Proof. For abelian varieties over \(\overline{\mathbb Q}\), Deligne’s absolute-Hodge theorem [6] and Blasius’s comparison theorem [3] give the assertion. The compatibility of the crystalline comparison with the de Rham comparison is part of [3]. The formulation for a complete mixed-characteristic discrete valuation ring with perfect residue field is given in [30]; the paragraph following that theorem explains removal of the number-field hypothesis. A finite extension of \(K_0\) is within that scope. The stated functorialities are those of the comparison isomorphisms. ◻

Proof. If \(r>\dim Y\), the input class is zero; if \(r>\dim X\), its pullback is zero. We may therefore suppose that \(f\) is an actual homomorphism and \(r\leq\min(\dim X,\dim Y)\). For \(\delta\in\mathop{\mathrm{Lef}}^{\dim X-r}(X)\), the class \(f_*\delta_v\) lies in the Lefschetz subspace of codimension \(\dim Y-r\) in every realization. Choose rational bases \(a_i\) of \(\mathop{\mathrm{Lef}}^{\dim Y-r}(Y)\) and \(b_j\) of \(\mathop{\mathrm{Lef}}^r(Y)\). The matrix \(M_{ij}=\deg(a_i b_j)\) is an invertible rational matrix. If \(f_*\delta_v=\sum_i c_{i,v}(a_i)_v\), the projection formula gives \[\sum_i c_{i,v}M_{ij}=\deg(\delta f^*b_j).\] The right side is a rational number independent of \(v\). Solving this one rational system shows that \(c_{i,v}=c_i\in\mathbb Q\) for every \(v\). Thus \(f_*\delta\) is simultaneous rational Lefschetz. Applying the projection formula once more proves the assertion. For rational \(f\), use (1). ◻

Proof. The crystalline map already commutes with Frobenius. Full faithfulness for rational filtered Dieudonné modules gives a rational morphism of the associated \(p\)-divisible groups; see [9] and the perfect-residue-field formulation in [11]. Equivalently, crystalline comparison first gives the rational map on Tate modules. Multiply by an integer \(N\) that clears both its Tate-lattice denominator and the denominator of \(f_0\). Tate’s extension theorem [34] extends the resulting integral generic-fiber map of \(p\)-divisible groups over \(R\). Faithfulness of the special-fiber Dieudonné functor identifies its reduction with the map induced by \(Nf_0\). Here equality after inverting \(p\) is already equality of integral maps, because the integral Dieudonné modules are torsion free. The integer \(N\) is fixed for the remainder of the construction.

The specified-map form of Serre–Tate deformation theory now gives a lift over each \(R/\mathfrak m_R^n\) [9]. One can also deduce it from the fixed-object formulation [11] as follows. On \(X_0\times Y_0\) form the shear \(s_0(x,y)=(x,y+Nf_0(x))\). The lifted \(p\)-divisible-group map gives the same shear on the product \(p\)-divisible group. View the existing product deformation with two special-fiber markings differing by \(s_0\). Full faithfulness lifts the shear to an algebraic isomorphism between these marked deformations; its second component restricted to \(X\) lifts \(Nf_0\). Uniqueness makes these maps compatible at every finite order.

Formal existence algebraizes the resulting formal map [33]: its source is proper over the complete noetherian ring \(R\), and its target is separated and of finite type. Dividing by the fixed \(N\) gives the desired rational lift. Rigidity gives uniqueness and injectivity of specialization on rational homomorphisms [18].

If \(f_0\) is symmetric, injectivity shows that its lift is symmetric. The usual identification \[\mathop{\mathrm{NS}}(X_K)\otimes\mathbb Q \simeq\mathop{\mathrm{Hom}}(X_K,X_K^\vee)^{\mathrm{sym}}\otimes\mathbb Q\] gives the rational divisor class, after a finite extension if necessary [18]. Finally an identity between the maps specializes to the given identity, so injectivity proves it upstairs as well. ◻

Proof. Choose a finite extension of \(\mathbb Q_p\) splitting \(G_{\mathbb Q_p}\). Polynomial approximation, together with Krasner’s lemma [26], gives a totally real number field \(P\) whose completions at \(p\) split \(G_{\mathbb Q_p}\). For example, approximate the minimal polynomial of a primitive element of a splitting extension at \(p\), and a polynomial with distinct real roots at the real place. The resulting number field can have just one place above \(p\); this additional condition is not needed.

Define the multiplier fiber product \[ J=(\mathop{\mathrm{Res}}_{P/\mathbb Q}G_P) \mathbin{\times}_{\mathop{\mathrm{Res}}_{P/\mathbb Q}\mathbf G_m}\mathbf G_m, \tag{6}\] where the last map is diagonal. The Hodge cocharacter, or its inverse according to convention, pairs to \(1\) with \(c\). Since \(c\) factors through \(G^{\mathrm{ab}}\), its character on this torus is primitive. Thus \(\ker(G^{\mathrm{ab}}\to\mathbf G_m)\) is a torus, and \(\ker(c:G\to\mathbf G_m)\) is connected reductive. The exact sequence \[1\longrightarrow\mathop{\mathrm{Res}}_{P/\mathbb Q}(\ker c)_P \longrightarrow J\xrightarrow{c}\mathbf G_m\longrightarrow1\] proves connectedness and reductivity of \(J\). Moreover, \[ J^{\mathrm{der}}=\mathop{\mathrm{Res}}_{P/\mathbb Q}G_P^{\mathrm{der}}. \tag{7}\]

The action on \(U=V\otimes_\mathbb QP\), with alternating form \(\mathop{\mathrm{Tr}}_{P/\mathbb Q}\psi_P\), gives a symplectic embedding of \(J\) with similitude character \(c\). The diagonal Hodge homomorphism gives \(X_J\). The adjoint Hodge-type and Cartan-involution axioms follow componentwise from those for \(X\). We also check that extending the totally real field introduces no rational adjoint factor on which the Hodge homomorphism is trivial. Write a rational simple adjoint factor as \(\mathop{\mathrm{Res}}_{L/\mathbb Q}H\), with \(L\) totally real. Every field factor \(L'\) of \(P\otimes_\mathbb QL\) is totally real, and every real embedding of \(L\) extends to a real embedding of \(L'\). Hence the existence of a nontrivial Hodge component for \(H\) persists in each resulting rational factor. Finally the diagonal weight is rational and the trace form is a polarization. This proves that \((J,X_J)\) is a Shimura datum of Hodge type.

