will give the asserted polynomial on Albanese differences.

Specialization multiplicities. Lift \(x\), using smoothness, to \(R=W(k)\), and write \(\mathcal G_w\) for the lifted active \(q\)-divisible group. The ring \(R\) is a complete henselian discrete valuation ring. In the lifted active \(q\)-torsion, the \(q^2\) open-and-closed special label components lift uniquely by lifting idempotents of its finite coordinate algebra. Each lifted component is finite flat of rank \(q\). The identity-specializing component is a subgroup, and multiplication respects the lifted labels by uniqueness. Thus there is an exact sequence \[0\longrightarrow\mathcal H^0 \longrightarrow\mathcal G_w[q] \longrightarrow(\mathbb{F}_q^2)_R\longrightarrow0\] with \(\mathcal H^0\) finite flat of rank \(q\). No splitting of this sequence over \(R\) is asserted.

Pass to a finite extension of \(R\) over which the generic torsion is split. Its active fiber is a three-dimensional \(\mathbb{F}_q\)-space with distinguished line \(\mathcal H^0_{\mathrm{gen}}\). For a fixed \(b\)-dimensional subspace of the label quotient, the generic subspaces disjoint from that line and mapping onto the label subspace are its graph lifts. They form a torsor under the \(b\)-dimensional space of maps to the line, so there are \(q^b\) of them. A generic subspace containing the line is the unique full inverse image of its label subspace.

The schematic closure of a generic subgroup in the finite flat torsion scheme is finite torsion-free over the new DVR, hence flat. It is still a subgroup, since the subgroup identities extend from the generic fiber on flat schemes. Intersection with a label component is an idempotent direct summand of its coordinate algebra, so its rank is its generic rank. A graph has rank one on each of its labels. A finite flat rank-one algebra over a local ring is that ring, and its special fiber is the unique reduced point of that label component. All graph lifts over a label subspace therefore specialize to the same étale subgroup of \(E_0[q]\). A subspace containing the distinguished line fills the corresponding rank-\(q\) components and specializes to \(M_0[q]\) plus that étale subgroup.

The active kernels of \(D_i\) are \(i\)-dimensional subspaces of the generic \(q\)-torsion. The preceding count gives: \[\begin{array}{c|c|c} \text{operator}&\text{special active kernel}&\text{multiplicity}\\ \hline D_1&M_0[q]&1\\ &\ell\subset E_0[q],\ \dim_{\mathbb{F}_q}\ell=1&q\text{ for each }\ell\\ D_2&M_0[q]\oplus\ell,\ \dim_{\mathbb{F}_q}\ell=1&1\text{ for each }\ell\\ &E_0[q]&q^2\\ D_3&M_0[q]\oplus E_0[q]&1 . \end{array}\] In the first two cases the total number is \(q^2+q+1\): there are \(q+1\) étale lines. The nonactive selected kernels are uniquely their connected \(q^i\)-torsion for \(D_i\).

Full tuples and Frobenius. We now show that the counted kernels give targets in \(S\), and identify the resulting quotient operations. Properness of the model of \(S\) extends each generic neighbor over a finite valuation-ring extension. The universal source and target are abelian schemes over that new DVR. Apply the Néron mapping property there, anew, to extend the generic isogeny (Bosch et al. 1990, sec. 1.2, Proposition 8). The polarization identity extends by separatedness. On each fiber the kernel of the extension is contained in the finite kernel of \(f^\vee\lambda'f=[q^i]\lambda\), so it has finite kernel and is an isogeny between equal-dimensional abelian varieties. Since source and target polarizations have the same degree, its degree is \(q^{3si}\) on every fiber. Properness and fiberwise quasi-finiteness make the extension finite. Fiberwise flatness of isogenies between the smooth fibers and the relative flatness criterion make it flat; see (Bosch et al. 1990, sec. 7.3, Lemma 1 and Proposition 6; §2.4, Proposition 2). Its finite flat kernel is therefore the schematic closure of its generic kernel.

Finite-locally-free Cartier duality commutes with base change. For a subgroup \(H\subset G\) in a finite group with a perfect pairing, the annihilator is \[\operatorname{ann}(H) =\ker(G^D\longrightarrow H^D)=(G/H)^D.\] It is finite locally free and commutes with the above specialization. Thus the opposite generic kernels specialize to the annihilators of the selected kernels just counted. The corresponding nonactive kernels are already prescribed. Equal total kernels give canonically isomorphic abelian quotients. Their descended \(\mathcal O_E\)-actions and transported markings are unique, and their polarizations are unique by cancelling \(f\) in \(f^\vee\lambda'f=q^i\lambda\). The model retains these full tuples, so equal quotients are equal target points of \(S\). In particular every counted kernel has such a target. It is ordinary, since ordinariness is invariant under isogeny.

The constructions \(X_0,Y_0,Z_0\) are therefore operations on the ordinary points of the same full moduli space. We use them as multiset operations on points; no global special-fiber algebraic correspondence is required.

The operation \(X_0\) is the \(q\)-power Frobenius on full tuples. Indeed its total kernel is the connected \(q\)-torsion on every block. On the active pair, duality pairs the multiplicative part with the opposite étale part and the étale part with the opposite multiplicative part, so the annihilator of \(M_0[q]\) is exactly the opposite connected part; the pure nonactive pairs have the same property. The total kernel is therefore the kernel of relative Frobenius \(F_B:B\to B^{(q)}\). Naturality of the polarization and Frobenius–Verschiebung duality give \[F_B^\vee\lambda^{(q)}F_B =V_{B^\vee}F_{B^\vee}\lambda=[q]\lambda.\] Here \(V_{B^\vee}\) denotes Verschiebung. The action on the target is \(\iota_E^{(q)}\). For the constant label group \(T=(\mathbb{Z}/N\mathbb{Z})^{6s}\), naturality gives \[F_B\eta_N=\eta_N^{(q)}F_T.\] Relative Frobenius on \(T\) fixes its labels, and \(F_B\) is an isomorphism on \(N\)-torsion. Thus the transported marking is exactly the twisted marking, with no multiplication-by-\(q\) marking change. Since the model and universal data are defined over \(\mathbb{F}_q\), this is its coordinate-\(q\)-power operation on geometric points, not its inverse.

