abstract: For a supersingular abelian variety X X X F ‾ p \overline{\mathbb{F}}_p F p T 4 ( X ) T^4(X) T 4 ( X ) S ( X ) S(X) S ( X ) p ≥ 5 p \ge5 p ≥ 5 G p \mathcal{G}_p G p p p p X X X X [ p ∞ ] ≅ G p X[p^\infty] \cong\mathcal{G}_p X [ p ∞ ] ≅ G p T 4 ( X ) / S ( X ) ≅ ( Z / p ) 4 T^4(X)/S(X) \cong(\mathbb{Z}/p)^4 T 4 ( X ) / S ( X ) ≅ ( Z / p ) 4 X X X p p p T 4 / S T^4/S T 4 / S ( Z / p ) 2 (\mathbb{Z}/p)^2 ( Z / p ) 2 ( Z / p ) 2 (\mathbb{Z}/p)^2 ( Z / p ) 2 p p p T 4 = S T^4=S T 4 = S G p \mathcal{G}_p G p
Introduction
Let X X X g g g F ‾ p \overline{\mathbb{F}}_p F p W = W ( F ‾ p ) W = W(\overline{\mathbb{F}}_p) W = W ( F p ) σ \sigma σ M ( X ) = H c r i s 1 ( X / W ) M(X) = H^1_{\mathrm{cris}}(X/W) M ( X ) = H cris 1 ( X / W ) W W W 2 g 2g 2 g σ \sigma σ F F F H c r i s n ( X / W ) H^n_{\mathrm{cris}}(X/W) H cris n ( X / W ) ⋀ n M ( X ) \bigwedge^n M(X) ⋀ n M ( X ) n n n
T n ( X ) = { x ∈ H c r i s n ( X / W ) : F x = p n / 2 x } , (1) T^n(X) = \{x \in H^n_{\mathrm{cris}}(X/W) : Fx = p^{n/2}x\}, \tag*{(1)} T n ( X ) = { x ∈ H cris n ( X / W ) : F x = p n /2 x } , (1) a free Z p \mathbb{Z}_p Z p ( 2 g n ) \binom{2g}{n} ( n 2 g ) X X X X X X F F F X = E × E ′ X = E \times E' X = E × E ′ E , E ′ E,E' E , E ′ T 2 ( X ) T^2(X) T 2 ( X ) § 2.1 and NS ( X ) \operatorname{NS}(X) NS ( X ) [1 Chapter I, Proposition 14.2] ). Let Alg i ( X ) ⊆ T 2 i ( X ) \operatorname{Alg}^i(X) \subseteq T^{2i}(X) Alg i ( X ) ⊆ T 2 i ( X ) Z p \mathbb{Z}_p Z p i i i S ( X ) ⊆ T 4 ( X ) S(X) \subseteq T^4(X) S ( X ) ⊆ T 4 ( X ) Z p \mathbb{Z}_p Z p T 2 ( X ) T^2(X) T 2 ( X ) i i i X X X Alg i ( X ) = T 2 i ( X ) \operatorname{Alg}^i(X) = T^{2i}(X) Alg i ( X ) = T 2 i ( X ) i = 1 i = 1 i = 1 NS ( X ) ⊗ Z p ≅ T 2 ( X ) \operatorname{NS}(X) \otimes\mathbb{Z}_p \cong T^2(X) NS ( X ) ⊗ Z p ≅ T 2 ( X ) § 2.1 ), and hence S ( X ) ⊆ Alg 2 ( X ) S(X) \subseteq\operatorname{Alg}^2(X) S ( X ) ⊆ Alg 2 ( X ) T 2 ( X ) ⊗ W T^2(X) \otimes W T 2 ( X ) ⊗ W ⋀ 2 M ( X ) \bigwedge^2 M(X) ⋀ 2 M ( X ) p − 1 ∧ 2 F p^{-1}\wedge^2 F p − 1 ∧ 2 F T 2 i ( X ) ⊗ Q T^{2i}(X) \otimes\mathbb{Q} T 2 i ( X ) ⊗ Q T 4 ( X ) / Alg 2 ( X ) T^4(X)/\operatorname{Alg}^2(X) T 4 ( X ) / Alg 2 ( X ) T 4 ( X ) / S ( X ) T^4(X)/S(X) T 4 ( X ) / S ( X ) Hypothesis 7.1 below, for the abelian fourfolds with one particular supersingular p p p
Let G p \mathcal{G}_p G p p p p F ‾ p \overline{\mathbb{F}}_p F p
M = W 2 ⊗ W 2 ⊗ W 2 , F ( x 1 ⊗ x 2 ⊗ x 3 ) = σ x 3 ⊗ A σ x 1 ⊗ σ x 2 , A = ( 0 p 1 0 ) . (2) M = W^2 \otimes W^2 \otimes W^2,\qquad F(x_1 \otimes x_2 \otimes x_3) = \sigma x_3 \otimes A\sigma x_1 \otimes\sigma x_2,\qquad A = \begin{pmatrix} 0 & p \\ 1 & 0 \end{pmatrix}. \tag*{(2)} M = W 2 ⊗ W 2 ⊗ W 2 , F ( x 1 ⊗ x 2 ⊗ x 3 ) = σ x 3 ⊗ A σ x 1 ⊗ σ x 2 , A = ( 0 1 p 0 ) . (2) It exists by (R1) and depends only on p p p E 0 [ p ∞ ] × H E_0[p^\infty]\times\mathcal{H} E 0 [ p ∞ ] × H E 0 E_0 E 0 p p p H \mathcal{H} H
Part (b) of the theorem below depends on Markman’s preprint [2 ] , which has not been refereed. Markman states that the Weil classes on polarized abelian sixfolds of Weil type with discriminant − 1 -1 − 1 [2 Theorem 1.5.1] , a statement with rational coefficients. We need an integral form of it with a multiplier prime to p p p d ≥ 3 d \ge3 d ≥ 3 Q ( − d ) \mathbb{Q}(\sqrt{-d}) Q ( − d ) − 1 -1 − 1 2 ( 8 d ) 8 κ 3 ( E ) 2(8d)^8\kappa_3(\mathcal{E}) 2 ( 8 d ) 8 κ 3 ( E ) E \mathcal{E} E 8 d 8d 8 d [2 Theorem 1.4.1] (Proposition 7.3). We prove it from three statements made in Markman’s proof of [2 Theorem 1.5.1] and from other statements of [2 ] , which together form Hypothesis 7.1 in § 7 ; we have not verified Markman’s proofs of them.
Theorem 1.3 (computer-assisted). Let p ≥ 5 p \ge5 p ≥ 5 X X X F ‾ p \overline{\mathbb{F}}_p F p X [ p ∞ ] ≅ G p X[p^\infty] \cong\mathcal{G}_p X [ p ∞ ] ≅ G p
(a) T 4 ( X ) / S ( X ) ≅ ( Z / p ) 4 T^4(X)/S(X) \cong(\mathbb{Z}/p)^4 T 4 ( X ) / S ( X ) ≅ ( Z / p ) 4
(b) Assume Hypothesis 7.1. Then Alg 2 ( X ) = T 4 ( X ) \operatorname{Alg}^2(X)=T^4(X) Alg 2 ( X ) = T 4 ( X ) X X X H c r i s 2 ( X ) ⊗ H c r i s 2 ( X ) H^2_{\mathrm{cris}}(X)\otimes H^2_{\mathrm{cris}}(X) H cris 2 ( X ) ⊗ H cris 2 ( X ) T 4 ( X × X ) T^4(X\times X) T 4 ( X × X )
Besides Hypothesis 7.1, the proof uses the standard facts of § 2.1 , a closed-immersion criterion of SGA 3 in Lemma 3.1, the exact computations of Appendix § 10 , and Lemma 7.2 on Chern classes in families, which rests on results of Atiyah and Hirzebruch, of Borel and Haefliger, and of Grothendieck.
Part (b) is proved first at the supersingular points of Mumford’s families. Let k k k B B B k k k Cor k / Q B ≅ Mat 8 ( Q ) \operatorname{Cor}_{k/\mathbb{Q}}B \cong\operatorname{Mat}_8(\mathbb{Q}) Cor k / Q B ≅ Mat 8 ( Q ) [3 Section 4] constructed from ( k , B ) (k,B) ( k , B ) B 1 B^1 B 1 B B B V = Q 8 V=\mathbb{Q}^8 V = Q 8 B × → ( Cor k / Q B ) × B^\times\to(\operatorname{Cor}_{k/\mathbb{Q}}B)^\times B × → ( Cor k / Q B ) × ( G , X ) (G,X) ( G , X ) G ⊂ GL ( V ) G\subset\operatorname{GL}(V) G ⊂ GL ( V ) G m × B 1 \mathbb{G}_{\mathrm{m}}\times B^1 G m × B 1 [4 Section 2] ), so that G Q ‾ G_{\overline{\mathbb{Q}}} G Q G m × SL 2 3 \mathbb{G}_{\mathrm{m}}\times\operatorname{SL}_2^3 G m × SL 2 3 U 1 ⊗ U 2 ⊗ U 3 U_1\otimes U_2\otimes U_3 U 1 ⊗ U 2 ⊗ U 3 dim U i = 2 \dim U_i=2 dim U i = 2 ψ \psi ψ V V V G ⊂ GSp ( V , ψ ) G\subset\operatorname{GSp}(V,\psi) G ⊂ GSp ( V , ψ ) Λ ⊂ V \Lambda\subset V Λ ⊂ V p p p K = K p K p K=K_pK^p K = K p K p K p K_p K p Λ ⊗ Z p \Lambda\otimes\mathbb{Z}_p Λ ⊗ Z p G ( Q p ) G(\mathbb{Q}_p) G ( Q p ) K p K^p K p k k k B B B S \mathcal{S} S [5 ] over the localization of its ring of integers at a prime above p p p A → S \mathcal{A}\to\mathcal{S} A → S
(h1) p ≥ 5 p\ge5 p ≥ 5
(h2) p p p k k k
(h3) B ⊗ k k p ≅ Mat 2 ( k p ) B\otimes_k k_p\cong\operatorname{Mat}_2(k_p) B ⊗ k k p ≅ Mat 2 ( k p ) k p = k ⊗ Q p k_p=k\otimes\mathbb{Q}_p k p = k ⊗ Q p O k p = O k ⊗ Z p \mathcal{O}_{k_p}=\mathcal{O}_k\otimes\mathbb{Z}_p O k p = O k ⊗ Z p V ⊗ Q p = Cor k p / Q p U p V\otimes\mathbb{Q}_p=\operatorname{Cor}_{k_p/\mathbb{Q}_p}U_p V ⊗ Q p = Cor k p / Q p U p k p k_p k p U p ≅ k p 2 U_p\cong k_p^2 U p ≅ k p 2 B ⊗ k k p B\otimes_k k_p B ⊗ k k p Cor \operatorname{Cor} Cor
(h4) Λ ⊗ Z p = p j Cor ( Λ 0 ) \Lambda\otimes\mathbb{Z}_p=p^j\operatorname{Cor}(\Lambda_0) Λ ⊗ Z p = p j Cor ( Λ 0 ) O k p \mathcal{O}_{k_p} O k p Λ 0 ⊂ U p \Lambda_0\subset U_p Λ 0 ⊂ U p j ∈ Z j\in\mathbb{Z} j ∈ Z
By Lemma 3.1 , (h4) implies that the closure of G G G GL ( Λ ⊗ Z ( p ) ) \operatorname{GL}(\Lambda\otimes\mathbb{Z}_{(p)}) GL ( Λ ⊗ Z ( p ) ) K p K_p K p ( k , B , Λ ) (k,B,\Lambda) ( k , B , Λ ) p p p p ≥ 5 p\ge5 p ≥ 5 Lemma 8.2 ).