Here are the local conditions explicitly. Put \(P\otimes\mathbb Q_p= \prod_{v\mid p}P_v\) and choose a split maximal torus \(T_v\) of \(G_{P_v}\). The character \(c\) on \(T_v\) has a cocharacter section: a conjugate of the Hodge cocharacter pairs to \(1\) with it, and all cocharacters of a split torus are defined over \(P_v\). The same argument applies to \(G_{P_v}^{\mathrm{ab}}\). Consequently the maximal torus of \(J_{\mathbb Q_p}\) obtained from the \(T_v\), and the abelianization of \(J_{\mathbb Q_p}\), have the forms \[\begin{align*} T_J&\simeq\mathbf G_m\times \prod_{v\mid p}\mathop{\mathrm{Res}}_{P_v/\mathbb Q_p}\mathbf G_m^{\operatorname{rk}G-1}, \tag{8}\\ J_{\mathbb Q_p}^{\mathrm{ab}}&\simeq\mathbf G_m\times \prod_{v\mid p}\mathop{\mathrm{Res}}_{P_v/\mathbb Q_p} \mathbf G_m^{\dim G^{\mathrm{ab}}-1}. \tag{9}\end{align*}\] The split Borels over the \(P_v\) give a Borel of \(J_{\mathbb Q_p}\), so this group is quasi-split. Its derived group is a product of restrictions of split groups by (7). The vertices from the hyperspecial vertices in those split groups give a very special parahoric of \(J_{\mathbb Q_p}\). The inertia coinvariants of the cocharacter module in (9) are torsion-free, since that module is a direct sum of permutation modules. The full stabilizer is therefore connected by [15].

The centralizer of a maximal \(K_0\)-split torus is obtained from (8) by unramified base extension; its torus factors are still induced split tori and \(\mathbf G_m\). It is \(R\)-smooth by [15]. Finally restrictions of split root systems give reduced relative root systems, also at \(p=2\). These are precisely the strong-admissibility checks in the construction of [16]. They use the stipulated Hodge embedding, not the assertion that \(G\) is the full Mumford–Tate group of a particular point.

Apply [16], starting with the representation \(U\) just constructed. Its proof first tensors \(U\) with an auxiliary totally real \(\mathbb Q\)-vector space, then takes direct sums, and finally applies the symplectic self-duality construction. The group acts trivially on that extra vector-space factor. In the last construction the dual representation twisted by \(c\) is identified with \(U\) by its symplectic form. Thus every resulting rational summand is a copy of \(U\), with the same multiplier. This gives (5), together with \(\psi_W\) and \(\Lambda_p\). The assertion of the cited proposition includes \(p=2\); its proof uses the reduced-relative-root condition through [16]. ◻

Proof. Write \(k=\mathbb F\) and let \(\breve E\) be the completed local reflex field. We work over a common finite extension \(K\) of \(K_0\) containing \(\breve E\) and a field of definition of \(\mathcal F\). Every further field extension below is finite. Put \(K_p=\mathcal J(\mathbb Z_p)\).

We first fix the relation between the group and cohomological conventions. Let \(\rho:J_{\mathbb Q_p}\to\mathop{\mathrm{GL}}(W_{\mathbb Q_p})\) be the adapted homological representation and let \(\rho^\vee\) be its dual. Choose the tensor-preserving identification \(\mathbb D_x\simeq\Lambda_p^\vee\otimes_{\mathbb Z_p}W(k)\) of [16]. The local shtuka parameter is \(b_x\in J(K_0)\), whereas crystalline Frobenius on this cohomological Dieudonné module is \[F_x=\rho^\vee(\sigma(b_x))\circ\sigma.\] The additional \(\sigma\) on \(b_x\) comes from the comparison with the vector shtuka: its special-fiber module is \((\sigma^{-1})^*(\mathbb D_x^\vee)\) [32]. Write \(\{\mu\}\) for the class prescribed by \(X_J\); then \([b_x]\in B(J,\mu^{-1})\).

For precision, identify the general linear crystalline period flag with the appropriate filtration variety on \(D_x\) by its marked Rapoport–Zink interpretation. If a deformation \(\mathcal H\) has special-fiber marking \(q:\mathcal H_k\dashrightarrow C_x[p^\infty]\), this identification sends its period to \((q^*)^{-1}\mathop{\mathrm{Fil}}^1H^1_{\mathrm{dR}}(\mathcal H)\). Pushout through the faithful Hodge representation gives the commutative period square \[\begin{array}{ccc} M_{\mathcal J,b_x,\mu,K_p}&\longrightarrow&M_{\mathop{\mathrm{GL}},\rho(b_x),\rho(\mu)}\\ \pi_{\mathrm{GM}}\downarrow&&\downarrow\pi_{\mathrm{GM}}\\ \mathcal F_{J,\mu}&\longrightarrow&\operatorname{Flag}(D_x). \end{array}\] The lower map is a closed immersion: the parabolic stabilizing the faithful filtration intersects \(J\) in the parabolic for \(\mu\). Define \(f_{\mathcal F}\) to be the unique \(J\)-flag with image \(\mathcal F\). It exists by condition (2). The square uses crystalline periods and the dual Frobenius descent above, so no extension of \(\sigma\) to the ramified filtration field is chosen. When \(q=b^{-1}\) its lower right period is exactly \(b^*\mathop{\mathrm{Fil}}^1\).

A framed point over the prescribed period. Admissibility of the faithful filtered isocrystal implies that the image of \(f_{\mathcal F}\) in the general linear period flag variety is in its admissible locus. To see this in the cohomological coordinates of the square, let \(U\) be a crystalline representation realizing \((D_x,\mathcal F)\). Its crystalline comparison identifies the associated vector bundle away from the untilt with \(U\otimes\mathcal O\). The filtered de Rham comparison identifies the modification at the untilt with the standard lattice \(U\otimes B_{\mathrm{dR}}^+\). They glue to the trivial bundle \(U\otimes\mathcal O\) on the Fargues–Fontaine curve, which is the general linear admissibility condition. Since \(\rho\) is faithful, \(\rho^{-1}(Z_{\mathop{\mathrm{GL}}(W)})\subset Z_J\): an element acting as a scalar commutes with the faithful image of \(J\). The admissible-locus identity in [32] therefore puts \(f_{\mathcal F}\) in \(\mathcal F_{J,\mu}^{\mathrm{adm}}\).

This locus is the image of the crystalline period map \[\pi_{\mathrm{GM}}: M_{\mathcal J,b_x,\mu,K_p} \longrightarrow\mathcal F_{J,\mu}^{\mathrm{adm}}.\] The map is étale [32]. Its fiber at the classical point \(f_{\mathcal F}\) is nonempty; an étale rigid space over \(K\) has a point over a finite extension of \(K\). After making that extension, choose \[ z\in M_{\mathcal J,b_x,\mu,K_p}(K), \qquad \pi_{\mathrm{GM}}(z)=f_{\mathcal F}. \tag{10}\] In particular \(z\) supplies the integral level as well as the rational isocrystal marking. If \(C\) is the completed algebraic closure of \(K\), its moduli description gives an actual \(\mathcal J\)-torsor \(\mathcal P_z\), a Frobenius isomorphism bounded by \(\mu\), and, for sufficiently large \(a\), an isomorphism \[i_a:\mathcal J_{Y_{[a,\infty)}(C^\flat)} \xrightarrow{\ \sim\ } \mathcal P_z|_{Y_{[a,\infty)}(C^\flat)}, \qquad \varphi_{\mathcal P_z}\circ\varphi^*i_a =i_a\circ\varphi_{b_x}.\] These are precisely the torsor and annular framing in [32]. Its associated pro-étale \(\mathcal J(\mathbb Z_p)\)-torsor gives integral tensor frames. Thus the integral structure and Frobenius-compatible framing are part of the chosen point, with all their tensor compatibilities.