Frobenius commutes with \(Y_0\) and with \(Z_0\) as multiset operations. For any of their quotient maps \(f:B\to B'\), \[F_{B'}f=f^{(q)}F_B.\] Frobenius is an isomorphism on the étale part and permutes its line choices bijectively; this square also respects the polarizations, actions, and markings. On the active block the composite \(X_0Y_0\) has kernel \(M_0[q]\oplus\ell\) with multiplier \(q^2\), and \(X_0Z_0\) has kernel \(M_0[q]\oplus E_0[q]\) with multiplier \(q^3\).

For clarity, the opposite kernels of these composites are indeed forced by their selected kernels and multipliers. On the \(q\)-divisible groups, write \(G,G'\) for source and target and define the polarization adjoint \(f^\dagger=\lambda^{-1}f^D\lambda'\); the polarizations are isomorphisms there because their degrees are prime to \(q\). For a composite with multiplier \(q^h\), \[f^\dagger f=[q^h],\qquad ff^\dagger=[q^h].\] The second identity follows by cancellation of the fppf epimorphism \(f\). As fppf subgroup sheaves, \(\ker f=f^\dagger(G'[q^h])\). For \(u,v\in G'[q^h]\), adjunction gives \[e_{q^h}\bigl(f^\dagger u,f^\dagger v\bigr) =e'_{q^h}\bigl(u,ff^\dagger v\bigr)=1,\] so the image is isotropic in \(G[q^h]\). The endomorphism idempotents split its kernel into conjugate-block parts. If \(K_j,K_{\bar j}\) are the parts in one conjugate pair of height-three blocks, then \[K_{\bar j}\subseteq K_j^\perp,\qquad \operatorname{rank}(K_j)\operatorname{rank}(K_{\bar j}) \leq q^{3h}.\] Cartier duality gives equal kernel ranks for \(f\) and \(f^\dagger\). Their composite has kernel rank \(q^{6sh}\), so \(\operatorname{rank}\ker f=q^{3sh}\). Multiplying the pairwise inequalities gives equality, and hence equality in each pair. Thus \(K_{\bar j}=K_j^\perp\). For \(X_0Z_0\), for example, the annihilator of the active \(q\)-torsion in \(q^3\)-torsion is the opposite block’s \(q^2\)-torsion. The pure nonactive connected kernels likewise compose to connected \(q^h\)-torsion. Uniqueness of the polarizations and markings now identifies the composite targets with the counted targets. We have proved the multiset identities \[ \begin{aligned} D_1&:\quad X_0+qY_0,\\ D_2&:\quad X_0Y_0+q^2Z_0,\\ D_3&:\quad X_0Z_0. \end{aligned} \tag{40}\]

The integral Albanese identity. The neighbor counts are now identities of multisets of full tuples. We pass from them to endomorphisms by applying the Albanese difference formula. Let \(\mathcal O\) be the set of ordinary \(k\)-points of the union of components and extend \(X_0,Y_0,Z_0\) linearly to the free abelian group \(\mathbb{Z}[\mathcal O]\). Define \[\alpha:\mathbb{Z}[\mathcal O]\longrightarrow\widetilde A_S(k), \qquad \alpha([x])=a_{j(x)}(x).\] The component Albanese maps reduce over \(\mathbb{F}_q\), so \(\Phi\alpha=\alpha X_0\). For ordinary \(x,y\) in one source component, lift them and specialize the integral identity (37). With \[N_1=X_0+qY_0,\qquad N_2=X_0Y_0+q^2Z_0,\qquad N_3=X_0Z_0,\] it gives \[D_i\bigl(a_j(x)-a_j(y)\bigr) =\alpha\bigl(N_i([x]-[y])\bigr).\] This is an equality in the actual group of geometric points, with all branch multiplicities retained. No rationalization of a finite-field point group is used.

Frobenius preserves each component and its ordinary locus. Applying the same formula to \(X_0^m x,X_0^m y\) shows that the endomorphism in (39) on this difference is \[\alpha\!\left( (X_0^3-N_1X_0^2+qN_2X_0-q^3N_3)([x]-[y]) \right).\] The operator inside this expression vanishes: \[\begin{aligned} &X_0^3-N_1X_0^2+qN_2X_0-q^3N_3\\ &\qquad=-qY_0X_0^2+qX_0Y_0X_0 +q^3Z_0X_0-q^3X_0Z_0=0. \end{aligned}\] Only the two proved commutations with \(X_0\) are used. Ordinary points are dense in every smooth component by Lemma 20. Therefore the morphism obtained by applying the endomorphism to \(a_j(x)-a_j(y)\) vanishes on all of the product of that component with itself. The finite-sum generation in the same lemma makes the endomorphism zero on each Albanese factor. It is zero on \(\widetilde A_S\) after base change to \(k\); faithful base change gives the integral identity over \(\mathbb{F}_q\). ◻

Proof. As a quasi-isogeny, such a position leaves the rational polarized period fixed. It changes only the active lattice and its opposite dual, keeping the marking and all other local data. This is also the description obtained by clearing the central shift in Lemma 21 and undoing its marking change. A determinant-exponent-zero lattice position has a representative in \(\mathop{\mathrm{SL}}_3(\mathbb{Q}_q)\). Adjusting determinant units in Smith bases shows that its double cosets and branches are exactly those for \(\mathop{\mathrm{SL}}_3(\mathbb{Z}_q)\backslash\mathop{\mathrm{SL}}_3(\mathbb{Q}_q)/\mathop{\mathrm{SL}}_3(\mathbb{Z}_q)\).