Under (h2)–(h4) every supersingular point of S ( F ‾ p ) \mathcal{S}(\overline{\mathbb{F}}_p) S ( F p ) p p p G p \mathcal{G}_p G p Proposition 3.3 ), so Theorem 1.3 applies to these points. Theorem 5.2 describes T 4 / S T^4/S T 4 / S ( Z / p ) 2 (\mathbb{Z}/p)^2 ( Z / p ) 2 ( Z / p ) 2 (\mathbb{Z}/p)^2 ( Z / p ) 2 T 4 T^4 T 4 S S S k k k B B B p p p T 4 = S T^4=S T 4 = S Proposition 8.4 ). In § 9 we compare G p \mathcal{G}_p G p X C X_{\mathbb{C}} X C X C × X C X_{\mathbb{C}} \times X_{\mathbb{C}} X C × X C [6 ] , and as far as we know their algebraicity is open; Remark 8.6 records the modest consequence of our results for them.
A preprint of Broe claims the Tate conjecture for abelian fourfolds over finite fields, through Markman’s Weil classes, and deduces the crystalline Tate conjecture for them [7 Theorem 1.1 and Corollary 8.1(ii)] ; it does not state an integral version. A preprint of Milne states that it deduces a similar statement for the integral Tate conjecture [8 ] from the failure of the integral Hodge conjecture for very general principally polarized abelian varieties of dimension at least 4, which is claimed in a preprint of Engel, de Gaay Fortman and Schreieder [9 ] . A preprint of Scavia and Suzuki claims applications of its results on direct summands of products of Jacobians to the integral Tate conjecture for divisors and 1-cycles on abelian varieties over finitely generated fields, and to a p p p [10 ] . The same authors constructed non-algebraic classes over finite fields that vanish over the algebraic closure [11 ] . The classes of [11 ] have degree 4, and neither abstract concerns integral Tate classes over F ‾ p \overline{\mathbb{F}}_p F p [12 ] . Gregory proved that torsion codimension-2 cycles on supersingular abelian varieties are algebraically trivial [13 ] , which concerns the kernel of the cycle class map rather than its image. For Mumford’s families, a preprint of Chen claims that some of their fourfolds have infinitely many supersingular primes [14 ] .
Use of large language models and status of the proofs
This note was produced by large language models, instances of the Claude models of Anthropic running as autonomous agents, in October 2026. They chose the question, found the proofs, wrote and ran the programs of Appendix A, and wrote the text. The proofs were checked only by other instances of the same models, which re-derived each result and repeated every computation with independently written programs. Such checks may share the blind spots of the models that wrote the proofs. The author has read the note and verified the bibliographic data of its references, but no human expert has checked the proofs. The least secure step is the dependence of part (b) on statements of [2 ] whose proofs we have not verified (§ 7 ).
Code availability
The programs for the computations of Appendix A are available under the MIT license at https://github.com/ahippelainen/integral-tate-supersingular-fourfolds-public.
Conventions
Throughout, p ≥ 5 p \ge5 p ≥ 5 W W W σ \sigma σ Q ~ p = W [ 1 / p ] \widetilde{\mathbb{Q}}_p=W[1/p] Q p = W [ 1/ p ] r ≥ 1 r \ge1 r ≥ 1 Z p r = W ( F p r ) \mathbb{Z}_{p^r}=W(\mathbb{F}_{p^r}) Z p r = W ( F p r ) Q p r = Z p r [ 1 / p ] \mathbb{Q}_{p^r}=\mathbb{Z}_{p^r}[1/p] Q p r = Z p r [ 1/ p ] X X X F ‾ p \overline{\mathbb{F}}_p F p M ( X ) M(X) M ( X ) F F F H c r i s n ( X / W ) = ⋀ n M ( X ) H^n_{\mathrm{cris}}(X/W)=\bigwedge^n M(X) H cris n ( X / W ) = ⋀ n M ( X ) V = p F − 1 V=pF^{-1} V = p F − 1 U U U X X X U ∗ U^* U ∗ H c r i s n H^n_{\mathrm{cris}} H cris n ⋀ n U ∗ \bigwedge^n U^* ⋀ n U ∗ Z p \mathbb{Z}_p Z p W W W [ ⋅ : ⋅ ] [\cdot:\cdot] [ ⋅ : ⋅ ] ( e j ) (e_j) ( e j ) Q ~ p \widetilde{\mathbb{Q}}_p Q p ⨁ j p ν j W e j \bigoplus_j p^{\nu_j}We_j ⨁ j p ν j W e j k k k L = k K L = kK L = k K ι , ι ˉ : Z p 2 → W \iota,\bar{\iota}: \mathbb{Z}_{p^2} \to W ι , ι ˉ : Z p 2 → W E 0 E_0 E 0
The method
At an inert prime, the crystalline tensors of the integral model force M ( X ) M(X) M ( X ) σ 3 \sigma^3 σ 3 M ( X ) M(X) M ( X ) F F F e s e_s e s s ∈ { 0 , 1 } 3 s \in\{0,1\}^3 s ∈ { 0 , 1 } 3 T 4 T^4 T 4 S S S p p p T 4 / S ≅ ( Z / p ) 4 T^4/S \cong(\mathbb{Z}/p)^4 T 4 / S ≅ ( Z / p ) 4 ⋀ 4 M \bigwedge^4 M ⋀ 4 M ( 2 , 2 ) (2,2) ( 2 , 2 ) T 4 / S T^4/S T 4 / S
To produce one algebraic Weil class with a unit coefficient, we choose a fourfold X c X_c X c K × L K \times L K × L K K K p p p O K O_K O K p p p X c X_c X c p p p Z Z Z X c X_c X c λ ω + λ ˉ ω ˉ \lambda\omega+\bar{\lambda}\bar{\omega} λω + λ ˉ ω ˉ λ \lambda λ p p p β ∈ O K \beta\in O_K β ∈ O K ( O K / p ) × (O_K/p)^\times ( O K / p ) × P P P Z ( p ) \mathbb{Z}_{(p)} Z ( p ) [ β ] ∗ [\beta]^* [ β ] ∗ Z 1 = P Z Z_1 = PZ Z 1 = P Z Z 2 = [ β ] ∗ Z 1 Z_2 = [\beta]^*Z_1 Z 2 = [ β ] ∗ Z 1
det ( deg ( Z i ⋅ Z j ) ) = − u 2 Nm ( λ ) 2 ( β 4 − β ˉ 4 ) 2 (3) \det\left(\deg(Z_i \cdot Z_j)\right)=-u^2\operatorname{Nm}(\lambda)^2(\beta^4-\bar{\beta}^4)^2 \tag*{(3)} det ( deg ( Z i ⋅ Z j ) ) = − u 2 Nm ( λ ) 2 ( β 4 − β ˉ 4 ) 2 (3) with u u u p p p β 4 ≢ β ˉ 4 m o d p \beta^4 \not\equiv\bar{\beta}^4 \bmod p β 4 ≡ β ˉ 4 mod p p > 5 p > 5 p > 5 p p p p p p p p p p p p G p \mathcal{G}_p G p
Section 2 collects standard facts and the monomial computation, Section 3 derives (1.2) at the supersingular points of Mumford’s families, Sections 4 and 5 prove part (a) and Theorem 5.2, Sections 6 and 7 construct and specialize the Weil cycle, Section 8 proves Theorems 1.3 and 8.3 and Proposition 8.4, and Section 9 treats the other monomial modules. Appendix A describes the computations. The proof of Theorem 1.3 does not use Proposition 8.4, Remark 8.6 or Section 9.
What is new
We searched arXiv, Hexagon and the web in October 2026 for integral and crystalline Tate classes of codimension 2 on supersingular abelian varieties, for integral Tate classes on abelian fourfolds over finite fields, and for Mumford’s fourfolds at supersingular points. Within that search we found no source that decides the integral crystalline Tate conjecture in codimension 2 for a supersingular abelian variety that is not superspecial, none that computes T 4 / S T^4/S T 4 / S p p p T 4 = S T^4 = S T 4 = S p p p
Preliminaries
Standard facts
We use the following facts.