Integral realization and its special fiber. Our adapted datum is of global Hodge type: the group scheme is a full connected Bruhat–Tits stabilizer, and the chosen Hodge embedding is integral on \(\Lambda_p\). The center also satisfies the split-rank hypothesis of [32]. Indeed its real subgroup of multiplier one preserves the positive definite form \(\psi_W(v,h(i)w)\) and is compact. Its real split rank is therefore at most one, while the rational weight torus supplies one split central direction over \(\mathbb Q\); both split ranks are one. The model here is the one in [16], obtained from this adapted Hodge embedding and satisfying the Pappas–Rapoport characterization. Condition \((U_x)\) holds by [8]; see also [32]. Consequently [32] represents the integral local Shimura variety by a normal formal scheme \(\mathfrak M_{\mathcal J,b_x,\mu}\) and gives the morphism \[\Theta_x:\mathfrak M_{\mathcal J,b_x,\mu} \longrightarrow\widehat{\mathcal S_K}.\] More precisely, [32] identifies this formal scheme with the tensor-compatible open and closed part of the pullback of the Siegel Rapoport–Zink space. This construction identifies the pullback of the universal \(\mathcal J\)-shtuka with its tautological framed shtuka and the faithful vector realization with the universal \(p\)-divisible group.

Its rigid generic fiber is the local Shimura variety [32]. The classical point \(z\) of this rigid generic fiber gives \(\operatorname{Spf}\mathcal O_K\to\mathfrak M_{\mathcal J,b_x,\mu}\). Indeed it lies in the generic fiber of a formal affine chart; for a special formal chart one may take an affinoid member of its standard generic-fiber covering containing \(z\). Evaluation of the formal chart’s ring is integral, and its ideal of definition maps to topologically nilpotent elements of \(\mathcal O_K\). Let \(z_0\) be the special point and put \(y=\Theta_x(z_0)\) and \(\widetilde y=\Theta_x(z)\). The formal map \(\widetilde y\) gives a point of the integral model over \(\mathcal O_K\) by restricting to an affine neighborhood of \(y\). Pullback of the universal abelian scheme gives \(C_{\widetilde y}\) and its \(p\)-divisible group \(\mathcal H_z\).

We record explicitly the integral comparison at this point. For a uniformizer \(\pi\) of \(K\), put \(\mathfrak S=W(k)[[u]]\), with \(u\mapsto\pi\), and let \(E_\pi(u)\) generate its kernel. The crystalline representation and integral level of \(z\) give the \(\mathcal J\)-Breuil–Kisin torsor of [32], with \[\Phi:\varphi^*\mathcal P_{\mathrm{BK}}[1/E_\pi] \xrightarrow{\ \sim\ } \mathcal P_{\mathrm{BK}}[1/E_\pi].\] Its pushout to \(\mathop{\mathrm{GL}}(\Lambda_p)\) is the Breuil–Kisin module of \(\mathcal H_z\). Reduction at \(u=0\), followed by the dual Frobenius descent fixed above, therefore gives the actual Dieudonné lattice of \(\mathcal H_{z,k}\). Proposition 3.5.1 of [32] compares the two specializations of this already constructed framed point. In the coordinates of [16] it gives \[g\in X_{\mathcal J}(\mu^{-1},b_x),\qquad \mathbb D(\mathcal H_{z,k}) =\rho^\vee(\sigma(g))\mathbb D_x\subset D_x.\] Indeed the Frobenius matrix on this lattice, written in its induced basis, is \[\rho^\vee\!\left( \sigma(g^{-1}b_x\sigma(g))\right)\circ\sigma,\] which explains the same Frobenius descent in the Deligne–Lusztig condition and in the lattice formula.

The abelian marking and the filtration. Keep the identity prime-to-\(p\) coordinate in the uniformization. The description of its special fiber in [32], or equivalently the isogeny-class map of [16], gives \(y=i_x(g)\) and a single abelian quasi-isogeny \(b:C_x\to C_y\) preserving the crystalline tensors and every prime-to-\(p\) tensor. Its crystalline marking is the one just used for \(\mathcal H_z\), because the uniformization is induced by the marked Siegel Rapoport–Zink space.

The crystalline period map of that space is the Hodge filtration of its universal deformation pulled back by its marking. Compatibility of \(\Theta_x\) with the tautological shtuka and (10) consequently give \[b^*\mathop{\mathrm{Fil}}^1H^1_{\mathrm{dR}}(C_{\widetilde y}/K) =\mathcal F^1.\] The deformation criterion [16] is satisfied concretely: the filtration is of type \(\mu\), and the crystalline structure tensors compare with the integral étale tensors in the \(\mathcal J(\mathbb Z_p)\)-frames of \(z\). The bounded integral shtuka and its annular framing were obtained before using this criterion or comparing specializations. All cited global Hodge-type and strongly admissible results apply at \(p=2\) as well.

Finally, under finite local-field descent, the weak-admissibility inequalities restrict to the descended Frobenius subobjects. Colmez–Fontaine’s theorem [4] makes those descended data admissible, and scalar extension supplies condition (1). ◻

Proof of Proposition 6. The original point and its reduction. Apply Lemma 7. Choose a rational \(J\)-equivariant identification \(W=(V\otimes_\mathbb QP)^{\oplus m}\) and set \(n=m[P:\mathbb Q]\). Choose a basis of \(P\) whose first vector is \(1\). The resulting identification of the Hodge structure at the diagonal point \(h\) with \(V^{\oplus n}\) makes the first-factor map equal to \[ v\longmapsto(v\otimes1,0,\ldots,0). \tag{11}\] The rational homology/isogeny correspondence [21] therefore makes the abelian variety \(C\) at that point isogenous to \(A^n\). Fix a rational isogeny \(f:A^n\to C\) realizing this identification.

We explain the field and reduction assertions for this starting point. The Hodge embedding maps it to the Siegel point of the polarized variety \(C\) with finite level. The abelian variety \(C\) is defined over \(\overline{\mathbb Q}\): clearing a denominator in \(f\) realizes it as a quotient of \(A^n\) by a finite subgroup of its torsion, and every such subgroup is defined over \(\overline{\mathbb Q}\). Its rational polarization and its finite level also descend to \(\overline{\mathbb Q}\), since homomorphisms between abelian varieties over \(\overline{\mathbb Q}\) do not enlarge after extending algebraically closed fields [18]. The Hodge embedding induces a finite morphism onto its image at sufficiently small finite level [21]. Its fiber over a \(\overline{\mathbb Q}\)-point therefore consists of \(\overline{\mathbb Q}\)-points. This places the diagonal point on the canonical model over \(\overline{\mathbb Q}\).

After a finite number-field extension the variety \(C\) has good reduction, since it is isogenous to \(A^n\) [23]. Its polarization and prime-to-\(p\) level extend after a further finite extension. The integral Hodge-type model is the normalization of the closure of the generic fiber in the adapted Siegel model. The valuative criterion for this finite normalization therefore extends our point to its integral model. Denote the reduction by \(x\). The rational isogeny \(f\) specializes to a quasi-isogeny \(f_0:A_0^n\to C_x\).