For each component apply strong approximation to the special-unitary group of its rational Hermitian space. This group is simply connected and absolutely almost simple over \(F\), and its factor at \(\mu\) is noncompact. Strong approximation away from all infinite places therefore makes its \(F\)-points dense in its finite adelic points (Milne 2005, Theorem 4.16). Put \(K_q=\mathop{\mathrm{SL}}_3(\mathbb{Z}_q)\). Write \(\Lambda^{(j)}\) for the marked lattice of the component under consideration. For a branch with new active lattice \(g^{-1}\Lambda_w^{(j)}\), write \((\Lambda^{(j)})'\) for the full changed lattice and choose a rational representative \(\gamma\) whose active component lies in \(K_qg\) and whose other finite components are integral and marking-preserving. Then \(\gamma(\Lambda^{(j)})'=\Lambda^{(j)}\), and \(\gamma\) identifies the target tuple with the transformed negative period line in the same component.

On the original component, Lemmas 12 and 14 give \[\gamma^*\Theta\bigl(\omega_f(\gamma)\phi\bigr)=\Theta(\phi).\] Thus fixed-input pullback by the branch acts through \(\omega_f(\gamma)^{-1}\). Away from \(w\) the chosen level fixes the input. At \(w\) our good-prime choice makes it invariant under \(K_q\); if \(\gamma_w=kg\), its inverse action on that input is therefore \(\omega_f(g^{-1})\). Summing the branches proves the inverse spherical formula. The sum is raw: no division by its degree and no additional power of \(q\) occurs. ◻

Proof. Principalize a positive power \(w^m=(a)\) and put \(r=a/\bar a\). It has norm one and the required valuations with \(M=m\). At every other finite place it is a unit. Powers of an element of a compact unit group enter every prescribed open subgroup: its image in the finite quotient by that subgroup has finite order. Smoothness of the bare local Weil representation gives an open stabilizer of the fixed input in the metaplectic group; its image meets the scalar unit group in an open subgroup. Replacing \(r\) by a common positive power therefore places it in the stabilizers needed at the finitely many exceptional places of the fixed input; replace \(M\) accordingly.

Multiplication by \(r\) on \(V\otimes_E P\) preserves its \(F\)-symplectic form because \(r\bar r=1\). We use its bare rational symplectic covariance, with paired metaplectic lifts, not the value of a chosen unitary splitting character. For a transformation \(h\) of a local position space, the normalized linear action is \[(\rho(h)b)(x)=|\det h|_q^{-1/2}b(h^{-1}x),\] up to the finite Weil phase of the bare lift. The linear-subgroup formula, rational summation invariance, and the double-cover control are in (Weil 1964, I, no. 13; III, no. 41; IV, nos. 43–45). At \(w\), the position action is the scalar \(r_w\), of valuation \(M\). Its unit part fixes a radial function, so the finite action there is \(\mathcal D_q^M\), up to a root of unity. At almost every other place the input is the unramified lattice vector and \(r\) is integral. At the remaining places the chosen power fixes the input under a suitable bare lift; the paired lift differs only by its finite phase. Their product is again a root of unity. An arbitrary lift in the circle pushout would not justify this conclusion, which is why the bare rational action is used here.

At infinity, multiplication by \(r\) acts on the \(\tau\)-embedding summand as the scalar \(\tau(r)\): this is the defining scalar action after tensoring the \(E\)-space through \(\tau\). On the \(c\tau\)-summand it acts as \(\tau(\bar r)=\tau(r)^{-1}\). It consequently fixes every chosen positive Lagrangian and the point of \(D\). In the identity \(g e_Z=e_{gZ}J\) of Lemma 12, \(J\) is this action on the homology filtration. The squared scalar on the projected first tensor is the scalar on \(\det\mathcal F_P\) times the square of the scalar on its projected \(c\)-summand. At \(\mu\), the filtration has scalar exponents \(1,-1,-1\). Its determinant contributes \(-1\), and the squared projected conjugate factor contributes \(-2\). The total is \(-3\). At a compact pair, the selected full rank-three summand contributes \(-3d(\tau)\), with no projected tangent factor. Thus the squared infinite scalar is \(\prod_{\tau\in\Sigma}\tau(r)^{-3d(\tau)}\). This calculation includes the scalar action on the determinant’s constant line, which is trivial only on \(\mathop{\mathrm{SU}}(V)\).

The covariance formula has the finite transformed input on its left and the tensor action on its right. Since \(r\) fixes the period, it gives \(\Theta(\mathcal D_q^M b)=\omega\Theta(b)\), with the finite phase absorbed in \(\omega\), and its square is exactly (42). This equality of one-forms implies the asserted equality of cohomology classes.

The right side of (42) is algebraic, so \(\omega\) is algebraic. Every complex embedding of \(E\) sends \(r\) to a number of modulus one, since \(E\) is CM and \(r\bar r=1\). Applying any scalar conjugation to the displayed identity shows that every complex conjugate of \(\omega\) has modulus one. The same is then true of each root of \(X^M-\omega\). ◻

Proof. The active spherical action commutes with \(\mathcal D_q\). The radial Laurent module modulo \(\mathcal D_q^M-\omega\) is a direct sum of one-dimensional spaces \(\mathcal D_q=e_0\), because \(X^M-\omega\) has distinct nonzero roots. It suffices to compute the spherical character on one of them. This decomposition concerns the radial quotient only; it asserts no semisimplicity for the ambient cohomological Hecke action.

