The rational Hodge conjecture for CM abelian varieties

Proof. Choose a finite Galois extension K/\Q containing the coefficients of t. For a\in K, the vector

is rational. Each automorphism of K extends to \Qbar, 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 [27, Proposition 6.5, Theorems 6.7–6.8, and Corollary 6.9].

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

Proof. The zero-dimensional case is immediate. Put H=H^1(A,\Q), and first assume that J(H) is a torus. Its associative envelope E_0\subset\End_{\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 \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,

The involution cannot exchange two primitive factors of the real finite étale algebra E_0\otimes\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 \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\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_{\Q}H. Lemma 2.2 realizes it in \End(A)\otimes\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

Hence every m_j=1. The centralizer of this action in \GL(H) is the torus \prod_j\Res_{K_j/\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 2.1 in \Hom_{\Q}(H,H'), followed, when applicable, by Lemma 2.2.□

Proof. Let A be a CM abelian variety and let h\in\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 2.3, choose a rational decomposition of H=H^1(A,\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,\Q)=\Lambda^*H, the exterior monomials in these lines give a direct-sum decomposition of H^*(A,\Qbar) into one-dimensional lines [27, 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

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

Repeated factors in these products are kept as separate slots. Lemma 2.4 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

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 2.5 supplies the pure algebraic tensor

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

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

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\Qbar^\times, since its vectors have algebraic Betti coefficients and \ell_i is rational. The codimension on the left of Equation (2.6) is

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,\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 (2.1) proves the rational assertion.

Finite products are covered because they are themselves CM, as observed after Lemma 2.3.□

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 (2.4) 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 (2.4) to g,g,gc,gc. The complete type function of the line labeled g is the right translate

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 2.4 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 (2.4).

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. By Lemma 2.2, there are positive integers m_i and morphisms f_i:S\to B(v_i) with f_i^*=m_i h_i. Therefore

Nonvanishing follows from Equation (3.1), since the new value is multiplied by \prod_i m_i. The value is algebraic because the pullbacks, cup product, and evaluation on [S] are rational operations on Betti cohomology and the \alpha_i have algebraic Betti coefficients.

Put A=\prod_iB(v_i), n=\dim A=4s, and f=(f_i):S\to A. Let

For t_\alpha=\alpha_1\otimes\cdots\otimes\alpha_4 in the four-H^1 Künneth summand of H^4(A,\Qbar), the projection formula gives

Thus Z gives a nonzero functional on the switch line; it is not itself a vector on that line. In particular f(S) cannot have dimension less than two, since then the proper pushforward of the two-cycle [S] and the displayed pairing would be zero.

For each i, take a copy B(v_i)' with its transported E-action, and choose a nonzero vector \beta_i\in U(v_i)'_{c\lambda_i} with algebraic Betti coefficients. Under the Künneth inclusion define the pure class

If \sigma\in\Gal(\Qbar/\Q) restricts to g\in G_E, then Equation (2.3) puts its two factors on the lines labeled

These have opposite Hodge types. This uses centrality of c in G_E only, and remains valid when the scalar coordinates of \alpha_i and \beta_i lie outside E. Every scalar conjugate of D_i therefore has type (1,1). Scalar descent and the Lefschetz (1,1) theorem give

The entire cohomology class D_i is algebraic here. It was defined in the pure H^1\otimes H^1 summand before applying descent, so there is no need to project an arbitrary divisor class onto that summand.

Set A'=\prod_iB(v_i)'. Let p:A\times A'\to A and p':A\times A'\to A' be the projections, and let q_i:A\times A'\to B(v_i)\times B(v_i)' select the i-th old and new slots. Write

Order the old factors first and the new factors second in the Künneth identification for A\times A'. Moving the new degree-one factor of D_i past the later old degree-one factors makes 3+2+1=6 sign changes. Hence

In particular K has degree four in the old factors and degree four in the new factors. Integration over A consequently gives the exact formula

Any other Künneth component of Z contributes zero to this integration. The codimension of the left side is (n-2)+4-n=2, equivalently its cohomological degree is (2n-4)+8-2n=4.