The trace tensor and its descent. Next extend the alternating form defining \(\gamma\) \(P\)-multilinearly and take its trace. Composing with projection onto the first copy in (5) gives \[ \eta=\mathop{\mathrm{Tr}}_{P/\mathbb Q}(\gamma_P)\circ (\operatorname{pr}_1)^{\otimes 2r} \in\bigwedge^{2r}W^*. \tag{12}\] For \(r=0\) this is interpreted as the scalar trace. The equal-multiplier condition implies that \(\eta\) has multiplier \(c^r\) under \(J\). On (11), its restriction is \[ j^*f^*\eta_C=[P:\mathbb Q]\gamma. \tag{13}\] The class \(\eta_C\) is understood with its Tate twist.

We turn this semi-invariant into the untwisted tensors to which the integral-model theorems directly apply. Write \(\psi_W^{-1}\in\bigwedge^2W\) for the inverse symplectic bivector, and include \[ u_\psi=\psi_W\otimes\psi_W^{-1},\qquad u_\eta=\eta\otimes(\psi_W^{-1})^{\otimes r} \tag{14}\] among the structure tensors. Both are \(J\)-invariant tensors in ordinary tensor constructions of \(W\) and \(W^*\); no separately trivialized Tate object is needed. They can be made integral and added to the defining family as explained above. Their crystalline and prime-to-\(p\) realizations at \(x\) are the specializations from the original point, with the comparison conventions fixed above.

The entire finite family of structure tensors, including \(u_\psi\) and \(u_\eta\), has finite local-field descent. To see this, choose a finite extension \(L/\mathbb Q_p\) over whose valuation ring the original number-field point is defined, and let \(L_0\) be its maximal unramified subfield. The canonical étale tensor sections of [16], pulled back to that point, are \(\mathop{\mathrm{Gal}}(\overline L/L)\)-invariant. Applying the tensor functor \(D_{\mathrm{cris},L}\) to these morphisms from the unit object places them in the crystalline tensor spaces over \(L_0\). Their extension to \(K_0\) is exactly the family \(s_{\alpha,0,x}\), by good-reduction comparison. The map \(f_0\) also descends after a finite residue-field extension, so this descent is retained under its marking. Taking a compositum with the finite field of definition of a prescribed filtration therefore gives one finite local field for all the data. No tensor-preserving trivialization over \(L_0\) is being asserted or needed.

The two choices of lift. For specification (1), apply [16] to \(x\). It gives a point \(y\in\mathcal I_x\) with a special-point lift and a tensor-preserving quasi-isogeny \(b:C_x\to C_y\). The special-point lift supplies a CM abelian variety \(B\). Special points on the canonical Shimura variety are algebraic over its reflex field by the canonical-model theory [21], so \(B\) and its level data can be taken over \(\overline{\mathbb Q}\). The given local lift specifies their embedding at \(w\) and their good reduction \(B_0=C_y\).

For specification (2), identify \(D_x\) with \(D_A^{\oplus n}\) using \(f_0^*\) and prescribe the direct-sum filtration \(\mathcal F_A^{\oplus n}\). It is weakly admissible, with the required finite local-field descent. It is also in the required \(J\)-cocharacter orbit. Indeed a geometric element of the transported \(G\) taking the original filtration on \(D_A\) to \(\mathcal F_A\) acts diagonally on all the copies of \(V\) in \(W\); its image in \(J\) takes the original filtration on \(D_x\) to the prescribed one. The tensor-preserving identifications used here come from the initial point and comparison, so they include the \(J\)-structure tensors as well. Geometric conjugacy can be realized over a finite extension, since the corresponding isomorphism scheme is of finite type. Lemma 8 now gives \(y\), its lift \(B\), and a tensor-preserving quasi-isogeny \(b:C_x\to B_0\) with the prescribed marked filtration. This proves (4) for \(a=bf_0\).

A common rational normalization. It remains, in both specifications, to recover the twisted class and normalize it by a single rational scalar. Choose a complex description of the lifted Shimura point. Its rational marking by \(W\) supplies a rational Hodge class \(\eta_B\) corresponding to \(\eta\), and a rational divisor class \(\theta_B\) corresponding to \(\psi_W\). Write \(\theta_C\) for the analogous class on \(C\) and put \[\theta_*=f_0^*\theta_{C,0}\in\mathop{\mathrm{NS}}(A_0^n)\otimes_\mathbb Z\mathbb Q.\] The class \(\theta_*\) is rational ample. Tensor preservation of \(u_\psi\) says that, in each realization, \[ a^*\theta_{B,0}=\lambda_v\theta_*, \tag{15}\] with a nonzero local scalar \(\lambda_v\). Let \(D=n\dim A\). Pairing with \(\theta_*^{D-1}\) gives \[ \lambda_v= \frac{\deg\bigl(a^*\theta_{B,0}\cdot\theta_*^{D-1}\bigr)} {\deg(\theta_*^D)}=:\lambda\in\mathbb Q^\times. \tag{16}\] Both intersection numbers refer to rational divisor classes on the same abelian variety; their cycle-class traces are identical in every realization. The denominator is positive because \(\theta_*\) is ample, and the numerator is nonzero by (15). Thus all the scalars in (15), including the crystalline one, are the single rational number \(\lambda\).

Preservation of \(u_\eta\), together with (16), then yields \[ a^*\eta_{B,0}=\lambda^r f_0^*\eta_{C,0} \tag{17}\] simultaneously. In fact the inverse bivector scales by \(\lambda^{-1}\), so that the equality for the untwisted tensor in (14) gives exactly (17). Set \[\kappa=\frac{\lambda^{-r}}{[P:\mathbb Q]}\eta_B.\] Specializing (13) and using (17) proves (3).

For the arbitrary local lift, the crystalline realization just used is unambiguous: the canonical model supplies the untwisted tensor \(u_\eta\), and the algebraic polarization supplies its Tate line and inverse bivector. The comparison isomorphism and these operations define \(\eta_B\) in the crystalline realization and agree with its canonical étale tensors. Consequently any identity expressing this class by algebraic divisor operations specializes with the same rational coefficients in all realizations. For the \(\overline{\mathbb Q}\)-valued CM lift, these are also the usual absolute-Hodge and de Rham realizations. This completes the two constructions. ◻

Proof. Choose a polarization \(\psi\) on \(V=H_1(A(\mathbb C),\mathbb Q)\), and let \(G=\mathop{\mathrm{MT}}(V)\). The Mumford–Tate group is connected reductive and is characterized by its Hodge tensors [6]. The Hodge homomorphism and its \(G(\mathbb R)\)-conjugacy class define the Mumford–Tate Shimura datum of Hodge type, with faithful symplectic representation \[G\hookrightarrow\mathop{\mathrm{GSp}}(V,\psi).\] The polarization supplies the rational weight and the Cartan involution condition. The definition of the Mumford–Tate group excludes an adjoint factor on which the Hodge homomorphism is trivial. The class \(\gamma\), with its Tate twist, is fixed by \(G\). Equivalently, its alternating form on \(V\) has multiplier \(c^r\), where \(c\) is the similitude character. Thus the first alternative of Proposition 6 applies and gives the asserted data and simultaneous equality. ◻

Proof. We prove the contrapositive. A nonzero pairing will produce a divisor form nondegenerate on one \(E\)-eigenspace. We will then show that a rational endomorphism in \(\mathcal U\) with this property is invertible.