For this computation, extend the substitutions to \[\rho(g)b(x)=|\det g|_q^{-1/2}b(g^{-1}x), \qquad g\in\mathop{\mathrm{GL}}_3(\mathbb{Q}_q),\] and put \(K_3=\mathop{\mathrm{GL}}_3(\mathbb{Z}_q)\). This is a computational extension of the active \(\mathop{\mathrm{SL}}_3\)-action, not a claim of theta equivariance for the full moduli similitude group. Let \(\mathsf T_i\) be the raw double coset of \(\mathop{\mathrm{diag}}(q,\ldots,q,1,\ldots,1)\), with \(i\) entries \(q\), and \(\operatorname{vol}(K_3)=1\). Its right cosets correspond to sublattices between \(q\mathbb{Z}_q^3\) and \(\mathbb{Z}_q^3\) of index \(q^i\). For \(i=1,2\) there are \(q^2+q+1\) such lattices. The numbers containing a fixed primitive vector are respectively \(q+1\) and \(1\). Since \(\rho(g)b_0\) is \(q^{i/2}\) times the characteristic function of its sublattice, we obtain \[ \begin{aligned} \mathsf T_1b_0&=q^{1/2}\bigl((q+1)b_0+q^2b_1\bigr),\\ \mathsf T_2b_0&=q\bigl(b_0+(q^2+q)b_1\bigr),\\ \mathsf T_3b_0&=\mathcal D_qb_0. \end{aligned} \tag{45}\] For example, vectors in \(q\mathbb{Z}_q^3\) belong to every sublattice, while a primitive vector is tested by its nonzero class in \(\mathbb{F}_q^3\); the displayed counts are the numbers of the corresponding planes or lines containing that class.

On the quotient \(\mathcal D_q=e_0\) we have \(b_1=q^{-3/2}e_0b_0\). Substitution into (45) gives the values \[q\,e_1(e_0,q^{1/2},q^{-1/2}),\qquad q\,e_2(e_0,q^{1/2},q^{-1/2}),\qquad e_3(e_0,q^{1/2},q^{-1/2}),\] where \(e_i\) is the \(i\)-th elementary symmetric polynomial. In the half-modulus Satake isomorphism the raw minuscule cosets have transforms \(q e_1,q e_2,e_3\); these generate the spherical algebra, with \(e_3\) invertible. See (Gross 1998, Equation (3.14) and §6), where the \(\mathop{\mathrm{GL}}_n\) formula is attributed to Tamagawa. The character just computed therefore has parameter \(\{e_0,q^{1/2},q^{-1/2}\}\).

The geometric action on theta inputs is inverse by Lemma 23. Inversion sends the half-modulus parameter to its reciprocal, without a degree factor. One can check this on the same generators: \(\mathsf T_1^\vee=\mathsf T_3^{-1}\mathsf T_2\), \(\mathsf T_2^\vee=\mathsf T_3^{-1}\mathsf T_1\), and \(\mathsf T_3^\vee=\mathsf T_3^{-1}\), where \(f^\vee(g)=f(g^{-1})\). Their transforms are the corresponding elementary symmetric functions of the reciprocal triple. After interchanging the first two entries, the active projective parameter is therefore \(\{q^{1/2},q^{-1/2},e_0^{-1}\}\).

The determinant-zero \(\mathop{\mathrm{GL}}_3\) and \(\mathop{\mathrm{SL}}_3\) double cosets agree as in Lemma 23. Their Satake unipotent integrals and modulus factors agree as well. The half-sum of positive roots is \((1,0,-1)\), integral on the relevant cocharacter lattices; the Satake comparison is consequently defined over rational coefficients (Gross 1998, Proposition 3.6). Scalar conjugation of an active numerical character therefore conjugates its projective parameter. It may change the two entries \(q^{1/2},q^{-1/2}\) by their common sign. Removing that sign projectively leaves third entry \(\pm\sigma e\).

It remains to calculate the valuation. Put \(g=\sigma|_E\). By the definition of the prime labels, \[\nu_\iota(\sigma\tau(r)) =\operatorname{val}_{\mathfrak p_{g\tau}}(r).\] This is nonzero only when \(g\tau=1\) or \(c\). Among the representatives \(\Sigma\), the contributing one is \(g^{-1}\) with valuation \(-M\), or \(cg^{-1}\) with valuation \(M\). Here \(c\) is central in \(G_E\), and oddness of \(d\) gives in either case \[\sum_{\tau\in\Sigma}d(\tau)\nu_\iota(\sigma\tau(r)) =-M d(g^{-1}).\] Taking valuations in (42) and in \(e_0^M=\omega\) yields \(\nu_\iota(\sigma e_0)=\tfrac32d(g^{-1})\). Since \(e=e_0^{-1}\) and \(d(g^{-1})=v(g)\), this proves (44). The unit-modulus assertion follows from Lemma 24. ◻

Proof. Each central position has a unique lattice choice. For \(T_3\), use the norm-one element \(r\) constructed in Lemma 24; only its valuations and norm are needed here. The scalar \(r\) takes the \(T_3^M\)-target lattice back to the source lattice on the active and opposite blocks. It preserves every polarized period because it is an \(E\)-scalar of norm one. Its denominators are supported at \(q\), so a further power is one on the \(N\)-marking. That power of \(T_3\) is the identity on every tuple.