(R1) For an abelian variety X X X g g g F ‾ p \overline{\mathbb{F}}_p F p M ( X ) M(X) M ( X ) 2 g 2g 2 g X [ p ∞ ] X[p^\infty] X [ p ∞ ] H c r i s ∗ ( X / W ) = ⋀ ∗ M ( X ) H^*_{\mathrm{cris}}(X/W)=\bigwedge^* M(X) H cris ∗ ( X / W ) = ⋀ ∗ M ( X ) F F F [15 Examples 1.8(1) and Section 3.4] ; for the contravariant Dieudonné module over finite fields, see e.g. [16 (14.3.1)] ). Dieudonné theory is an anti-equivalence between p p p F ‾ p \overline{\mathbb{F}}_p F p [17 Corollary 3.16, Definition 3.17 and Proposition 3.18] ), and V M / p M = ker ( F m o d p ) VM/pM=\ker(F\mathbin{\bmod}p) V M / pM = ker ( F mod p ) H 0 ( X , Ω 1 ) H^0(X,\Omega^1) H 0 ( X , Ω 1 ) [17 Corollary 5.11] .
(R2) A σ r \sigma^r σ r φ \varphi φ W W W R R R p p p p r p^r p r F ‾ p \overline{\mathbb{F}}_p F p [18 ] ). If ( e i ) (e_i) ( e i ) φ e i ≡ e i ( m o d p n ) \varphi e_i\equiv e_i\pmod{p^n} φ e i ≡ e i ( mod p n ) φ e i = e i + p n ∑ j a i j e j \varphi e_i=e_i+p^n\sum_j a_{ij}e_j φ e i = e i + p n ∑ j a ij e j b i j ∈ W b_{ij}\in W b ij ∈ W σ r ( b i j ) − b i j ≡ − a i j ( m o d p ) \sigma^r(b_{ij})-b_{ij}\equiv-a_{ij}\pmod p σ r ( b ij ) − b ij ≡ − a ij ( mod p ) F ‾ p \overline{\mathbb{F}}_p F p e i + p n ∑ j b i j e j e_i+p^n\sum_j b_{ij}e_j e i + p n ∑ j b ij e j p n + 1 p^{n+1} p n + 1 σ r \sigma^r σ r F F F R R R F n R ⊂ p R F^nR\subset pR F n R ⊂ pR n n n p p p [19 Section 4.5] , which treats σ \sigma σ r > 1 r>1 r > 1 σ \sigma σ ( R , F ) (R,F) ( R , F ) F F F r r r
(R3) For abelian varieties X , X ′ X,X' X , X ′ F ‾ p \overline{\mathbb{F}}_p F p Hom ( X ′ , X ) ⊗ Z p → Hom ( M ( X ) , M ( X ′ ) ) F \operatorname{Hom}(X',X)\otimes\mathbb{Z}_p\to\operatorname{Hom}(M(X),M(X'))^F Hom ( X ′ , X ) ⊗ Z p → Hom ( M ( X ) , M ( X ′ ) ) F [19 proof of Theorem 2.6] ). A supersingular abelian variety of dimension g g g E g E^g E g [20 Theorem 4.2] , so for supersingular X , X ′ X,X' X , X ′ 4 g 2 4g^2 4 g 2 NS ( X ) ⊗ Z p ≅ T 2 ( X ) \operatorname{NS}(X)\otimes\mathbb{Z}_p\cong T^2(X) NS ( X ) ⊗ Z p ≅ T 2 ( X ) X X X F ‾ p \overline{\mathbb{F}}_p F p
(R4) For smooth projective Y Y Y F ‾ p \overline{\mathbb{F}}_p F p cl : CH i ( Y ) → H c r i s 2 i ( Y / W ) ⊗ Q \operatorname{cl}\colon\operatorname{CH}^i(Y)\to H^{2i}_{\mathrm{cris}}(Y/W)\otimes\mathbb{Q} cl : CH i ( Y ) → H cris 2 i ( Y / W ) ⊗ Q [21 I, Theorem 4.3.1] and with push-forwards along morphisms of smooth projective varieties [21 II, Theorem 1.1.3] , with F cl ( Z ) = p i cl ( Z ) F\operatorname{cl}(Z)=p^i\operatorname{cl}(Z) F cl ( Z ) = p i cl ( Z ) Z Z Z i i i [21 Lemma B.2.2(a) and Remark B.2.3], [22, Proposition 7.13] . If Y Y Y ∫ : H c r i s 2 g ( Y / W ) → W \int\colon H^{2g}_{\mathrm{cris}}(Y/W)\to W ∫ : H cris 2 g ( Y / W ) → W g = dim Y g=\dim Y g = dim Y ∫ F x = p g σ ( ∫ x ) \int Fx=p^g\sigma(\int x) ∫ F x = p g σ ( ∫ x ) [15 Section 1.5(2) and Example 3.2] , the latter applied to the pairing of 1 1 1 x x x [21 II, proof of Lemma 1.1.9] ), and ∫ cl ( Z ) = deg Z \int\operatorname{cl}(Z)=\deg Z ∫ cl ( Z ) = deg Z 0 0 0 Z Z Z 2 2 2 cl ( Z ) = − c 2 ( O Z ) \operatorname{cl}(Z)=-c_2(\mathcal{O}_Z) cl ( Z ) = − c 2 ( O Z ) Z Z Z 2 2 2 [21 I, Theorem 4.3.1] , so Alg 2 ( Y ) ⊆ T 4 ( Y ) \operatorname{Alg}^2(Y)\subseteq T^4(Y) Alg 2 ( Y ) ⊆ T 4 ( Y ) [22 Sections 5–7] for the corresponding statements in rigid cohomology and their comparison with crystalline Chern classes.
(R5) Let Y \mathcal{Y} Y O \mathcal{O} O 0 0 0 Y \mathcal{Y} Y O \mathcal{O} O 0 0 0 Y \mathcal{Y} Y O \mathcal{O} O [21 Appendix B, B.3] , and [23 Tag 0H4H] for specialization of cycles over a discrete valuation ring).
(R6) For t ∈ S ( F ‾ p ) t\in\mathcal{S}(\overline{\mathbb{F}}_p) t ∈ S ( F p ) t t t S \mathcal{S} S W [ 1 / p ] W[1/p] W [ 1/ p ] S \mathcal{S} S [5 ] , and there is an isomorphism M ( A t ) ≃ Λ ∨ ⊗ W M(\mathcal{A}_t)\simeq\Lambda^\vee\otimes W M ( A t ) ≃ Λ ∨ ⊗ W F F F Λ ∨ \Lambda^\vee Λ ∨ G G G Λ ∨ ⊗ W \Lambda^\vee\otimes W Λ ∨ ⊗ W [24 Section 4.2] . The crystalline tensors correspond to the étale tensors of a lift under the p p p [24 4.2.1] , which is functorial (see e.g. [25 Remark 3.3.10] ), so endomorphisms of a lift that fix the étale tensors fix the crystalline tensors. Points of Sh K ( G , X ) \operatorname{Sh}_K(G,X) Sh K ( G , X ) p p p S \mathcal{S} S S \mathcal{S} S Sh K ( G , X ) \operatorname{Sh}_K(G,X) Sh K ( G , X ) p p p [5 (2.3.3) and Theorem (2.3.8)] , hence finite over it, as that closure is of finite type over a Dedekind domain of characteristic $0. (see e.g. [23 Tags 0335 and 035S] ), and the point extends to that moduli scheme by the Néron property (see e.g. [1 Chapter I, Remark 17.2] ). Abelian varieties with complex multiplication have potentially good reduction (see e.g. [26 Proposition 10.5] , applied to the simple isogeny factors).
(R7) Skolem–Noether: two homomorphisms from a simple algebra over a field to a central simple algebra Δ \Delta Δ Δ × \Delta^{\times} Δ × [27 Main Theorem 7.7.1] ).