Fix one realization and extend its coefficient field to an algebraically closed field \(\Omega\). At the crystalline component we also include the coefficient extension used for the specialized class. In this proof, \(V_\tau\) also denotes the corresponding eigenspace in the dual of first cohomology of \(A_0\), identified by specialization and comparison. Write \(L^j_\Omega(A_0)\) for the \(\Omega\)-span of products of \(j\) divisor classes. Since \([E:\mathbb Q]\geq2\), we have \(g-2s\geq0\). Hard Lefschetz on the Lefschetz algebra gives \[ L^{g-s}_\Omega(A_0) =\psi_0^{g-2s}L^s_\Omega(A_0). \tag{28}\] The restriction of hard Lefschetz to this algebra, including for crystalline cohomology, follows from the Lefschetz correspondences of [19].

Suppose a Weil class pairs nontrivially with \(D_1\cdots D_s\psi_0^{g-2s}\), and choose a pure term, supported on \(V_\tau\), whose contribution is nonzero. This term already has degree \(2s\) on \(V_\tau\), so no further factor in a nonzero top-degree product can use that space. A factor from \(\psi_0\) which uses \(V_{\bar\tau}\) also uses \(V_\tau\); hence the polarization factors contribute to neither space in this pair. All \(2s\) degrees on \(V_{\bar\tau}\) must therefore come from the \(s\) divisor factors. Each has degree two, so each must contribute its restriction to \(V_{\bar\tau}\), and those restrictions have nonzero product. The coefficient of \(t_1\cdots t_s\) in \[(t_1D_1+\cdots+t_sD_s)^s\big|_{V_{\bar\tau}}\] is \(s!\) times that product. Some \(\Omega\)-linear combination of the divisors therefore restricts to a nondegenerate alternating form on \(V_{\bar\tau}\).

We identify which endomorphisms can give these restrictions. Let \(p_\xi\in E\otimes_\mathbb Q\Omega\) be the eigenspace projectors. Rosati satisfies \(p_\xi^\dagger=p_{\bar\xi}\). A divisor corresponds under \(\psi_0\) to a Rosati-symmetric endomorphism \(t\). Define \[ \Pi(t)=\sum_\xi p_{\bar\xi}t p_\xi. \tag{29}\] This operation takes place inside \(\mathop{\mathrm{End}}^0(A_0)\otimes_\mathbb Q\Omega\). Moreover, \[(p_{\bar\xi}t p_\xi)^\dagger =p_{\bar\xi}t^\dagger p_\xi, \qquad \Pi(t)e=\bar e\Pi(t).\] Thus \(\Pi(t)\in\mathcal U\otimes_\mathbb Q\Omega\). Conversely, these are exactly the blocks allowed by the second equation in (27). For \(x,y\in V_\xi\), the form \(\psi_0(x,ty)\) depends only on \(p_{\bar\xi}t p_\xi\). It follows that projecting by (29) preserves all the same-embedding restrictions in question.

Nondegeneracy on \(V_{\bar\tau}\) is a polynomial open condition on \(\mathcal U\otimes_\mathbb Q\Omega\), and we have shown this open is nonempty. The rational points of the rational vector space \(\mathcal U\) are Zariski dense after extension to \(\Omega\). Hence there is a rational \(u\in\mathcal U\) for which \(\psi_0(\cdot,u\cdot)\) is nondegenerate on \(V_{\bar\tau}\).

To prove invertibility, we use the fact that a nonzero \(E\)-stable abelian subvariety has a nonzero homology space at every embedding of \(E\). Suppose that \(u\) is not invertible, and choose an integer \(N>0\) such that \(Nu\) is an endomorphism. The reduced identity component \(C=((\ker(Nu))^0)_{\mathrm{red}}\) is a positive-dimensional abelian subvariety. The relation \(ue=\bar e u\) makes \(C\) stable under the rational action of \(E\). This action is unital and therefore faithful. Choose a primitive element \(e\) of \(E\) and an integer \(M>0\) such that \(Me\) acts by an integral endomorphism of \(C\). The exterior algebra description of cohomology and normalized top-degree pullback give, in every realization [19], \[\det\bigl(n-e\mid H^1(C)\bigr) =M^{-2\dim C}\deg\bigl([Mn]-Me\bigr)\] for every integer \(n\) for which the map on the right is an isogeny. All but finitely many integers have this property. The right side is rational and independent of the realization. Polynomial interpolation therefore gives one characteristic polynomial in \(\mathbb Q[T]\), identical also in crystalline cohomology. Its roots lie among the conjugates of \(e\), and rationality forces their multiplicities to be equal. Since \(C\ne0\), every embedding of \(E\) occurs with positive multiplicity. The same statement holds on first homology, in the chosen realization. Rational Poincaré reducibility [18] makes the inclusion \(C\hookrightarrow A_0\) injective on first homology in every realization. This inclusion is \(E\)-equivariant, and its image is killed by \(u\). Thus the nonzero \(\bar\tau\)-eigenspace of \(H_1(C)\) shows that the kernel of \(u\) meets \(V_{\bar\tau}\) nontrivially. Such a vector lies in the radical of \(\psi_0(\cdot,u\cdot)|_{V_{\bar\tau}}\), a contradiction.

A nonzero pairing would therefore give an invertible element of \(\mathcal U\). Taking the contrapositive proves the lemma. The argument applies separately to every realization, so all the pairings in this case have the common rational value zero. ◻

Proof. For \(b\in E\), let \(\widetilde b\in\mathop{\mathrm{End}}^0(B)\) act by \(b\) on \(T\) and by zero on the complement. Its pullback of a rational divisor is a rational divisor: clear a denominator in \(\widetilde b\), and use that multiplication by \(N\) acts as \(N^2\) on \(H^2\). Over a splitting field \(\Omega\), choosing Tate-line bases, we have \[(\widetilde b^*D)^s =\left(\sum_\tau\tau(b)^2D_\tau\right)^s.\] The coordinates \(x_\tau=\tau(b)\) are independent coordinates on the scalar extension of the rational vector group underlying \(E\). In the corresponding vector-valued polynomial \[ P((x_\tau)_\tau)=\left(\sum_\tau x_\tau^2D_\tau\right)^s, \tag{31}\] the coefficient of \(x_\tau^{2s}\) is \(D_\tau^s\). Nondegeneracy makes this coefficient nonzero, so it spans \(\bigwedge^{2s}T_\tau^\vee\otimes\mathbb Q(s)_\Omega\).

Let \(M\) be the rational span of the Lefschetz classes \((\widetilde b^*D)^s\) for \(b\in E\). The set \(E\) of rational points is Zariski dense in its vector group, also after extension to \(\Omega\). Any linear functional annihilating \(M\otimes_\mathbb Q\Omega\) annihilates all evaluations of (31) on those points. It therefore annihilates the polynomial, and hence each coefficient. Every \(D_\tau^s\) belongs to \(M\otimes_\mathbb Q\Omega\).