For \(C_0\), let \(I\) be the product of the primes where its full exponent is one. There is exactly one from each conjugate pair, including \(\bar w\), and \(I\bar I=(q)\). Choose \(I^m=(a)\). Then \[a\bar a=q^m u\] for a totally positive unit \(u\in\mathcal O_F^\times\). The scalar \(b=a^2/u\) has exactly the valuations of \(I^{2m}\) and satisfies \[b\bar b=q^{2m}.\] It identifies the corresponding powered target lattice with the source, with exactly the required polarization multiplier. A further power fixes the marking. This argument works on every rational Hermitian component because an \(E\)-scalar preserves each of its embedding filtrations. Thus a power of \(C_0\) is also the identity on the full tuple space. The induced Albanese homomorphisms have finite order as well. ◻

Proof. Good-reduction crystalline comparison identifies \[D_{\mathrm{cris}}\! \left(H^1_{\mathrm{\acute et}}(A_{\overline{\mathbb{Q}}_q},\mathbb{Q}_q)\right) \simeq H^1_{\mathrm{cris}}(\mathcal A_{\mathbb{F}_q}/\mathbb{Z}_q)[1/q]\] and identifies the filtered de Rham realization with \(D_A\). These identifications are functorial for extending endomorphisms and compatible with Frobenius and filtration (Bhatt et al. 2018, Theorem 1.1(i), Theorem 12.1, and §12.2).¹ As in Subsection 5.4, crystalline Frobenius is the pullback of the special fiber’s \(q\)-power endomorphism: Witt Frobenius on \(W(\mathbb{F}_q)=\mathbb{Z}_q\) is the identity (The Stacks Project Authors 2026, Tags 07N3 and 07N5). Crystalline representations give weakly admissible filtered modules (Brinon and Conrad 2009, Theorem 9.3.4). The degree-one Hodge filtration has only indices \(0\) and \(1\), and functoriality gives the assertion about extending endomorphisms.

The extension by \(L\) changes coefficients, not the geometric base field. With Frobenius acting trivially on this coefficient factor, the underlying \(\mathbb{Q}_q\)-filtered Frobenius module \[\bigl(D_A\otimes_{\mathbb{Q}_q}L,\ \varphi_A\otimes1,\ F^iD_A\otimes_{\mathbb{Q}_q}L\bigr)\] is a direct sum of \([L:\mathbb{Q}_q]\) copies of \(D_A\), and is weakly admissible. Since \(p\) and \(1-p\) both preserve the filtration and commute with Frobenius, their images split it as filtered Frobenius modules. Weak admissibility gives \(t_H\leq t_N\) on each image. The Hodge and Newton numbers are additive, and are equal for the whole module. They are therefore equal on each image separately.

Let \(m=\dim_{\mathbb{Q}_q}p(D_A\otimes L)\). A pure slope \(t\) gives \(t_N=tm\) after restriction of scalars: all \(\mathbb{Q}_q\)-conjugates of a \(q\)-adic algebraic number have the same valuation, including when \(L/\mathbb{Q}_q\) is ramified. On the other hand, \[t_H=\dim_{\mathbb{Q}_q}F^1\bigl(p(D_A\otimes L)\bigr),\] because the only filtration indices are \(0\) and \(1\). Equality of these two numbers proves the assertion. The general coefficient formalism is also described in (Hellmann 2011, Proposition 2.17(ii) and Corollary 2.22). ◻

Proof. Fix a full character \(\chi\) extending \(\eta\); if there is none, the assertion is vacuous. We first use Weil weight to select the Frobenius root. We then apply Lemma 27 and compare the resulting filtration ranks with those on the complex Betti realization.

At the fixed completion \(K_v=\mathbb{Q}_q\), write \[D_S=H^1_{\mathrm{dR}}((A_S)_{K_v}/\mathbb{Q}_q)\] and let \(\varphi\) be its crystalline Frobenius. If \(\Phi\) is the \(q\)-power Frobenius endomorphism of \(\widetilde A_S/\mathbb{F}_q\) used in Proposition 22, then \(\varphi=\Phi^*\) under crystalline comparison. The Hecke endomorphisms extend over \(\mathbb{Z}_q\); their reductions commute with \(\Phi\), so their pullbacks commute with \(\varphi\).

By additivity of pullback on abelian degree-one cohomology, the integral identity of Proposition 22 gives \[ \varphi^3-D_1^*\varphi^2+qD_2^*\varphi-q^3D_3^*=0. \tag{50}\] Pullback reverses composition, but all operators in this formula commute.

Choose a prime \(\ell\ne q\) and put \(V_\ell\widetilde A_S=T_\ell\widetilde A_S\otimes_{\mathbb{Z}_\ell}\mathbb{Q}_\ell\). Weil’s theorem for abelian varieties over \(\mathbb{F}_q\) gives \[P_\Phi(X)=\det\bigl(X-\Phi\mid V_\ell\widetilde A_S\bigr)\in\mathbb{Z}[X],\] all of whose complex roots have absolute value \(q^{1/2}\). The prime-to-\(q\) Tate-module action on endomorphisms is faithful. Cayley–Hamilton therefore gives \(P_\Phi(\Phi)=0\) in the endomorphism ring, and degree-one additivity gives \(P_\Phi(\varphi)=0\). These statements hold for an arbitrary abelian variety over \(\mathbb{F}_q\); see (Milne 1986, Lemma 12.2, Proposition 12.9, and Theorem 19.1(c) with its proof). No CM or ordinary hypothesis on \(\widetilde A_S\) is involved. This uses the \(q\)-power endomorphism and its pullback, not a reciprocal Galois-Frobenius convention.