(R8) Unimodular Hermitian lattices of equal rank over O K , p \mathcal{O}_{K,p} O K , p p p p K K K [28 Section 2] , where this is attributed to [29 ] ); a Hermitian space over K p K_p K p K p × K_p^{\times} K p × Q p × \mathbb{Q}_p^{\times} Q p × [28 Section 2] and [27 13.2.5] ). Let U V U_{\mathcal V} U V V \mathcal V V K K K g p ∈ U V ( Q p ) g_p \in U_{\mathcal V}(\mathbb{Q}_p) g p ∈ U V ( Q p ) Υ ⊆ U V ( Q p ) \Upsilon\subseteq U_{\mathcal V}(\mathbb{Q}_p) Υ ⊆ U V ( Q p ) 1 1 1 O K , p × \mathcal{O}_{K,p}^{\times} O K , p × g p Υ ∩ U V ( Q ) ≠ ∅ g_p\Upsilon\cap U_{\mathcal V}(\mathbb{Q}) \ne\varnothing g p Υ ∩ U V ( Q ) = ∅ g p Υ g_p\Upsilon g p Υ g p g_p g p 1 1 1 O K , p × \mathcal{O}_{K,p}^{\times} O K , p × − 1 -1 − 1 g p g_p g p det ( 1 + g p ) ≠ 0 \det(1+g_p) \ne0 det ( 1 + g p ) = 0 g ↦ ( 1 − g ) ( 1 + g ) − 1 g \mapsto(1-g)(1+g)^{-1} g ↦ ( 1 − g ) ( 1 + g ) − 1 det ( 1 + g ) ≠ 0 \det(1+g) \ne0 det ( 1 + g ) = 0 U V ( Q p ) U_{\mathcal V}(\mathbb{Q}_p) U V ( Q p ) Q p \mathbb{Q}_p Q p Q \mathbb{Q} Q u V \mathfrak{u}_{\mathcal V} u V K K K u V ( Q ) \mathfrak{u}_{\mathcal V}(\mathbb{Q}) u V ( Q ) U V ( Q ) U_{\mathcal V}(\mathbb{Q}) U V ( Q ) u V ( Q ) \mathfrak{u}_{\mathcal V}(\mathbb{Q}) u V ( Q ) u V ( Q p ) \mathfrak{u}_{\mathcal V}(\mathbb{Q}_p) u V ( Q p ) g p Υ g_p\Upsilon g p Υ g p g_p g p U V ( Q ) U_{\mathcal V}(\mathbb{Q}) U V ( Q )
Monomial Dieudonné modules
Let M M M W W W ( e s ) s ∈ C (e_s)_{s\in C} ( e s ) s ∈ C F F F σ \sigma σ F e s = p a ( s ) e π ( s ) F e_s = p^{a(s)}e_{\pi(s)} F e s = p a ( s ) e π ( s ) π \pi π C C C a ( s ) ∈ { 0 , 1 } a(s) \in\{0,1\} a ( s ) ∈ { 0 , 1 } π \pi π a ( s ) a(s) a ( s ) F q = p q / 2 σ q F^q = p^{q/2}\sigma^q F q = p q /2 σ q ( e s ) (e_s) ( e s ) q q q π \pi π J ⊆ C J \subseteq C J ⊆ C ∣ J ∣ = n |J|=n ∣ J ∣ = n e J ∈ ⋀ n M e_J \in\bigwedge^n M e J ∈ ⋀ n M e s e_s e s s ∈ J s \in J s ∈ J a ( J ) = ∑ s ∈ J a ( s ) a(J)=\sum_{s\in J}a(s) a ( J ) = ∑ s ∈ J a ( s ) ⋀ n F ( e J ) = ± p a ( J ) e π ( J ) \bigwedge^n F(e_J)=\pm p^{a(J)}e_{\pi(J)} ⋀ n F ( e J ) = ± p a ( J ) e π ( J )
Lemma 2.1 (descent). Let n n n T n = { x ∈ ⋀ n M : ⋀ n F ( x ) = p n / 2 x } T^n=\{x\in\bigwedge^n M:\bigwedge^n F(x)=p^{n/2}x\} T n = { x ∈ ⋀ n M : ⋀ n F ( x ) = p n /2 x } T n ⊗ Z p W = ⨁ ∣ J ∣ = n p ν T ( J ) W e J T^n\otimes_{\mathbb{Z}_p}W=\bigoplus_{|J|=n}p^{\nu_T(J)}We_J T n ⊗ Z p W = ⨁ ∣ J ∣ = n p ν T ( J ) W e J
ν T ( J ) = max 0 ≤ j < q ∑ i = 1 j ( a ( π − i ( J ) ) − n 2 ) . (4) \nu_T(J)=\max_{0\le j<q}\sum_{i=1}^{j}\left(a(\pi^{-i}(J))-\frac{n}{2}\right). \tag*{(4)} ν T ( J ) = 0 ≤ j < q max i = 1 ∑ j ( a ( π − i ( J )) − 2 n ) . (4) and T n T^n T n ( ∣ C ∣ n ) \binom{|C|}{n} ( n ∣ C ∣ ) S n S^n S n x ∧ y x\wedge y x ∧ y x ∈ T n 1 x\in T^{n_1} x ∈ T n 1 y ∈ T n 2 y\in T^{n_2} y ∈ T n 2 n 1 + n 2 = n n_1+n_2=n n 1 + n 2 = n S n ⊗ W = ⨁ ∣ J ∣ = n p ν S ( J ) W e J S^n\otimes W=\bigoplus_{|J|=n}p^{\nu_S(J)}We_J S n ⊗ W = ⨁ ∣ J ∣ = n p ν S ( J ) W e J ν S ( J ) = min J = J 1 ⊔ J 2 ( ν T ( J 1 ) + ν T ( J 2 ) ) \nu_S(J)=\min_{J=J_1\sqcup J_2}(\nu_T(J_1)+\nu_T(J_2)) ν S ( J ) = min J = J 1 ⊔ J 2 ( ν T ( J 1 ) + ν T ( J 2 )) ∣ J i ∣ = n i |J_i|=n_i ∣ J i ∣ = n i [ T n : S n ] = p ∑ J ( ν S ( J ) − ν T ( J ) ) [T^n:S^n]=p^{\sum_J(\nu_S(J)-\nu_T(J))} [ T n : S n ] = p ∑ J ( ν S ( J ) − ν T ( J ))
Proof. Put φ = p − n / 2 ⋀ n F \varphi=p^{-n/2}\bigwedge^n F φ = p − n /2 ⋀ n F ⋀ n M ⊗ Q \bigwedge^n M\otimes\mathbb{Q} ⋀ n M ⊗ Q σ \sigma σ φ ( e J ) = ± p a ( J ) − n / 2 e π ( J ) \varphi(e_J)=\pm p^{a(J)-n/2}e_{\pi(J)} φ ( e J ) = ± p a ( J ) − n /2 e π ( J ) φ q = σ q \varphi^q=\sigma^q φ q = σ q F q = p q / 2 σ q F^q=p^{q/2}\sigma^q F q = p q /2 σ q M M M Λ φ = ⋂ 0 ≤ j < q φ j ( ⋀ n M ) \Lambda_\varphi=\bigcap_{0\le j<q}\varphi^j(\bigwedge^n M) Λ φ = ⋂ 0 ≤ j < q φ j ( ⋀ n M ) φ q ( ⋀ n M ) = ⋀ n M \varphi^q(\bigwedge^n M)=\bigwedge^n M φ q ( ⋀ n M ) = ⋀ n M φ ( Λ φ ) = Λ φ \varphi(\Lambda_\varphi)=\Lambda_\varphi φ ( Λ φ ) = Λ φ φ \varphi φ σ \sigma σ Λ φ \Lambda_\varphi Λ φ Λ φ = Λ φ φ ⊗ Z p W \Lambda_\varphi=\Lambda_\varphi^{\varphi}\otimes_{\mathbb{Z}_p}W Λ φ = Λ φ φ ⊗ Z p W φ \varphi φ ⋀ n M \bigwedge^n M ⋀ n M φ j ( ⋀ n M ) \varphi^j(\bigwedge^n M) φ j ( ⋀ n M ) Λ φ φ = T n \Lambda_\varphi^{\varphi}=T^n Λ φ φ = T n T n ⊗ W = Λ φ T^n\otimes W=\Lambda_\varphi T n ⊗ W = Λ φ φ j ( ⋀ n M ) \varphi^j(\bigwedge^n M) φ j ( ⋀ n M ) ∑ i = 1 j ( a ( π − i J ) − n / 2 ) \sum_{i=1}^{j}(a(\pi^{-i}J)-n/2) ∑ i = 1 j ( a ( π − i J ) − n /2 ) e J e_J e J p ν T ( J 1 ) e J 1 p^{\nu_T(J_1)}e_{J_1} p ν T ( J 1 ) e J 1 p ν T ( J 2 ) e J 2 p^{\nu_T(J_2)}e_{J_2} p ν T ( J 2 ) e J 2 ± p ν T ( J 1 ) + ν T ( J 2 ) e J 1 ⊔ J 2 \pm p^{\nu_T(J_1)+\nu_T(J_2)}e_{J_1\sqcup J_2} ± p ν T ( J 1 ) + ν T ( J 2 ) e J 1 ⊔ J 2 J 1 ∩ J 2 = ∅ J_1\cap J_2=\varnothing J 1 ∩ J 2 = ∅ 0 0 0 ⊗ W \otimes W ⊗ W ν S ( J ) \nu_S(J) ν S ( J ) W W W Z p \mathbb{Z}_p Z p T n / S n T^n/S^n T n / S n Z p \mathbb{Z}_p Z p ( T n / S n ) ⊗ W (T^n/S^n)\otimes W ( T n / S n ) ⊗ W W W W
The same argument applies to Hom-lattices: End ( M , F ) ⊗ W \operatorname{End}(M,F)\otimes W End ( M , F ) ⊗ W F j End W ( M ) F − j F^j\operatorname{End}_W(M)F^{-j} F j End W ( M ) F − j 0 ≤ j < q 0\le j<q 0 ≤ j < q ( e s ) (e_s) ( e s ) U 1 , … , U 4 U_1,\ldots,U_4 U 1 , … , U 4 D ( U 1 , … , U 4 ) D(U_1,\ldots,U_4) D ( U 1 , … , U 4 ) U ↦ ⋀ 4 U U\mapsto\bigwedge^4 U U ↦ ⋀ 4 U e J e_J e J 4 ! 4! 4 ! Z p \mathbb{Z}_p Z p ⋀ 4 U ( x ) \bigwedge^4 U(x) ⋀ 4 U ( x ) U ∈ End ( M , F ) U \in\operatorname{End}(M,F) U ∈ End ( M , F ) x x x Z p \mathbb{Z}_p Z p D D D ⊗ W \otimes W ⊗ W a ( s ) a(s) a ( s ) p ≥ 5 p \ge5 p ≥ 5