By Equation (3.4), the left side of Equation (3.5) is algebraic with \Qbar-coefficients. Since k\in\Qbar^\times, division by k proves algebraicity of the line generated by t_\beta, whose labels are (c,c,1,1). Ordinary scalar complex conjugation fixes the rational cycle span on the same A' and changes these labels to (1,1,c,c). Identifying the copied factors with the original ones gives the asserted line on A.□

Proof. Fix a nonzero algebraic Betti generator of each selected line. Each \eta_i is a finite complex linear combination of its images under rational Hodge maps. Expanding the nonzero cup product selects a nonzero term of the form in Equation (3.1). Thus Proposition 3.1 applies. This argument uses only finite sums; all four maps have the same target H^1(S,\Q).□

Proof. Write \tau(\delta)=it_\tau, where t_\tau\in\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

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

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

For general P the active determinant is therefore

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 (4.6); 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

The matrix K(g,Z) is symmetric; this follows from JC^t=CJ^t, a symplectic identity. The formula to prove is

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 (4.13) with j=1. For m(A)=\diag(A,A^{-t}), the chosen lift acts by

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

The Gaussian integral is

where the square root of \det(-iZ) equals 1 at Z=iI. Taking

gives j(w,Z)^2=\det Z and proves (4.13) 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

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 (4.13). 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 (4.13), the transformed sum is

Equating this with the original sum gives (4.11). 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

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 (4.11), so

Comparison of homogeneous terms yields (4.12). Finally,

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 [29, §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 \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 [12, §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 [5, §§2.1 and 3.1] is exactly (4.4). We take the same global additive character as in Lemma 4.2. The determinant twists from changing the characters vanish on G=\SU(V). The product of the finite local splittings is therefore an honest action \omega_{P,f} on G(\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(\U(2)\times\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 \SU(3). On the maximal compact \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_\Q\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_\Q\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_\Q\R) into the double cover. Its half-determinant action is the one on L^n in (4.6). The original circle lift and this double lift, pushed out to the circle extension, differ by a continuous character of G(F\otimes_\Q\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 \diag(h^{-t},h). The Schrödinger linear formula gives (4.15), up to the scalar of the chosen bare lift. On \SL_3(F_v), the normalized substitution lift and the unitary lift have a character as their ratio. This character is trivial because \SL_3(F_v) is perfect: elementary matrices generate it, and for distinct i,j,k,

The determinant factor is also 1 there. This proves the split-place assertion without removing scalars from lifts outside \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(\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 (4.3). Descent follows from Lemma 4.4.

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

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

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,

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

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 (4.16), 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 4.4.

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 4.3, the total reference input becomes another finite Schwartz input in the adapted coordinates. Such an input is a finite sum

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

The coefficient of the mixed product is 1, by the 1/k! normalization in (4.10). 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 (4.12). 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 4.4.

For (ii), fix a point of D. Proposition 4.5 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 \SU(3), since all changes are in the multiplicity factor. Their determinant characters on \SU(3) are trivial. The active pieces have not changed, so their determinant character remains \ell^4. The infinite comparison in Lemma 4.4 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 (4.6) 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 (4.10) now give uniform convergence of every fixed order of derivatives in the chart coordinates and the parameters. Projection to (4.8) 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,

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

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 4.6(iii) now hold with the same rational W, the same G, the same total finite \phi, and the same compact X. Thus

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 (4.18) for all sufficiently close rational choices. At each such endpoint holomorphy follows from Proposition 4.6(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

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 [31, Chapter 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

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 (\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 [8, §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 (5.2) 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 [6]; see [27, 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 5.2. 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

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 \Qbar. 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\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

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 (5.7) 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 (5.5). 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 [14, §3].□

Proof. Let k=\overline{\Fq}, and let x be an ordinary geometric point of the special fiber of S. Its active q-divisible group has the canonical splitting

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

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

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 \Fq-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:

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 [8, §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 [8, §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

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

Here V_{B^\vee} denotes Verschiebung. The action on the target is \iota_E^{(q)}. For the constant label group T=(\Z/N\Z)^{6s}, naturality gives

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 \Fq, 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',

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,

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

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

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

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 \Z[\mathcal O]. Define

The component Albanese maps reduce over \Fq, so \Phi\alpha=\alpha X_0. For ordinary x,y in one source component, lift them and specialize the integral identity (5.7). With

it gives

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 (5.9) on this difference is