Transport the algebraic coefficients of \(p_\chi\) and \(\chi\) by \(\iota\). On the corresponding generalized summand of \(D_S\otimes_{\mathbb{Q}_q}\overline{\mathbb{Q}}_q\), the commuting Hecke maps and \(\varphi\) can be simultaneously triangularized. Equations (50) and (49) imply that every diagonal Frobenius eigenvalue is a root of \[X^3-qe_1(\alpha)X^2+q^2e_2(\alpha)X-q^3e_3(\alpha) =\prod_{j=1}^3(X-q\alpha_j).\] Thus the candidates, before applying \(\iota\), are \[ bq^{3/2},\qquad bq^{1/2},\qquad qbe'. \tag{51}\] Each is algebraic. If its image under \(\iota\) is a root of the rational polynomial \(P_\Phi\), the algebraic number itself is a root of \(P_\Phi\). We may therefore apply the complex absolute-value condition. The three absolute values are respectively \[q^{3/2},\qquad q^{1/2},\qquad q,\] because every complex conjugate of \(b\) and \(e'\) has modulus one. Only \(bq^{1/2}\) remains.

The weight condition has now selected one Newton slope on the entire generalized summand. Equation (48) gives \[ \nu_\iota(bq^{1/2}) =\frac12-\frac13\nu_\iota(e')=t_\eta. \tag{52}\] This conclusion concerns every Frobenius eigenvalue; Jordan blocks do not change it. Choose a finite extension \(L/\mathbb{Q}_q\) containing the coefficients of the transported idempotent and numerical character. The idempotent \(p_\chi\) is a polynomial in extending Hecke maps, so it preserves the Hodge filtration and commutes with \(\varphi\). Lemma 27 makes its filtration pure of index \(t_\eta\).

It remains to relate this filtered rank to the Hodge type of the same complex generalized space. Put \[V_{\mathrm{dR}}=H^1_{\mathrm{dR}}(A_S/K).\] Its filtration and Hecke operators are defined over the fixed field \(K\). For a numerical character \(\chi\), choose a polynomial expression for \(p_\chi\), and let \(k_\chi\subset\overline{\mathbb{Q}}\) be the actual compositum of \(K\) and its algebraic coefficients. Evaluating that polynomial in the fixed operators gives an idempotent on \(V_{\mathrm{dR}}\otimes_K k_\chi\). Its induced filtration is its projected filtration, since the idempotent preserves \(F^\bullet\). The dimensions of its image and of this filtration are ranks of matrices over \(k_\chi\); they are unchanged under the restriction of the fixed embedding \(\iota\) to \(k_\chi\).

Over the given embedding \(K\subset\mathbb{C}\), Betti–de Rham comparison intertwines the same operators and polynomial. It identifies these ranks with the filtration ranks on \(H_\chi\otimes_{\overline{\mathbb{Q}}}\mathbb{C}\). For a scalar-conjugate numerical character \(\sigma\chi\), repeat this argument with \(p_{\sigma\chi}\) in the same \(K\)-defined operators. On Betti cohomology their rational matrices give \(\sigma H_\chi=H_{\sigma\chi}\). We have therefore computed the type of the scalar-conjugated Betti space on the same complex variety. Each conjugate projector is evaluated separately in the fixed \(K\)-defined operators. This works even when \(K\) intersects the character field, since it uses ranks over their actual compositum.

Finally the complex Hecke operators preserve both \(H^{1,0}\) and \(H^{0,1}\). Hence a full projected \(F^1\) means that the whole generalized space is holomorphic, and zero projected \(F^1\) means that it is antiholomorphic. Equation (52) depends only on \(\eta\), so summing all full extensions gives the assertion for \(H^{\mathrm{act}}_\eta\). ◻

Proof of Theorem 18. Use the Albanese identification \(H^1(A_S,\mathbb{Q})\simeq H^1(S,\mathbb{Q})\). Extend a theta class from \(S_0\) by zero on the other components of \(S\). This is compatible with the active algebra, because Lemma 23 makes that algebra component-preserving. The class is holomorphic by Proposition 15. Decompose it into the active generalized character spaces of Equation (46). For each nonzero component, let \(\eta\) be its active character and \(e\) its parameter from Proposition 25. For every \(\sigma\in\mathop{\mathrm{Gal}}(\overline{\mathbb{Q}}/\mathbb{Q})\), put \(g=\sigma|_E\). That proposition and Proposition 28 give \[ \nu_\iota(\sigma e)=-\frac32v(g),\qquad t_{\sigma\eta} =\frac12-\frac13\nu_\iota(\sigma e) =\frac{1+v(g)}2. \tag{53}\] The possible sign in the conjugate parameter \(e'=\pm\sigma e\) has valuation zero. Consequently the entire active space for \(\sigma\eta\) is of type \((1,0)\) when \(v(g)=1\) and of type \((0,1)\) when \(v(g)=-1\). This applies to all of its vectors, not just to theta vectors. At the identity it gives type \((1,0)\), in agreement with \(v(1)=1\) and the holomorphy of the theta class.