The Dieudonné module at a supersingular point
We show that all supersingular points of S ( F ‾ p ) \mathcal{S}(\overline{\mathbb{F}}_p) S ( F p )
Lemma 3.1. Assume ( h 2 ) (h2) ( h 2 ) ( h 4 ) (h4) ( h 4 ) G \mathcal{G} G G G G GL ( Λ ⊗ Z ( p ) ) \operatorname{GL}(\Lambda\otimes\mathbb{Z}_{(p)}) GL ( Λ ⊗ Z ( p ) ) G ( W ) = G ( W ) G(W)=\mathcal{G}(W) G ( W ) = G ( W ) Λ 1 , Λ 2 , Λ 3 \Lambda_1,\Lambda_2,\Lambda_3 Λ 1 , Λ 2 , Λ 3 Λ 0 \Lambda_0 Λ 0 W W W O k p → W \mathcal{O}_{k_p} \to W O k p → W G \mathcal{G} G G W \mathcal{G}_W G W G m ⋅ SL 2 3 \mathbb{G}_m \cdot\operatorname{SL}_2^3 G m ⋅ SL 2 3 Λ 1 ⊗ Λ 2 ⊗ Λ 3 \Lambda_1 \otimes\Lambda_2 \otimes\Lambda_3 Λ 1 ⊗ Λ 2 ⊗ Λ 3 G ( Q ˘ p ) G(\breve{\mathbb{Q}}_p) G ( Q ˘ p ) G ( W ) G(W) G ( W ) GL 2 ( Q ˘ p ) 3 \operatorname{GL}_2(\breve{\mathbb{Q}}_p)^3 GL 2 ( Q ˘ p ) 3 GL 2 ( W ) 3 \operatorname{GL}_2(W)^3 GL 2 ( W ) 3 G ( W ) G(W) G ( W ) Λ ⊗ Q ˘ p \Lambda\otimes\breve{\mathbb{Q}}_p Λ ⊗ Q ˘ p c ( Λ ⊗ W ) c(\Lambda\otimes W) c ( Λ ⊗ W ) c ∈ Q ˘ p × c \in\breve{\mathbb{Q}}_p^\times c ∈ Q ˘ p × Λ ∨ \Lambda^\vee Λ ∨ Λ 0 \Lambda_0 Λ 0
Proof. Since p p p k k k O k p ⊗ Z p W = W 3 \mathcal{O}_{k_p} \otimes_{\mathbb{Z}_p} W = W^3 O k p ⊗ Z p W = W 3 Λ ⊗ W = p j Λ 1 ⊗ Λ 2 ⊗ Λ 3 \Lambda\otimes W = p^j\Lambda_1 \otimes\Lambda_2 \otimes\Lambda_3 Λ ⊗ W = p j Λ 1 ⊗ Λ 2 ⊗ Λ 3 G Q ˘ p G_{\breve{\mathbb{Q}}_p} G Q ˘ p GL 2 3 \operatorname{GL}_2^3 GL 2 3 GL 2 , W 3 → GL ( Λ 1 ⊗ Λ 2 ⊗ Λ 3 ) \operatorname{GL}_{2,W}^3 \to\operatorname{GL}(\Lambda_1 \otimes\Lambda_2 \otimes\Lambda_3) GL 2 , W 3 → GL ( Λ 1 ⊗ Λ 2 ⊗ Λ 3 ) T T T γ 1 ⊗ γ 2 ⊗ γ 3 \gamma_1 \otimes\gamma_2 \otimes\gamma_3 γ 1 ⊗ γ 2 ⊗ γ 3 GL 2 , W 3 / T \operatorname{GL}_{2,W}^3/T GL 2 , W 3 / T GL ( Λ ⊗ W ) \operatorname{GL}(\Lambda\otimes W) GL ( Λ ⊗ W ) [30 Exp. XVI, Corollaire 1.5(a)] . Its image is flat and closed with generic fiber G Q ˘ p G_{\breve{\mathbb{Q}}_p} G Q ˘ p G Q ˘ p G_{\breve{\mathbb{Q}}_p} G Q ˘ p G W \mathcal{G}_W G W Z ( p ) → W \mathbb{Z}_{(p)} \to W Z ( p ) → W [23 Tag 081I] ); so G W \mathcal{G}_W G W G \mathcal{G} G T T T Pic ( W ) = 0 \operatorname{Pic}(W)=0 Pic ( W ) = 0 GL 2 ( W ) 3 → G ( W ) \operatorname{GL}_2(W)^3 \to\mathcal{G}(W) GL 2 ( W ) 3 → G ( W ) GL 2 ( Q ˘ p ) 3 → G ( Q ˘ p ) \operatorname{GL}_2(\breve{\mathbb{Q}}_p)^3 \to G(\breve{\mathbb{Q}}_p) GL 2 ( Q ˘ p ) 3 → G ( Q ˘ p ) Λ 1 ⊗ Λ 2 ⊗ Λ 3 \Lambda_1 \otimes\Lambda_2 \otimes\Lambda_3 Λ 1 ⊗ Λ 2 ⊗ Λ 3 p p p SL 2 ( F ‾ p ) 3 \operatorname{SL}_2(\overline{\mathbb{F}}_p)^3 SL 2 ( F p ) 3 GL 2 ( W ) 3 → GL 2 ( F ‾ p ) 3 \operatorname{GL}_2(W)^3 \to\operatorname{GL}_2(\overline{\mathbb{F}}_p)^3 GL 2 ( W ) 3 → GL 2 ( F p ) 3 G ( W ) G(W) G ( W ) Λ ⊗ W \Lambda\otimes W Λ ⊗ W p ( Λ ⊗ W ) p(\Lambda\otimes W) p ( Λ ⊗ W ) Λ ⊗ F ‾ p \Lambda\otimes\overline{\mathbb{F}}_p Λ ⊗ F p Λ ⊗ W \Lambda\otimes W Λ ⊗ W Λ ∨ \Lambda^\vee Λ ∨ Λ ∨ ⊗ W = p − j Λ 1 ∨ ⊗ Λ 2 ∨ ⊗ Λ 3 ∨ \Lambda^\vee\otimes W = p^{-j}\Lambda_1^\vee\otimes\Lambda_2^\vee\otimes\Lambda_3^\vee Λ ∨ ⊗ W = p − j Λ 1 ∨ ⊗ Λ 2 ∨ ⊗ Λ 3 ∨
Lemma 3.2. Let R R R W W W F R F_R F R σ r \sigma^r σ r p R ⊂ F R R ⊂ R pR \subset F_RR \subset R pR ⊂ F R R ⊂ R R / F R R R/F_RR R / F R R ( R , F R ) ≅ ( W 2 , A σ r ) (R,F_R) \cong(W^2,A\sigma^r) ( R , F R ) ≅ ( W 2 , A σ r ) A A A
Proof. Put V = p F R − 1 V=pF_R^{-1} V = p F R − 1 F R R / p R F_RR/pR F R R / pR V R / p R VR/pR V R / pR F R F_R F R p p p R / p R R/pR R / pR F R F_R F R p p p F R R = V R F_RR=VR F R R = V R F R 2 R = p R F_R^2R=pR F R 2 R = pR Π = p − 1 F R 2 \Pi=p^{-1}F_R^2 Π = p − 1 F R 2 σ 2 r \sigma^{2r} σ 2 r R R R R = R 0 ⊗ Z p 2 r W R=R_0\otimes_{\mathbb{Z}_{p^{2r}}}W R = R 0 ⊗ Z p 2 r W R 0 = R Π R_0=R^\Pi R 0 = R Π F R F_R F R e 0 ∈ R 0 ∖ F R R 0 e_0 \in R_0\setminus F_RR_0 e 0 ∈ R 0 ∖ F R R 0 e 1 = F R e 0 e_1=F_Re_0 e 1 = F R e 0 F R e 1 = p e 0 F_Re_1=pe_0 F R e 1 = p e 0 e 1 ∉ p R 0 e_1\notin pR_0 e 1 ∈ / p R 0 e 0 ∈ V R 0 = F R R 0 e_0\in VR_0=F_RR_0 e 0 ∈ V R 0 = F R R 0 ( e 0 , e 1 ) (e_0,e_1) ( e 0 , e 1 )
Proposition 3.3. Assume ( h 2 ) (h2) ( h 2 ) ( h 4 ) (h4) ( h 4 ) t ∈ S ( F ‾ p ) t \in\mathcal{S}(\overline{\mathbb{F}}_p) t ∈ S ( F p ) M = M ( A t ) M=M(\mathcal{A}_t) M = M ( A t ) μ \mu μ 0 , 1 3 , 2 3 , 1 0,\frac{1}{3},\frac{2}{3},1 0 , 3 1 , 3 2 , 1 1 , 3 , 3 , 1 1,3,3,1 1 , 3 , 3 , 1 ( M , F ) (M,F) ( M , F ) A t \mathcal{A}_t A t t ∈ S ( F ‾ p ) t \in\mathcal{S}(\overline{\mathbb{F}}_p) t ∈ S ( F p ) p p p G p \mathbb{G}_p G p
For Mumford’s families, that the Newton polygon jumps only at supersingular points is stated, under a mild assumption on p p p [4 ] . We include a proof because the model (1.2) is needed.
Proof. By (R6) and Lemma 3.1, M ≅ Λ ∨ ⊗ W = p − j Λ 1 ∨ ⊗ Λ 2 ∨ ⊗ Λ 3 ∨ M \cong\Lambda^\vee\otimes W = p^{-j}\Lambda_1^\vee\otimes\Lambda_2^\vee\otimes\Lambda_3^\vee M ≅ Λ ∨ ⊗ W = p − j Λ 1 ∨ ⊗ Λ 2 ∨ ⊗ Λ 3 ∨ p p p k k k σ \sigma σ O k p → W \mathcal{O}_{k_p} \to W O k p → W Λ ∨ ⊗ W \Lambda^\vee\otimes W Λ ∨ ⊗ W σ Λ ( x 1 ⊗ x 2 ⊗ x 3 ) = σ x 3 ⊗ σ x 1 ⊗ σ x 2 \sigma\Lambda(x_1 \otimes x_2 \otimes x_3)=\sigma x_3 \otimes\sigma x_1 \otimes\sigma x_2 σ Λ ( x 1 ⊗ x 2 ⊗ x 3 ) = σ x 3 ⊗ σ x 1 ⊗ σ x 2 Λ i ∨ \Lambda_i^\vee Λ i ∨ σ \sigma σ F F F σ Λ \sigma\Lambda σ Λ F = b σ Λ F=b\sigma\Lambda F = bσ Λ b ∈ G ( Q ˘ p ) b \in G(\breve{\mathbb{Q}}_p) b ∈ G ( Q ˘ p ) b = b 1 ⊗ b 2 ⊗ b 3 b=b_1\otimes b_2\otimes b_3 b = b 1 ⊗ b 2 ⊗ b 3 b i ∈ GL 2 ( Q ˘ p ) b_i \in\operatorname{GL}_2(\breve{\mathbb{Q}}_p) b i ∈ GL 2 ( Q ˘ p ) dim A t = 4 \dim\mathcal{A}_t=4 dim A t = 4 F F F ( 1 4 , p 4 ) (1^4,p^4) ( 1 4 , p 4 ) b i b_i b i ( ν i , ν i ′ ) (\nu_i,\nu_i') ( ν i , ν i ′ ) F F F ν i , ν i ′ \nu_i,\nu_i' ν i , ν i ′ i i i ∑ i ∣ ν i − ν i ′ ∣ = 1 \sum_i |\nu_i-\nu_i'|=1 ∑ i ∣ ν i − ν i ′ ∣ = 1 b i b_i b i GL 2 ( W ) \operatorname{GL}_2(W) GL 2 ( W ) b 2 b_2 b 2 b 2 ∈ GL 2 ( W ) diag ( 1 , p ) GL 2 ( W ) b_2\in\operatorname{GL}_2(W)\operatorname{diag}(1,p)\operatorname{GL}_2(W) b 2 ∈ GL 2 ( W ) diag ( 1 , p ) GL 2 ( W ) b 1 , b 3 ∈ GL 2 ( W ) b_1,b_3\in\operatorname{GL}_2(W) b 1 , b 3 ∈ GL 2 ( W )