The operator inside this expression vanishes:

Only the two proved commutations with X_0 are used. Ordinary points are dense in every smooth component by Lemma 5.3. 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 \Fq.□

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 5.4 and undoing its marking change. A determinant-exponent-zero lattice position has a representative in \SL_3(\Q_q). Adjusting determinant units in Smith bases shows that its double cosets and branches are exactly those for \SL_3(\Z_q)\backslash\SL_3(\Q_q)/\SL_3(\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 [27, Theorem 4.16]. Put K_q=\SL_3(\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 4.2 and 4.4 give

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

up to the finite Weil phase of the bare lift. The linear-subgroup formula, rational summation invariance, and the double-cover control are in [37, 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 4.2, 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 \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 (6.2). This equality of one-forms implies the asserted equality of cohomology classes.

The right side of (6.2) 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

and put K_3=\GL_3(\Z_q). This is a computational extension of the active \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 \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\Z_q^3 and \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

For example, vectors in q\Z_q^3 belong to every sublattice, while a primitive vector is tested by its nonzero class in \Fq^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 (6.5) gives the values

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 [14, Equation (3.14) and §6], where the \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 6.1. 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 \GL_3 and \SL_3 double cosets agree as in Lemma 6.1. 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 [14, 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,

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

Taking valuations in (6.2) 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 (6.4). The unit-modulus assertion follows from Lemma 6.2.□

Proof. Each central position has a unique lattice choice. For T_3, use the norm-one element r constructed in Lemma 6.2; 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

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

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

and identifies the filtered de Rham realization with D_A. These identifications are functorial for extending endomorphisms and compatible with Frobenius and filtration [4, 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(\Fq)=\Z_q is the identity [36, Tags 07N3 and 07N5]. Crystalline representations give weakly admissible filtered modules [9, 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 \Q_q-filtered Frobenius module

is a direct sum of [L:\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_{\Q_q}p(D_A\otimes L). A pure slope t gives t_N=tm after restriction of scalars: all \Q_q-conjugates of a q-adic algebraic number have the same valuation, including when L/\Q_q is ramified. On the other hand,

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 [17, 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 6.5 and compare the resulting filtration ranks with those on the complex Betti realization.

At the fixed completion K_v=\Q_q, write

and let \varphi be its crystalline Frobenius. If \Phi is the q-power Frobenius endomorphism of \widetilde A_S/\Fq used in Proposition 5.5, then \varphi=\Phi^* under crystalline comparison. The Hecke endomorphisms extend over \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 5.5 gives

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_{\Z_\ell}\Q_\ell. Weil’s theorem for abelian varieties over \Fq gives

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 \Fq; see [24, 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_{\Q_q}\overline{\Q}_q, the commuting Hecke maps and \varphi can be simultaneously triangularized. Equations (6.10) and (6.9) imply that every diagonal Frobenius eigenvalue is a root of

Thus the candidates, before applying \iota, are

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

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 (6.8) gives

This conclusion concerns every Frobenius eigenvalue; Jordan blocks do not change it. Choose a finite extension L/\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 6.5 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

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\Qbar 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\C, Betti–de Rham comparison intertwines the same operators and polynomial. It identifies these ranks with the filtration ranks on H_\chi\otimes_{\Qbar}\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 (6.12) depends only on \eta, so summing all full extensions gives the assertion for H^{\mathrm{act}}_\eta.□

Proof of Theorem 5.1. Use the Albanese identification H^1(A_S,\Q)\simeq H^1(S,\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 6.1 makes that algebra component-preserving. The class is holomorphic by Proposition 4.5. Decompose it into the active generalized character spaces of Equation (6.6). For each nonzero component, let \eta be its active character and e its parameter from Proposition 6.3. For every \sigma\in\Gal(\Qbar/\Q), put g=\sigma|_E. That proposition and Proposition 6.6 give

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 \Qbar 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(\Qbar). Define a \Qbar-linear map

by h(u)=x and by setting it equal to zero on every other embedding line of U(v)_{\Qbar}. This is an element of

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 (6.13) 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 2.1, applied to the complex-linear subspace of Hodge maps, expresses h as a \Qbar-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 (5.1).