This proves the required inclusion after scalar extension. Both subspaces are rational; applying faithful scalar extension to their quotient proves the inclusion over \(\mathbb Q\). In particular it gives rational coefficients for every rational class in the asserted subspace, without requiring an individual embedding line to be defined over \(\mathbb Q\). ◻

Proof. The polarization pairs \(V_\tau\) with \(V_{\bar\tau}\), and no other pair of eigenspaces. Over an algebraically closed field, choose one embedding in each conjugate pair. An element of \(G\) is specified by \(c\) and matrices \(a_\tau\in\mathop{\mathrm{GL}}_{2s}\) with \(\det(a_\tau)=c^s\); the matrices on the opposite spaces are then determined by the polarization. In particular, \[\ker(c:G\longrightarrow\mathbb G_m) \simeq\prod_{\{\tau,\bar\tau\}}\mathop{\mathrm{SL}}_{2s}.\] The multiplier is surjective geometrically. Thus \(G\) is connected reductive, with the same adjoint group as the group of \(E\)-linear polarization similitudes.

The two Hodge characters occur \(s\) times at every embedding. Their product is the polarization multiplier, so the Hodge homomorphism satisfies the determinant condition in (32). Its weight is rational and central. On the adjoint Lie algebra the Hodge types are among \((-1,1),(0,0),(1,-1)\), as follows from the weight-one representation. Conjugation by the image of \(i\) is a Cartan involution, by the polarization. At every real embedding of the maximal totally real subfield of \(E\), the adjoint group has unitary signature \((s,s)\). Its Hodge homomorphism is nontrivial there, so no rational simple adjoint factor has trivial Hodge homomorphism. These are the Shimura-datum conditions for the given Hodge embedding.

On \(H^1\) the polarization still pairs opposite eigenspaces. An \(E\)-stable Lagrangian is specified by an \(s\)-dimensional subspace at one embedding of each pair; the subspace at the opposite embedding is its annihilator. The factor \(\mathop{\mathrm{SL}}_{2s}\) acts transitively on these subspaces, as in the cohomological orbit calculation of Theorem 10. Finally, an alternating determinant form evaluated on \(a_\tau v_1,\ldots,a_\tau v_{2s}\) is multiplied by \(\det(a_\tau)=c^s\). This proves the asserted transformation law. ◻

Proof of Proposition 17. The vanishing alternative is Lemma 18. Suppose that \(u\) is invertible, and use the divisor \(D^u\) from (30).

A filtration preserving the divisor.

Let \(\mathcal D=H^1_{\mathrm{cris}}(A_0/W(\mathbb F))[1/p]\), with its contravariant convention, and let \(\beta\) be the dual polarization pairing, taking values in the isocrystal line of slope one. The actions on \(\mathcal D\) satisfy \[\beta(ux,y)=\beta(x,uy),\qquad \beta(ex,y)=\beta(x,\bar e y),\qquad ue=\bar e u.\] All these endomorphisms and the polarization have finite field descent. Theorem 10 supplies, over a finite extension with finite local field descent, a weakly admissible filtration \(\mathop{\mathrm{Fil}}^1\mathcal D=L\) which is \(E\)- and \(u\)-stable, is Lagrangian for \(\beta\), and has dimension \(s\) at every embedding of \(E\). By Lemma 20 it lies geometrically in the required \(G\)-orbit. The finite-descent argument in Proposition 6 shows that its full finite family of Hodge-type structure tensors descends from the original number-field point by crystalline comparison. Taking a compositum with the local field of definition of \(L\) gives the common finite local-field descent required by its prescribed-filtration clause.

Transport and the first rational summand. Apply Proposition 6 to \(G\) and \(\gamma\) with this filtration. It gives a lift \(B\), a quasi-isogeny \(a:A_0^n\longrightarrow B_0\), and a rational Hodge class \(\kappa\) on \(B\), with \[f_0=aj:A_0\longrightarrow B_0,\qquad f_0^*\kappa_0=\gamma_0,\] simultaneously in the realizations under consideration. The filtration on \(B\), marked by \(a\), is the direct sum of \(n\) copies of the chosen filtration. Here \(j\) includes the first factor. The map \(f_0\) identifies \(A_0\) with a rational direct summand of \(B_0\). Its retraction \(q_0\) and the corresponding projector \(R_0\) are \[ q_0=\operatorname{pr}_1a^{-1},\qquad R_0=f_0q_0. \tag{33}\] Thus \(q_0f_0=\mathop{\mathrm{id}}\) and \(R_0^2=R_0\). Pulling back \(f_0^*\kappa_0=\gamma_0\) by \(q_0\) gives \[R_0^*\kappa_0=q_0^*\gamma_0.\] Act by \(E\) on \(\operatorname{im}(R_0)\) using \(e\mapsto f_0e q_0\), and by zero on the complement. Also put \(D_0=q_0^*D^u\).

Lifting the projector, action, and divisor. The projector and this \(E\)-action preserve the repeated filtration. By Corollary 15, \(D^u\) satisfies the divisor lifting condition for the chosen filtration on the first summand. The crystalline map induced by \(q_0=\operatorname{pr}_1a^{-1}\) respects that filtration and the marked repeated filtration on \(B_0\). Hence \(D_0=q_0^*D^u\) satisfies the same condition on \(B\). Apply Lemma 5 to \(R_0\), to a finite \(\mathbb Q\)-basis of the action of \(E\) on \(\operatorname{im}(R_0)\), and to the symmetric homomorphism \(h_{D_0}:B_0\to B_0^\vee\) defined by \(D_0\). Denote their lifts by \(R\), the corresponding basis of maps, and \(h_D\). Injectivity of specialization preserves \(R^2=R\), the rational multiplication table of that basis, and its unit \(R\). It therefore gives an algebra action \(E\to R\mathop{\mathrm{End}}^0(B)R\). The symmetric map \(h_D\) defines a rational divisor class \(D\). Moreover, \(q_0R_0=q_0\), so \[R_0^\vee h_{D_0}R_0=h_{D_0}.\] Injectivity lifts this identity to \(R^\vee h_DR=h_D\), which is exactly \(R^*D=D\). Thus the required projector, action, and divisor all exist on the one auxiliary variety \(B\).

The rational identity and its specialization. Consider now rational Betti cohomology at a complex realization of \(B\). The summand \(T=\operatorname{im}(R)\) has dimension \(2s\) at each embedding of \(E\). The form of \(D\) is supported on this summand and pairs each embedding with itself nondegenerately. Also \(R^*\kappa\) is supported on the pure determinant spaces of this summand. These assertions can be checked after extension to any one prime-to-\(p\) realization: for the divisor they follow from its specified specialization, and for the Hodge class they follow from the equality \(R_0^*\kappa_0=q_0^*\gamma_0\) and the pure support of \(\gamma_0\). Compatibility of the transported realizations then gives these support and nondegeneracy assertions in Betti cohomology.

Lemma 19 now expresses the rational Betti class \(R^*\kappa\) as a finite rational linear combination of products of rational divisor classes on \(B\). The realization and specialization compatibilities in Proposition 6, together with those for divisors and homomorphisms, specialize this identity with the same rational coefficients in every component. Hence \(R_0^*\kappa_0\) is a simultaneous rational Lefschetz class. Finally, \[\gamma_0=f_0^*R_0^*\kappa_0,\] because \(R_0f_0=f_0\). Pullback by a rational homomorphism preserves simultaneous rational Lefschetz classes, so the same is true of \(\gamma_0\). Its complementary traces are therefore one common rational intersection number, proving the second alternative and the proposition. ◻

Proof. We first identify the eigenspace components that can support a rational Hodge class. For each such component we construct a balanced abelian factor of a power of \(A\), whose Weil classes pull back onto that component. Taking the sum of these rational pullback spaces will then give the decomposition over \(\mathbb Q\).

Balanced eigenspace components. Set \(V=H_1(A(\mathbb C),\mathbb Q)\). We work throughout up to isogeny. There is a CM algebra \(E'\), a product of CM fields, acting on \(A\) such that \(V\) is free of rank one over \(E'\). To obtain it, decompose \(A\) up to isogeny into simple CM factors and take the product of their CM fields, with a separate label for each repeated factor. Transporting this action back along an isogeny gives the asserted action on \(A\) itself.

Choose a finite Galois CM field \(K\subset\overline{\mathbb Q}\) containing the images of all the factors of \(E'\), and put \(S=\mathop{\mathrm{Hom}}_{\mathbb Q\text{-alg}}(E',K)\). Then \[V_K:=V\otimes_{\mathbb Q}K=\bigoplus_{\rho\in S}V_\rho, \qquad \dim_K V_\rho=1.\] The Hodge decomposition is diagonal for this decomposition after extension to \(\mathbb C\). We suppress the common Tate twist in the following linear-algebra calculations. Alternating forms of degree \(2r\) decompose as \[\bigwedge^{2r}V_K^* =\bigoplus_{\substack{\Delta\subset S\\ |\Delta|=2r}}L_\Delta, \qquad L_\Delta= \bigwedge^{2r}\left(\bigoplus_{\rho\in\Delta}V_\rho\right)^*,\] where each form on the right is extended by zero on the other summands. Every \(L_\Delta\) is one-dimensional and has a definite Hodge type.

Suppose the component of \(\gamma\) in \(L_\Delta\) is nonzero. Since \(\gamma\) is rational, its component in \(L_{\sigma\Delta}\) is the semilinear conjugate of that component and is nonzero for every \(\sigma\in\mathop{\mathrm{Gal}}(K/\mathbb Q)\). The Hodge condition therefore implies that every \(\sigma\Delta\) contains \(r\) lines of each of the two Hodge types. We call such a subset balanced under all conjugates.

An abelian factor with balanced signatures. Fix one of these subsets \(\Delta\). Form the abelian variety \(A\otimes_{\mathbb Q}K\) in the category up to isogeny: it is a power of \(A\) whose index space is the underlying rational vector space of \(K\). The regular action of \(K\) on this index space is an action by rational endomorphisms of the power. Its rational homology is \[T=V\otimes_{\mathbb Q}K,\qquad \dim_\mathbb QT=[K:\mathbb Q]\dim_\mathbb QV.\] In contrast with the coefficient space \(V_K\), we regard \(T\) here as a \(\mathbb Q\)-vector space, with \(K\) acting on its second factor. The abelian variety remains over \(\overline{\mathbb Q}\). The actions on the two factors give an action of the rational algebra \[E'\otimes_{\mathbb Q}K\simeq\prod_{\rho\in S}K, \qquad e\otimes k\longmapsto\bigl(\rho(e)k\bigr)_{\rho\in S}.\] Let \(e_\Delta\) be the idempotent with coordinates \(1\) on \(\Delta\) and \(0\) elsewhere. It is a rational endomorphism of \(A\otimes_{\mathbb Q}K\), so its image is an abelian variety \(B_\Delta\) up to isogeny. In particular no Galois invariance of \(\Delta\) is required for this construction. Write \[U=H_1(B_\Delta(\mathbb C),\mathbb Q)=e_\Delta T.\] The index-field action of \(K\) preserves \(U\).

Now extend the coefficients of \(U\) to \(K\), and use the fixed inclusion \(K\subset\mathbb C\) to identify the embeddings of \(K\) with \(\mathop{\mathrm{Gal}}(K/\mathbb Q)\). Evaluation of the index factor at \(\tau\) identifies its eigenspace in \(T\otimes_\mathbb QK\) with \(V_K\). In that eigenspace the idempotent retains exactly the summands indexed by \(\tau\Delta\). Thus \[ U_\tau=\bigoplus_{\rho\in\Delta}V_{\tau\rho}, \tag{35}\] and the same description holds after extension to \(\mathbb C\). Explicitly, evaluation of the idempotent at the pair of embeddings \((\xi,\tau)\) is \(1\) exactly when \(\xi=\tau\rho\) for some \(\rho\in\Delta\). Thus \(\dim_K U_\tau=2r\) for every \(\tau\). Since \(K\) is Galois and \(\Delta\) is balanced under all conjugates, each \(U_\tau\) has \(r\) lines of each Hodge type. Consequently \(U\) has rank \(2r\) over \(K\), \(\dim B_\Delta=r[K:\mathbb Q]\), and the required Hodge signatures are \((r,r)\).

A compatible polarization. Let \(\psi\) be a rational polarization form on \(V\), and equip the index space \(K\) with the positive definite rational symmetric form \[t(k,l)=\mathop{\mathrm{Tr}}_{K/\mathbb Q}(k\bar l).\] The form \(\Psi=\psi\otimes t\) is a rational polarization form on the power \(A\otimes_{\mathbb Q}K\): its alternating and Hodge properties follow from those of \(\psi\), and its positivity follows by tensoring the positive symmetric form associated to \(\psi\) with \(t\). A positive integral multiple gives an actual polarization. For \(a\in K\), \[t(ak,l)=t(k,\bar a l),\] so the Rosati involution of \(\Psi\) induces conjugation on the index-field action. Restrict \(\Psi\) to \(B_\Delta\). This is again a polarization, and because \(U\) is \(K\)-stable its adjoint identity still induces conjugation on \(K\). This restriction does not require \(e_\Delta\) to be self-adjoint.

Pullback and descent of the rational span. The factor \(B_\Delta\) now has the field action, balanced signatures, and polarization needed to define its Weil classes. The rational homomorphism \(f_\Delta\colon A\to B_\Delta\) defined on homology by \[v\longmapsto e_\Delta(v\otimes 1)\] has the needed effect on Weil classes. After splitting coefficients, its component at the identity embedding of \(K\) is the projection \[\bigoplus_{\rho\in S}V_\rho \longrightarrow \bigoplus_{\rho\in\Delta}V_\rho.\] Thus pullback maps \(\bigwedge^{2r}U_{\mathop{\mathrm{id}}}^*\) isomorphically onto \(L_\Delta\). More generally the component indexed by \(\tau\) pulls back onto \(L_{\tau\Delta}\). The rational Weil-class space \(W_K^r(B_\Delta)\) is characterized, with the twist restored, by \[W_K^r(B_\Delta)\otimes_{\mathbb Q}K =\bigoplus_{\tau:K\hookrightarrow K} \bigwedge^{2r}U_\tau^*(r).\] In particular \(f_\Delta^*W_K^r(B_\Delta)\) is a rational subspace whose extension to \(K\) contains \(L_\Delta(r)\).