Conjugation by γ = γ 1 ⊗ γ 2 ⊗ γ 3 ∈ G ( W ) \gamma=\gamma_1\otimes\gamma_2\otimes\gamma_3\in G(W) γ = γ 1 ⊗ γ 2 ⊗ γ 3 ∈ G ( W ) b i b_i b i γ i b i σ ( γ i − 1 ) − 1 \gamma_i b_i\sigma(\gamma_{i-1})^{-1} γ i b i σ ( γ i − 1 ) − 1 γ 2 = 1 \gamma_2=1 γ 2 = 1 γ 3 = σ ( γ 2 ) b 3 − 1 \gamma_3=\sigma(\gamma_2)b_3^{-1} γ 3 = σ ( γ 2 ) b 3 − 1 γ 1 = σ ( γ 3 ) b 1 − 1 \gamma_1=\sigma(\gamma_3)b_1^{-1} γ 1 = σ ( γ 3 ) b 1 − 1 b 1 = b 3 = 1 b_1=b_3=1 b 1 = b 3 = 1 F 3 F^3 F 3 Λ ∨ \Lambda^\vee Λ ∨ F 2 = b 2 σ 3 F_2=b_2\sigma^3 F 2 = b 2 σ 3 F 3 = σ 2 F 2 ⊗ F 2 ⊗ σ F 2 F^3=\sigma^2F_2\otimes F_2\otimes\sigma F_2 F 3 = σ 2 F 2 ⊗ F 2 ⊗ σ F 2 σ 3 \sigma^3 σ 3 ( Λ ∨ , F 2 ) (\Lambda^\vee,F_2) ( Λ ∨ , F 2 ) 0 , 1 0,1 0 , 1 θ , 1 − θ \theta,1-\theta θ , 1 − θ θ ∈ { 0 , 1 2 } \theta\in\{0,\frac12\} θ ∈ { 0 , 2 1 } F F F 1 3 { 3 θ , 1 + θ , 2 − θ , 3 − 3 θ } \frac13\{3\theta,1+\theta,2-\theta,3-3\theta\} 3 1 { 3 θ , 1 + θ , 2 − θ , 3 − 3 θ } 1 , 3 , 3 , 1 1,3,3,1 1 , 3 , 3 , 1 θ = 0 \theta=0 θ = 0 μ \mu μ θ = 1 2 \theta=\frac12 θ = 2 1 r = 3 r=3 r = 3 γ 2 ∈ GL 2 ( W ) \gamma_2\in\operatorname{GL}_2(W) γ 2 ∈ GL 2 ( W ) γ 2 b 2 σ 3 ( γ 2 ) − 1 = A \gamma_2b_2\sigma^3(\gamma_2)^{-1}=A γ 2 b 2 σ 3 ( γ 2 ) − 1 = A σ 2 ( γ 2 ) ⊗ γ 2 ⊗ σ ( γ 2 ) \sigma^2(\gamma_2)\otimes\gamma_2\otimes\sigma(\gamma_2) σ 2 ( γ 2 ) ⊗ γ 2 ⊗ σ ( γ 2 ) b 1 = b 3 = 1 b_1=b_3=1 b 1 = b 3 = 1 b 2 = A b_2=A b 2 = A p j p^j p j F F F ( M , F ) (M,F) ( M , F ) □ \square □
Tate lattices of the model
In the model (1.2) put e s = e s 1 ⊗ e s 2 ⊗ e s 3 e_s=e_{s_1}\otimes e_{s_2}\otimes e_{s_3} e s = e s 1 ⊗ e s 2 ⊗ e s 3 s ∈ C = { 0 , 1 } 3 s\in C=\{0,1\}^3 s ∈ C = { 0 , 1 } 3 A e 0 = e 1 Ae_0=e_1 A e 0 = e 1 A e 1 = p e 0 Ae_1=pe_0 A e 1 = p e 0
F e s = p s 1 e π ( s ) , π ( s 1 , s 2 , s 3 ) = ( s 3 , 1 − s 1 , s 2 ) , (5) F e_s=p^{s_1}e_{\pi(s)},\qquad\pi(s_1,s_2,s_3)=(s_3,1-s_1,s_2), \tag*{(5)} F e s = p s 1 e π ( s ) , π ( s 1 , s 2 , s 3 ) = ( s 3 , 1 − s 1 , s 2 ) , (5) and π \pi π ( 000 , 010 , 011 , 111 , 101 , 100 ) (000,010,011,111,101,100) ( 000 , 010 , 011 , 111 , 101 , 100 ) ( 001 , 110 ) (001,110) ( 001 , 110 ) a ( s ) = s 1 a(s)=s_1 a ( s ) = s 1 000111 000111 000111 01 01 01 q = 6 q=6 q = 6 ker ( F m o d p ) \ker(F\bmod p) ker ( F mod p ) e s e_s e s s 1 = 1 s_1=1 s 1 = 1 C C C
T + = { 000 , 011 , 101 , 110 } , T − = { 001 , 010 , 100 , 111 } , M + = { 000 , 001 , 011 , 101 } , M − = { 010 , 100 , 110 , 111 } . (6) \begin{aligned}
T_+&=\{000,011,101,110\},\qquad T_-=\{001,010,100,111\},\\
M_+&=\{000,001,011,101\},\qquad M_-=\{010,100,110,111\}. \tag*{(6)}
\end{aligned} T + M + = { 000 , 011 , 101 , 110 } , T − = { 001 , 010 , 100 , 111 } , = { 000 , 001 , 011 , 101 } , M − = { 010 , 100 , 110 , 111 } . (6) Here T + T_+ T + T − T_- T − M + , M − M_+,M_- M + , M − ( − 1 ) s (-1)^s ( − 1 ) s + 1 +1 + 1 − 1 -1 − 1 s ↦ ( s 1 , s 2 , 1 − s 3 ) s\mapsto(s_1,s_2,1-s_3) s ↦ ( s 1 , s 2 , 1 − s 3 )
Proposition 4.3 (computer-assisted). For the model (1.2), ν S ( J ) = ν T ( J ) \nu_S(J)=\nu_T(J) ν S ( J ) = ν T ( J ) J ⊆ C J\subseteq C J ⊆ C ( ν T ( J ) , ν S ( J ) ) = ( 0 , 1 ) (\nu_T(J),\nu_S(J))=(0,1) ( ν T ( J ) , ν S ( J )) = ( 0 , 1 ) J = T + , T − , M − J=T_+,T_-,M_- J = T + , T − , M − ( ν T ( J ) , ν S ( J ) ) = ( 1 , 2 ) (\nu_T(J),\nu_S(J))=(1,2) ( ν T ( J ) , ν S ( J )) = ( 1 , 2 ) J = M + J=M_+ J = M + T 4 / S ≅ ( Z / p ) 4 T^4/S\cong(\mathbb{Z}/p)^4 T 4 / S ≅ ( Z / p ) 4 p ν T ( J ) e J p^{\nu_T(J)}e_J p ν T ( J ) e J J ∈ { T + , T − , M + , M − } J\in\{T_+,T_-,M_+,M_-\} J ∈ { T + , T − , M + , M − } ( T 4 / S ) ⊗ W (T^4/S)\otimes W ( T 4 / S ) ⊗ W T 6 T^6 T 6 T 2 T^2 T 2
Proof. Lemma 2.1 reduces the statement to the values of (2.2) and of ν S ( J ) \nu_S(J) ν S ( J ) ( 8 2 ) = 28 \binom{8}{2}=28 ( 2 8 ) = 28 ( 8 4 ) = 70 \binom{8}{4}=70 ( 4 8 ) = 70 T 4 / S T^4/S T 4 / S p p p ν S ( J ) − ν T ( J ) \nu_S(J)-\nu_T(J) ν S ( J ) − ν T ( J ) □ \square □
Lemma 4.4. Let K K K p p p K K K O K , p = Z p \mathcal{O}_{K,p}=\mathbb{Z}_p O K , p = Z p ( M , F ) (M,F) ( M , F ) ( e s ) (e_s) ( e s ) ι s ∈ { ι , ι ‾ } \iota_s\in\{\iota,\overline{\iota}\} ι s ∈ { ι , ι } J ι = { s : ι s = ι } J_\iota=\{s:\iota_s=\iota\} J ι = { s : ι s = ι }
(i) J ι J_\iota J ι π ( J ι ) \pi(J_\iota) π ( J ι )
(ii) If the action is the reduction of an action of O K O_K O K ( 2 , 2 ) (2,2) ( 2 , 2 ) J ι ∈ { T + , T − } J_\iota\in\{T_+,T_-\} J ι ∈ { T + , T − }
(iii) The Weil part ∧ W 4 M ι ⊕ ∧ W 4 M ι ‾ \wedge^4_W M_\iota\oplus\wedge^4_W M_{\overline{\iota}} ∧ W 4 M ι ⊕ ∧ W 4 M ι ∧ 4 M \wedge^4M ∧ 4 M M ι M_\iota M ι M ι ‾ M_{\overline{\iota}} M ι ι \iota ι ι ‾ \overline{\iota} ι W e T + ⊕ W e T − W e_{T_+}\oplus W e_{T_-} W e T + ⊕ W e T − L = T 4 ∩ ( W e T + ⊕ W e T − ) \mathcal{L}=T^4\cap(W e_{T_+}\oplus W e_{T_-}) L = T 4 ∩ ( W e T + ⊕ W e T − ) L ⊗ W = W e T + ⊕ W e T − \mathcal{L}\otimes W=W e_{T_+}\oplus W e_{T_-} L ⊗ W = W e T + ⊕ W e T − x ∈ L ∖ p L x\in\mathcal{L}\setminus p\mathcal{L} x ∈ L ∖ p L λ e T + ± σ ( λ ) e T − \lambda e_{T_+}\pm\sigma(\lambda)e_{T_-} λ e T + ± σ ( λ ) e T − λ ∈ W × \lambda\in W^\times λ ∈ W ×