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 (5.1) proves persistence.□

Proof. Apply Theorem 5.1 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 5.2 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 5.1 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 of Theorem 2.5. By Lemma 2.7, it suffices to treat the case s\geq2 and v_i(1)=1 for all four functions. Use the common Galois CM field E, the active embedding \mu=1, and the line signs

They satisfy d_1+d_2=d_3+d_4 and d_i(\mu)=1. Choose the rank-three Hermitian space V, the ball D, and the rational orthogonal decomposition W=W_1\perp W_2 prescribed by d_1,d_2, as in Section 4.

Proposition 4.6(ii) supplies a total finite theta input whose form for this decomposition is nonzero. With that input fixed, Corollary 4.7 supplies a second rational orthogonal decomposition and a nonzero mixed period. Write X' for the individual-factor cover constructed there, and \xi_1,\xi_2 for the two pulled-back endpoint forms. On this common compact principal quotient they have finite expansions

where the constants account for the chosen constant-line trivializations, and

All four families of individual inputs here are finite. The quotient and these expansions may be pulled to any deeper principal level; the integral is then multiplied by the degree of the cover.

Apply Corollary 6.7 to these four families, with X_{\mathrm{base}}=X' and the fixed E,V,D,\Lambda. It gives a common smooth connected projective principal cover \pi:Y\to X' on which all four U(v_i)_1-span conclusions hold; the auxiliary primes may differ.

Denote the pulled-back one-form classes by

with a\in I for i=1,2 and a\in J for i=3,4. These classes are represented by holomorphic one-forms. Their cup products compute the integrals of the wedge products, and the covering formula gives

Choose a,b for which the displayed cup-product period is nonzero, and write its four holomorphic factors as \vartheta_i.

Corollary 6.7 gives

The maps are rational, so they commute with scalar complex conjugation, which sends U(v_i)_1 to U(v_i)_c. Consequently

The four classes \vartheta_1,\vartheta_2,\overline{\vartheta_3}, \overline{\vartheta_4} thus satisfy Corollary 3.2 on the same projective surface Y, with the source labels (1,1,c,c) and the nonzero period selected from Equation (7.1). The normalized switch line is therefore algebraic. Lemma 2.7 returns every switch to this case or to a product of algebraic two-factor classes, and proves the theorem.□

Proof of Theorem 1.1. The four-factor switch theorem is now proved for every Galois CM field. Proposition 2.6 consequently places every rational Hodge class on a CM abelian variety in its rational cycle span. Its final scalar descent takes place on the original variety. Finite products and powers are CM, so the same conclusion covers them.□

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\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 \Q-subalgebra of \End_\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

is positive definite on U_\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_\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_\R is then a product of copies of \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_\R^\times; its image in each real scalar factor is a compact connected subgroup of \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=\Hom(E,\C), and write \bar\sigma for the conjugate of \sigma. Since E acts by Hodge endomorphisms and U is an E-line,

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

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 (2.2), applies to any CM field, with no Galois hypothesis. It gives a CM abelian variety B_j with H^1(B_j,\Q)\simeq H_j.

Identify U and the H_j with E as E-lines. The rational multiplication map

is surjective. Over \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,\Q) of H^n(B,\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 8.1, and write b=\dim B. Composing the map of that lemma with W\hookrightarrow H^k(A,\Q) gives

By Poincaré duality, f corresponds to a rational Hodge class of type (b+r_0,b+r_0) in

The product B\times A is CM. Theorem 1.1 therefore represents this exact class by \Gamma\in\CH^{b+r_0}(B\times A)_\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

The projection is proper, so Z is closed and \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) [25], with the correction in [22]: 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 1.1. 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 \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\CH^r(A_{\mathbb F_{q^m}})\otimes_\Z\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

Thus m^{-1}(\pi_m)_*z_m is the required class over \Fq. Division by m is valid in \Q_\ell, even when m is divisible by \ell or by the characteristic.□

Proof. Apply Milne’s Theorem 3.3 [26] using Theorem 1.1. 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 [23] and the corrected auxiliary-field choice from [22]. 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.□