Take the sum of these rational pullback spaces over the finitely many subsets supporting \(\gamma\). Its scalar extension contains every component of \(\gamma\), and therefore contains \(\gamma\) itself. Since extension of scalars is injective on the quotient by a rational subspace, the sum already contains \(\gamma\) over \(\mathbb Q\). This proves (34) with rational Weil classes. Notice that an individual embedding component need not descend to \(\mathbb Q\); only the entire Weil-class space and its pullback are used in this descent argument.

Good reduction and specialization. All the constructions use finitely many rational endomorphisms of powers of \(A\) and their images. They therefore exist over \(\overline{\mathbb Q}\) and descend, together with the maps, to a common number field after a finite extension. Choose that extension so that \(A\) has good reduction at the place specified by \(w\). For any \(\ell\ne p\), the rational Tate module of \(B_\Delta\) is a Galois-stable direct summand of the rational Tate module of the power of \(A\). It is unramified at this place. The Néron–Ogg–Shafarevich criterion gives good reduction of \(B_\Delta\) there; it also shows that the isogenies used in the construction do not affect this conclusion [23]. Rational homomorphisms between the resulting good-reduction varieties specialize after clearing denominators. The equality of rational Hodge classes (34) is respected by their compatible realizations and by specialization, with the same rational coefficients in every component. This proves the last assertion. ◻

Proof. In codimension zero the class is a rational multiple of the unit, so the assertion follows from the compatibility of divisor intersection numbers with the normalized traces. In positive codimension use Lemma 21. Proposition 17 gives the Lefschetz pairing property for the specialization of each \(\eta_i\). Lemma 4 preserves this property under rational pullback and rational linear combinations. Applying it to (34) proves the assertion. ◻

Proof of Theorem 1. The case \(r=0\) follows from divisor intersection numbers as in the preceding corollary. This includes \(\dim A=0\), where the abelian variety is a point. If \(r=\dim A\), a rational top class is a rational multiple of the point class; its normalized trace is preserved by specialization, and the complementary product is \(1\). We may therefore assume \(0<r<\dim A\).

Corollary 9 gives a CM abelian variety \(B/\overline{\mathbb Q}\) with good reduction at \(w\), a rational Hodge class \(\kappa\) on \(B\), a quasi-isogeny \(a\colon A_0^n\to B_0\), and the first-factor inclusion \(j\colon A_0\to A_0^n\), such that \[\gamma_0=(aj)^*\kappa_0\] simultaneously in all the realizations under consideration. Corollary 22 gives the Lefschetz pairing property for \(\kappa_0\), and Lemma 4 gives it for \(\gamma_0\). Hence, for each prescribed product of complementary divisor classes on \(A_0\), all the normalized traces in Theorem 1 equal one rational number. This proves the theorem. ◻

Proof. In codimension zero, take the corresponding rational multiple of the unit, which belongs to the theory by pullback from the point. This also covers \(d=0\). In top codimension \(r=d>0\), take \(q\) times the point class, where \(q\in\mathbb Q\) is the normalized trace of \(\gamma\). The point class belongs to \(\mathcal R^d(A_0)\) [24], and its normalized trace is one. Compatibility of traces with specialization gives the required components in these cases.

Suppose now that \(0<r<d\). Choose the CM variety \(B\), Hodge class \(\kappa\), and rational homomorphism \[b=aj\in\mathop{\mathrm{Hom}}^0(A_0,B_0),\qquad \gamma_0=b^*\kappa_0\] from Corollary 9. The equality holds in all the stated realizations, with the usual absolute-Hodge specializations of \(\kappa\). Axiom (R3) gives one class \(\rho_\kappa\in\mathcal R^r(B_0)\) with \[(\rho_\kappa)_\ell=\kappa_{0,\ell}\quad(\ell\ne p),\qquad (\rho_\kappa)_p\otimes1=\kappa_{0,p}.\]

Choose an integer \(N>0\) such that \(h=Nb\) is an actual homomorphism \(A_0\to B_0\). Pullback by \(h\) preserves the theory by axiom (R1) of [24]. Since \(\mathcal R^r(A_0)\) is a \(\mathbb Q\)-vector space, we may set \[\rho_\gamma=N^{-2r}h^*\rho_\kappa\in\mathcal R^r(A_0).\] Functoriality of the component maps and (1) give \[\begin{align*} (\rho_\gamma)_\ell &=N^{-2r}h^*\kappa_{0,\ell} =b^*\kappa_{0,\ell}=\gamma_{0,\ell} &&(\ell\ne p),\\ (\rho_\gamma)_p\otimes1 &=N^{-2r}h^*\kappa_{0,p} =b^*\kappa_{0,p}=\gamma_{0,p}. \end{align*}\] The second line is taken after extension from \(K_0\) to \(C_w\). Pullback in (R1) applies to any homomorphism \(h\), and rational scalar multiplication allows any denominator \(N\). Thus the argument requires neither an isogeny nor separability, and remains valid when \(p\mid N\), including at \(p=2\).

Finally, two classes with the required components have the same \(\ell\)-adic component for any one \(\ell\ne p\). Injectivity of that component map by (R4) makes them equal. Thus the class is unique in the fixed theory, independently of the auxiliary choices. ◻

Proof. In codimension zero or top codimension, use the corresponding rational multiple of the unit or the point class, as in the preceding proof. This includes dimension zero. For \(0<r<d\), use the CM variety \(B\), rational Hodge class \(\kappa\), and map \[b=aj\in\mathop{\mathrm{Hom}}^0(A_0,B_0),\qquad \gamma_0=b^*\kappa_0\] given by Corollary 9, with the equality in every stated realization.

By [31], \(\kappa\) is the Betti class of a rational algebraic cycle on \(B_{\mathbb C}\). A representative can be chosen over \(\overline{\mathbb Q}\): spread each of its finitely many components in the Hilbert scheme of the fixed projective variety \(B\), and choose a \(\overline{\mathbb Q}\)-point on the same geometrically connected component. The universal flat family preserves the Betti cycle class. Thus there is \(Z\in\operatorname{CH}^r(B)_{\mathbb Q}\) with class \(\kappa\). Descend \(B\) and \(Z\) to a number field and extend the chosen place so that \(B\) has an abelian scheme model. Specialization gives \(Z_0\in\operatorname{CH}^r(B_0)_{\mathbb Q}\). Compatibility of cycle classes with smooth proper specialization and the de Rham–crystalline comparison identifies its classes with \(\kappa_{0,\ell}\) and \(\kappa_{0,p}\), respectively. In particular the same rational cycle \(Z_0\) gives every realization.

Choose \(N>0\) with \(h=Nb\) an actual homomorphism \(A_0\to B_0\), and put \[z_\gamma=N^{-2r}h^*Z_0\in\operatorname{CH}^r(A_0)_{\mathbb Q}.\] Here \(h^*\) is the Chow pullback between smooth varieties, defined by the graph and refined intersection; it requires neither flatness nor separability of \(h\). Functoriality of cycle classes and (1) give the required equalities. Rational coefficients permit this denominator even if \(p\mid N\); the crystalline realization has already inverted \(p\). Finally, compatibility of cycle classes with products and normalized traces gives the asserted intersection number for every \(y\). ◻