Each active space is defined over \(\overline{\mathbb{Q}}\) in the rational Betti structure, because its idempotent has algebraic coefficients. Choose \(0\ne x\in H^{\mathrm{act}}_\eta\) and \(0\ne u\in U(v)_1(\overline{\mathbb{Q}})\). Define a \(\overline{\mathbb{Q}}\)-linear map \[h:U(v)_{\overline{\mathbb{Q}}}\longrightarrow H^1(S,\overline{\mathbb{Q}})\] by \(h(u)=x\) and by setting it equal to zero on every other embedding line of \(U(v)_{\overline{\mathbb{Q}}}\). This is an element of \[\mathop{\mathrm{Hom}}_{\mathbb{Q}}\bigl(U(v),H^1(S,\mathbb{Q})\bigr)\otimes_{\mathbb{Q}}\overline{\mathbb{Q}}.\] Scalar conjugation by \(\sigma\) moves its nonzero source line to \(U(v)_g\) and its target vector to \(\sigma x\in H^{\mathrm{act}}_{\sigma\eta}\). The source line has the type prescribed by \(v(g)\), and Equation (53) gives the same type for the target. All other source lines map to zero. Hence every scalar conjugate of \(h\) is a Hodge map.

Lemma 2, applied to the complex-linear subspace of Hodge maps, expresses \(h\) as a \(\overline{\mathbb{Q}}\)-linear combination of rational Hodge maps. Algebraic-Betti vectors span each complex active space, so taking complex spans covers every active component of the original theta class, including classes with transcendental coefficients. Restriction from \(S\) to \(S_0\) is a rational Hodge map and gives Equation (31).

For a finite collection of inputs, all level choices are made before the auxiliary prime is chosen, and only finitely many primes are excluded. The preceding argument therefore applies to that collection at one sufficiently deep level. If \(\pi:S'_0\to S_0\) is a deeper principal cover, then \(\pi^*\) is a rational Hodge map and the theta classes pull back to their classes on \(S'_0\). Applying \(\pi^*\) to Equation (31) proves persistence. ◻

Proof. Apply Theorem 18 to each finite input family. Choose a principal level contained in the resulting finitely many principal levels and in the level of \(X_{\mathrm{base}}\), and deepen it until Lemma 19 applies. Its quotient \(Y\) is smooth, connected, and projective. The reference moduli construction depends on \(V\), not on the multiplicity line, so it applies to this common level. The persistence in Theorem 18 pulls all the rational-map spans to \(Y\) and then to every deeper principal level. Each application chooses its auxiliary prime only after its own finite family and level have been fixed; the primes need not agree. ◻

Proof. Set \(U=W(r_0)\). Hodge symmetry shows that \(U\) is effective of weight \(n\ge0\), with smallest Hodge index zero. If \(n=0\), every type of \(U\) is \((0,0)\). Simplicity then gives \(U\simeq\mathbb{Q}\), and we take \(B\) to be a point.

Suppose \(n>0\). We first show that \(U\) is a line over a CM field; we will then decompose its Hodge type into weight-one CM types. Write \(T=J(U)\). Let \(E_0\) be the \(\mathbb{Q}\)-subalgebra of \(\mathop{\mathrm{End}}_\mathbb{Q}(U)\) generated by the image of \(T\). After splitting the torus, its action is diagonal by characters, so \(E_0\) is a commutative finite étale algebra. Its field idempotents and every subspace linear over one field factor are \(T\)-stable. Simplicity of \(U\) therefore forces \(E_0\) to be a field \(E\) and \(\dim_E U=1\).

We check that \(E\) is a CM field. Let the Weil operator \(C_U\) act as \(i^{p-q}\) on \(U^{p,q}\). By polarizability there is a rational Hodge pairing \(\psi\), with the conventional weight-dependent sign absorbed in its definition, for which \[q(x,y)=\psi(C_Ux,y)\] is positive definite on \(U_\mathbb{R}\). The adjoint \(\dagger\) for \(\psi\) sends \(t\in T\) to \(t^{-1}\), hence preserves \(E\). Every element of \(E\) commutes with \(C_U\), so \(\dagger\) is also the adjoint for \(q\). On the real finite étale algebra \(E_\mathbb{R}\), it must fix each primitive idempotent: if an idempotent \(e\) were sent to a different one, then \(e^\dagger e=0\), contrary to \(q(ex,ex)=q(x,e^\dagger ex)>0\) for some \(x\). On a complex factor it must act by complex conjugation, since the identity would give \(i^\dagger i=-1\), again contrary to positivity. Thus if \(\dagger\) is trivial on \(E\), the field is totally real. If it is nontrivial, a real embedding would force \(\dagger\) to be trivial on \(E\), so there are no real embeddings; the fixed algebra in \(E_\mathbb{R}\) is then a product of copies of \(\mathbb{R}\). Its fixed field is therefore totally real, and \(E\) is CM.

The totally real case cannot occur here. The Hodge circle acts through \(E_\mathbb{R}^\times\); its image in each real scalar factor is a compact connected subgroup of \(\mathbb{R}^\times\), hence is trivial. All Hodge types of \(U\) would then have equal indices. Since their smallest index is zero, this would force \(n=0\). Consequently \(E\) is CM, and \(\dagger\) is its complex conjugation.

Let \(\Sigma=\mathop{\mathrm{Hom}}(E,\mathbb{C})\), and write \(\bar\sigma\) for the conjugate of \(\sigma\). Since \(E\) acts by Hodge endomorphisms and \(U\) is an \(E\)-line, \[U_\mathbb{C}=\bigoplus_{\sigma\in\Sigma}U_\sigma,\qquad U_\sigma\text{ has type }(a_\sigma,n-a_\sigma),\] where \(0\le a_\sigma\le n\) and \(a_\sigma+a_{\bar\sigma}=n\). Choose \(n\) CM types \(\Phi_1,\ldots,\Phi_n\) of \(E\) with \[\#\{j:\sigma\in\Phi_j\}=a_\sigma.\] This is possible independently for each conjugate pair: put \(\sigma\) into any \(a_\sigma\) lists and \(\bar\sigma\) into the remaining lists.

For each \(j\), give the rational vector space \(H_j=E\) the weight-one Hodge structure whose \(\sigma\)-line has type \((1,0)\) exactly when \(\sigma\in\Phi_j\). The polarization and Riemann construction of Section 2, surrounding Equation (2), applies to any CM field, with no Galois hypothesis. It gives a CM abelian variety \(B_j\) with \(H^1(B_j,\mathbb{Q})\simeq H_j\).

Identify \(U\) and the \(H_j\) with \(E\) as \(E\)-lines. The rational multiplication map \[E^{\otimes_\mathbb{Q}n}\longrightarrow E,\qquad x_1\otimes\cdots\otimes x_n\longmapsto x_1\cdots x_n\] is surjective. Over \(\mathbb{C}\) it kills tensors with unequal embedding labels, while the tensor with all labels equal to \(\sigma\) has type \((a_\sigma,n-a_\sigma)\) and maps to \(U_\sigma\). It is therefore a Hodge map to \(U\). For \(B=\prod_{j=1}^nB_j\), its source is the rational Künneth summand \(\bigotimes_j H^1(B_j,\mathbb{Q})\) of \(H^n(B,\mathbb{Q})\). Extending by zero on the other summands and twisting by \((-r_0)\) proves the lemma. ◻

Proof. The zero substructure permits \(Z=\varnothing\). Otherwise, semisimplicity of polarizable rational Hodge structures reduces the claim to a simple summand \(W\), after which we take the finite union of the resulting supports. Let \(r_0\) be its smallest Hodge index. Then \(r_0\ge r\). Also \(r_0\le k/2\le\dim A\), since \(W\ne0\). Put \(n=k-2r_0\), take \(B\) from Lemma 30, and write \(b=\dim B\). Composing the map of that lemma with \(W\hookrightarrow H^k(A,\mathbb{Q})\) gives \[f:H^n(B,\mathbb{Q})(-r_0)\longrightarrow H^k(A,\mathbb{Q}), \qquad \mathop{\mathrm{im}}(f)=W.\] By Poincaré duality, \(f\) corresponds to a rational Hodge class of type \((b+r_0,b+r_0)\) in \[H^{2b-n}(B,\mathbb{Q})\otimes H^k(A,\mathbb{Q}) \subset H^{2b+2r_0}(B\times A,\mathbb{Q}).\] The product \(B\times A\) is CM. Theorem 1 therefore represents this exact class by \(\Gamma\in\mathop{\mathrm{CH}}^{b+r_0}(B\times A)_\mathbb{Q}\), whose action in degree \(n\) is \(f\).

Choose a finite rational cycle representing \(\Gamma\), and let \(Z\) be the projection of its support to \(A\). Every component of that cycle has dimension \[\dim(B\times A)-(b+r_0)=\dim A-r_0.\] The projection is proper, so \(Z\) is closed and \(\mathop{\mathrm{codim}}_A Z\ge r_0\ge r\). The correspondence restricts to zero on \(B\times(A\setminus Z)\); functoriality of its cohomological action therefore makes its image \(W\) restrict to zero on \(A\setminus Z\). ◻

Proof. Apply Milne’s Theorem 7.1 and his formulation (0.1) (Milne 1999), with the correction in (Milne, n.d.-a): the cofinal auxiliary Galois CM fields are enlarged to contain a real subfield of degree \(>2\) inert at the residue prime. The theorem’s hypothesis is supplied by Theorem 1. It gives the stated equality of equivalence relations and algebraic spanning of the geometric Tate classes: those fixed over some finite field of definition.

Let \(\alpha\) lie in the displayed \(G_q\)-invariant space. The geometric conclusion writes \(\alpha\) as a finite \(\mathbb{Q}_\ell\)-linear combination of cycle classes. All the cycles in this expression descend to \(A_{\mathbb F_{q^m}}\) for one \(m\); let \(z_m\in\mathop{\mathrm{CH}}^r(A_{\mathbb F_{q^m}})\otimes_\mathbb{Z}\mathbb{Q}_\ell\) denote the resulting combination. For the finite projection \(\pi_m:A_{\mathbb F_{q^m}}\to A\), proper pushforward and base change for cycle classes give \[\operatorname{cl}_\ell\!\left(\frac1m(\pi_m)_*z_m\right) =\frac1m\sum_{\sigma\in \mathop{\mathrm{Gal}}(\mathbb F_{q^m}/\mathbb{F}_q)}\sigma(\alpha)=\alpha.\] Thus \(m^{-1}(\pi_m)_*z_m\) is the required class over \(\mathbb{F}_q\). Division by \(m\) is valid in \(\mathbb{Q}_\ell\), even when \(m\) is divisible by \(\ell\) or by the characteristic. ◻

Proof. Apply Milne’s Theorem 3.3 (Milne 2002) using Theorem 1. In Milne’s proof, the same Hodge hypothesis supplies the compatibility and positivity of the relevant polarizations through Theorem 2.1, read with the author’s addendum (Milne, n.d.-b) and the corrected auxiliary-field choice from (Milne, n.d.-a). Theorem 3.3 gives both the equality of equivalence relations and positivity on the primitive space of cycles modulo homological equivalence. The former identifies that primitive space with the numerical one above.

The case \(g=0\) is immediate. For \(g>0\), choose a positive integer \(m\) so that \(mh\) is the class of a smooth hyperplane section; clearing denominators, taking a sufficiently large very ample multiple, and Bertini’s theorem permit this. Multiplication by the nonzero rational scalar \(m^{g-2r+1}\) does not change the primitive kernel, and the form for \(mh\) is \(m^{g-2r}\) times the form for \(h\). This positive factor preserves positive definiteness. ◻