Proof. For α ∈ Z p \alpha\in\mathbb{Z}_p α ∈ Z p F ( ι s ( α ) e s ) = σ ( ι s ( α ) ) p s 1 e π ( s ) F(\iota_s(\alpha)e_s)=\sigma(\iota_s(\alpha))p^{s_1}e_{\pi(s)} F ( ι s ( α ) e s ) = σ ( ι s ( α )) p s 1 e π ( s ) ι π ( s ) ( α ) p s 1 e π ( s ) \iota_{\pi(s)}(\alpha)p^{s_1}e_{\pi(s)} ι π ( s ) ( α ) p s 1 e π ( s ) ι π ( s ) = σ ∘ ι s = ι s ‾ \iota_{\pi(s)}=\sigma\circ\iota_s=\iota_{\overline{s}} ι π ( s ) = σ ∘ ι s = ι s p p p K K K J ι J_\iota J ι π \pi π J ι J_\iota J ι { 000 , 011 , 101 } \{000,011,101\} { 000 , 011 , 101 } { 010 , 111 , 100 } \{010,111,100\} { 010 , 111 , 100 } 001 001 001 110 110 110 ker ( F m o d p ) = V M / p M ≅ H 0 ( Ω 1 ) \ker(F\mathbin{\bmod}p)=VM/pM\cong H^0(\Omega^1) ker ( F mod p ) = V M / pM ≅ H 0 ( Ω 1 ) O K O_K O K O K O_K O K p p p K K K ∣ J ι ∩ { s 1 = 1 } ∣ = 2 |J_\iota\cap\{s_1=1\}|=2 ∣ J ι ∩ { s 1 = 1 } ∣ = 2 2 , 2 , 1 , 3 2,2,1,3 2 , 2 , 1 , 3 T + , T − , M + , M − T_+,T_-,M_+,M_- T + , T − , M + , M − M ι M_\iota M ι e s e_s e s s ∈ J ι s\in J_\iota s ∈ J ι ⋀ 4 M ι = W e J ι \bigwedge^4M_\iota=W e_{J_\iota} ⋀ 4 M ι = W e J ι W W W W e T + ⊕ W e T − W e_{T_+}\oplus W e_{T_-} W e T + ⊕ W e T − φ = p − 2 ⋀ 4 F \varphi=p^{-2}\bigwedge^4F φ = p − 2 ⋀ 4 F e T + e_{T_+} e T + e T − e_{T_-} e T − a ( T + ) = a ( T − ) = 2 a(T_+)=a(T_-)=2 a ( T + ) = a ( T − ) = 2 ν T ( T + ) = ν T ( T − ) = 0 \nu_T(T_+)=\nu_T(T_-)=0 ν T ( T + ) = ν T ( T − ) = 0 φ \varphi φ λ e T + + λ ′ e T − \lambda e_{T_+}+\lambda'e_{T_-} λ e T + + λ ′ e T − λ ′ = ± σ ( λ ) \lambda'=\pm\sigma(\lambda) λ ′ = ± σ ( λ ) p ( L ⊗ W ) p(\mathcal{L}\otimes W) p ( L ⊗ W ) λ ∈ W × \lambda\in W^\times λ ∈ W ×
Let N = T 4 ∩ ( W e M + ⊕ W e M − ) \mathcal{N}=T^4\cap(We_{M_+}\oplus We_{M_-}) N = T 4 ∩ ( W e M + ⊕ W e M − ) N ⊗ W = p W e M + ⊕ W e M − \mathcal{N}\otimes W=pWe_{M_+}\oplus We_{M_-} N ⊗ W = p W e M + ⊕ W e M − π ( M + ) = M − \pi(M_+)=M_- π ( M + ) = M −
Endomorphisms and the extension
By (R3), End ( X ) ⊗ Z p = End ( M , F ) \operatorname{End}(X)\otimes\mathbb{Z}_p=\operatorname{End}(M,F) End ( X ) ⊗ Z p = End ( M , F ) X X X M ( X ) ≅ M M(X)\cong M M ( X ) ≅ M Z p \mathbb{Z}_p Z p { U ∗ x } \{U^*x\} { U ∗ x } U ∈ End ( X ) U\in\operatorname{End}(X) U ∈ End ( X ) U ∈ End ( X ) ⊗ Z p U\in\operatorname{End}(X)\otimes\mathbb{Z}_p U ∈ End ( X ) ⊗ Z p U ↦ ⋀ 4 U ( x ) U\mapsto\bigwedge^4U(x) U ↦ ⋀ 4 U ( x ) Z p \mathbb{Z}_p Z p End ( M , F ) \operatorname{End}(M,F) End ( M , F )
Proposition 5.1 (computer-assisted). In the model ( 1.2 ) (1.2) ( 1.2 ) with End = End ( M , F ) \operatorname{End}=\operatorname{End}(M,F) End = End ( M , F ) acting on ⋀ 4 M \bigwedge^4M ⋀ 4 M by U ↦ ⋀ 4 U U\mapsto\bigwedge^4U U ↦ ⋀ 4 U
( i ) (i) ( i ) End ∗ S ⊆ S \operatorname{End}^*S\subseteq S End ∗ S ⊆ S
( i i ) (ii) ( ii ) [ T 4 : S + L ] = p 2 [T^4:S+\mathcal{L}]=p^2 [ T 4 : S + L ] = p 2 and S + End ∗ L = T 4 S+\operatorname{End}^*\mathcal{L}=T^4 S + End ∗ L = T 4
( i i i ) (iii) ( iii ) End ∗ N ⊆ S + N \operatorname{End}^*\mathcal{N}\subseteq S+\mathcal{N} End ∗ N ⊆ S + N and [ T 4 : S + N ] = p 2 [T^4:S+\mathcal{N}]=p^2 [ T 4 : S + N ] = p 2
( i v ) (iv) ( i v ) let T 22 T_{22} T 22 be the image of T 4 ( X × X ) T^4(X\times X) T 4 ( X × X ) under the Künneth projection to ⋀ 2 M ⊗ ⋀ 2 M \bigwedge^2M\otimes\bigwedge^2M ⋀ 2 M ⊗ ⋀ 2 M Then T 22 T_{22} T 22 is spanned by the Künneth ( 2 , 2 ) (2,2) ( 2 , 2 ) components of S ( X × X ) S(X\times X) S ( X × X ) and of the classes ( U 1 , U 2 ) ∗ x (U_1,U_2)^*x ( U 1 , U 2 ) ∗ x x ∈ L x\in\mathcal{L} x ∈ L for the homomorphisms ( U 1 , U 2 ) : X × X → X (U_1,U_2):X\times X\to X ( U 1 , U 2 ) : X × X → X ( y 1 , y 2 ) ↦ U 1 y 1 + U 2 y 2 (y_1,y_2)\mapsto U_1y_1+U_2y_2 ( y 1 , y 2 ) ↦ U 1 y 1 + U 2 y 2 U i ∈ End U_i\in\operatorname{End} U i ∈ End
Proof. By Section 2.2 every lattice here is monomial with exponents given by min-plus expressions in the a ( s ) a(s) a ( s )
Theorem 5.2. Let X X X F ‾ p \overline{\mathbb{F}}_p F p M ( X ) ≅ M M(X)\cong M M ( X ) ≅ M ( 1.2 ) (1.2) ( 1.2 ) T 4 / S ≅ ( Z / p ) 4 T^4/S\cong(\mathbb{Z}/p)^4 T 4 / S ≅ ( Z / p ) 4 ( S + N ) / S ≅ ( Z / p ) 2 (S+\mathcal{N})/S\cong(\mathbb{Z}/p)^2 ( S + N ) / S ≅ ( Z / p ) 2 End ( X ) \operatorname{End}(X) End ( X ) T 4 / ( S + N ) T^4/(S+\mathcal{N}) T 4 / ( S + N ) L \mathcal{L} L x ∈ T 4 ∖ ( S + N ) x\in T^4\setminus(S+\mathcal{N}) x ∈ T 4 ∖ ( S + N ) S S S U ∗ x U^*x U ∗ x U ∈ End ( X ) U\in\operatorname{End}(X) U ∈ End ( X ) T 4 T^4 T 4 T 4 / S T^4/S T 4 / S T 4 / ( S + N ) ≅ ( Z / p ) 2 T^4/(S+\mathcal{N})\cong(\mathbb{Z}/p)^2 T 4 / ( S + N ) ≅ ( Z / p ) 2 ( S + N ) / S (S+\mathcal{N})/S ( S + N ) / S End ( X ) \operatorname{End}(X) End ( X ) L \mathcal{L} L Z p 2 \mathbb{Z}_p^2 Z p 2 T + T_+ T + ( ( S + N ) / S ) ⊗ W ((S+\mathcal{N})/S)\otimes W (( S + N ) / S ) ⊗ W p e M + pe_{M_+} p e M + e M − e_{M_-} e M − M + M_+ M + M − M_- M − { s : s 1 = 1 } \{s:s_1=1\} { s : s 1 = 1 }
Proof. The first statement is Proposition 4.3, the stability is Proposition 5.1(i), (iii), and ( S + N ) / S ≅ ( Z / p ) 2 (S+\mathcal{N})/S\cong(\mathbb{Z}/p)^2 ( S + N ) / S ≅ ( Z / p ) 2 N \mathcal{N} N ( S + N ) ⊗ W (S+\mathcal{N})\otimes W ( S + N ) ⊗ W S ⊗ W S\otimes W S ⊗ W e M + e_{M_+} e M + e M − e_{M_-} e M − p p p S ⊗ W S\otimes W S ⊗ W e T + e_{T_+} e T + e T − e_{T_-} e T − ( S + N ) ∩ L = p L (S+\mathcal{N})\cap\mathcal{L}=p\mathcal{L} ( S + N ) ∩ L = p L S + N + L = T 4 S+\mathcal{N}+\mathcal{L}=T^4 S + N + L = T 4 x ∈ T 4 ∖ ( S + N ) x\in T^4\setminus(S+\mathcal{N}) x ∈ T 4 ∖ ( S + N ) x ≡ y ( m o d S + N ) x\equiv y\pmod{S+\mathcal{N}} x ≡ y ( mod S + N ) y ∈ L ∖ p L y\in\mathcal{L}\setminus p\mathcal{L} y ∈ L ∖ p L y = λ e T + ± σ ( λ ) e T − y=\lambda e_{T_+}\pm\sigma(\lambda)e_{T_-} y = λ e T + ± σ ( λ ) e T − λ ∈ W × \lambda\in W^\times λ ∈ W × Z p 2 \mathbb{Z}_p^2 Z p 2 π \pi π T + T_+ T + π \pi π π ( T + ) \pi(T_+) π ( T + ) End ( M , F ) \operatorname{End}(M,F) End ( M , F ) P ℓ ∈ End ( M , F ) ⊗ W P_\ell\in\operatorname{End}(M,F)\otimes W P ℓ ∈ End ( M , F ) ⊗ W e s e_s e s s ∈ T + s\in T_+ s ∈ T + ⋀ 4 P ℓ \bigwedge^4P_\ell ⋀ 4 P ℓ W e T + We_{T_+} W e T + S ⊗ W S\otimes W S ⊗ W e M + , e M − e_{M_+},e_{M_-} e M + , e M − ⋀ 4 P ℓ ( x ) ≡ λ e T + ( m o d S ⊗ W ) \bigwedge^4P_\ell(x)\equiv\lambda e_{T_+}\pmod{S\otimes W} ⋀ 4 P ℓ ( x ) ≡ λ e T + ( mod S ⊗ W ) ⋀ 4 P ℓ ( x ) \bigwedge^4P_\ell(x) ⋀ 4 P ℓ ( x ) W W W ⋀ 4 U ( x ) \bigwedge^4U(x) ⋀ 4 U ( x ) U ∈ End ( M , F ) U\in\operatorname{End}(M,F) U ∈ End ( M , F ) e T − e_{T_-} e T − ( S + End ∗ x ) ⊗ W ⊇ L ⊗ W (S+\operatorname{End}^*x)\otimes W\supseteq\mathcal{L}\otimes W ( S + End ∗ x ) ⊗ W ⊇ L ⊗ W S + End ∗ x ⊇ L S+\operatorname{End}^*x\supseteq\mathcal{L} S + End ∗ x ⊇ L S + End ∗ x = T 4 S+\operatorname{End}^*x=T^4 S + End ∗ x = T 4 ( S + N ) / S (S+\mathcal{N})/S ( S + N ) / S End ( X ) \operatorname{End}(X) End ( X ) p 2 p^2 p 2 x ∉ S + N x\notin S+\mathcal{N} x ∈ / S + N End ( X ) \operatorname{End}(X) End ( X ) x x x T 4 / S T^4/S T 4 / S
A fourfold with complex multiplication
Assume (h1)–(h4). The proof of Theorem 8.3 needs one supersingular point with an integral action of an inert imaginary quadratic field of signature ( 2 , 2 ) (2, 2) ( 2 , 2 )
Proposition 6.1. There are an imaginary quadratic field K K K p p p c ∈ Sh K ( G , X ) ( Q ‾ ) c \in\operatorname{Sh}_{K}(G, X)(\overline{\mathbb{Q}}) c ∈ Sh K ( G , X ) ( Q ) X c X_c X c K × L K \times L K × L End ( X c ) ⊗ Q \operatorname{End}(X_c) \otimes\mathbb{Q} End ( X c ) ⊗ Q L = k K L = kK L = k K
(a) ( O K × O L ) ⊗ Z p (\mathcal{O}_K \times\mathcal{O}_L) \otimes\mathbb{Z}_p ( O K × O L ) ⊗ Z p H 1 ( X c , Z p ) H_1(X_c, \mathbb{Z}_p) H 1 ( X c , Z p )
(b) an embedding K ↪ K × L K \hookrightarrow K \times L K ↪ K × L X c X_c X c K K K ( 2 , 2 ) (2, 2) ( 2 , 2 )
(c) c c c t 0 ∈ S ( F ‾ p ) t_0 \in\mathcal{S}(\overline{\mathbb{F}}_p) t 0 ∈ S ( F p ) X t 0 = A t 0 X_{t_0} = A_{t_0} X t 0 = A t 0 K × L K \times L K × L X t 0 X_{t_0} X t 0
(d) after an automorphism of ( M ( X t 0 ) , F ) (M(X_{t_0}), F) ( M ( X t 0 ) , F ) Proposition 3.3 , ( O K × O L ) ⊗ Z p (\mathcal{O}_K \times\mathcal{O}_L) \otimes\mathbb{Z}_p ( O K × O L ) ⊗ Z p ( e s ) (e_s) ( e s ) ι \iota ι O K , p \mathcal{O}_{K,p} O K , p T + T_+ T + T − T_- T −
Proof. Choice of K K K Let Ram ( B ) \operatorname{Ram}(B) Ram ( B ) k k k B B B p p p ℓ \ell ℓ Ram ( B ) \operatorname{Ram}(B) Ram ( B ) K ℓ K_\ell K ℓ Q ℓ \mathbb{Q}_\ell Q ℓ k q k_q k q q ∣ ℓ q \mid\ell q ∣ ℓ q ∈ Ram ( B ) q \in\operatorname{Ram}(B) q ∈ Ram ( B ) k q k_q k q Q ℓ \mathbb{Q}_\ell Q ℓ Q ℓ \mathbb{Q}_\ell Q ℓ K K K p p p K ℓ K_\ell K ℓ ℓ \ell ℓ L ⊗ k k q L \otimes_k k_q L ⊗ k k q q q q B B B C \mathbb{C} C L L L B B B [27 Proposition 14.6.7] ). Let L 1 L^1 L 1 L × → k × L^\times\to k^\times L × → k × T = G m ⋅ Res k / Q L 1 ⊂ G T = \mathbb{G}_m \cdot\operatorname{Res}_{k/\mathbb{Q}} L^1 \subset G T = G m ⋅ Res k / Q L 1 ⊂ G T R T_{\mathbb{R}} T R h ∈ X h \in X h ∈ X T R T_{\mathbb{R}} T R G G G G m × B 1 \mathbb{G}_m \times B^1 G m × B 1 X X X G ( R ) G(\mathbb{R}) G ( R ) h 0 h_0 h 0 R × × S 1 → R × × B 1 ( R ) \mathbb{R}^\times\times S^1 \to\mathbb{R}^\times\times B^1(\mathbb{R}) R × × S 1 → R × × B 1 ( R ) R × \mathbb{R}^\times R × S 1 S^1 S 1 B 1 ( R ) = SL 2 ( R ) × SU ( 2 ) 2 B^1(\mathbb{R}) = \operatorname{SL}_2(\mathbb{R}) \times\operatorname{SU}(2)^2 B 1 ( R ) = SL 2 ( R ) × SU ( 2 ) 2 ∞ 1 \infty_1 ∞ 1 B B B S 1 S^1 S 1 C → Mat 2 ( R ) \mathbb{C} \to\operatorname{Mat}_2(\mathbb{R}) C → Mat 2 ( R ) x + i y ↦ ( x y − y x ) x + iy \mapsto\begin{pmatrix} x & y \\ -y & x \end{pmatrix} x + i y ↦ ( x − y y x ) [4 ] ). By (R7), the embedding of L ⊗ k , ∞ 1 R ≅ C L \otimes_{k,\infty_1} \mathbb{R} \cong\mathbb{C} L ⊗ k , ∞ 1 R ≅ C B ⊗ k , ∞ 1 R ≅ Mat 2 ( R ) B \otimes_{k,\infty_1} \mathbb{R} \cong\operatorname{Mat}_2(\mathbb{R}) B ⊗ k , ∞ 1 R ≅ Mat 2 ( R ) γ ∞ ∈ GL 2 ( R ) \gamma_{\infty} \in\operatorname{GL}_2(\mathbb{R}) γ ∞ ∈ GL 2 ( R ) diag ( 1 , − 1 ) \operatorname{diag}(1,-1) diag ( 1 , − 1 ) γ ∞ \gamma_{\infty} γ ∞ SL 2 ( R ) \operatorname{SL}_2(\mathbb{R}) SL 2 ( R ) h 0 h_0 h 0 G ( R ) G(\mathbb{R}) G ( R ) ( 1 , γ ∞ , 1 , 1 ) (1,\gamma_{\infty},1,1) ( 1 , γ ∞ , 1 , 1 ) h ∈ X h \in X h ∈ X h ( S 1 ) h(S^1) h ( S 1 ) ( L ⊗ k , ∞ 1 R ) 1 × { 1 } × { 1 } (L \otimes_{k,\infty_1} \mathbb{R})^1 \times\{1\} \times\{1\} ( L ⊗ k , ∞ 1 R ) 1 × { 1 } × { 1 } h h h T R T_{\mathbb{R}} T R
The algebra K × L K \times L K × L Over Q ‾ \overline{\mathbb{Q}} Q T T T U 1 ⊗ U 2 ⊗ U 3 U_1 \otimes U_2 \otimes U_3 U 1 ⊗ U 2 ⊗ U 3 ϵ ∈ { ± } 3 \epsilon\in\{\pm\}^3 ϵ ∈ { ± } 3 i i i L L L U i U_i U i k k k K K K { + + + , − − − } \{+ + +, - - -\} { + + + , − − − } Gal ( Q ‾ / K ) \operatorname{Gal}(\overline{\mathbb{Q}}/K) Gal ( Q / K ) Gal ( Q ‾ / L ) \operatorname{Gal}(\overline{\mathbb{Q}}/L) Gal ( Q / L ) Q [ T ( Q ) ] ≅ K × L \mathbb{Q}[T(\mathbb{Q})] \cong K \times L Q [ T ( Q )] ≅ K × L K ↪ K × L K \hookrightarrow K \times L K ↪ K × L ϵ \epsilon ϵ ι \iota ι ι ˉ \bar{\iota} ι ˉ { + + + , − − − } \{+ + +, - - -\} { + + + , − − − } K K K ι \iota ι { ϵ : ϵ 1 ϵ 2 ϵ 3 = + } \{\epsilon: \epsilon_1\epsilon_2\epsilon_3 = +\} { ϵ : ϵ 1 ϵ 2 ϵ 3 = + } { ϵ : majority = + } \{\epsilon: \text{majority} = +\} { ϵ : majority = + } H 1 , 0 H^{1,0} H 1 , 0 ( 2 , 2 ) (2,2) ( 2 , 2 ) ( 1 , 3 ) (1,3) ( 1 , 3 )
Integrality. Since p p p k k k K K K L p = L ⊗ Q p ≅ Q p 6 L_p = L \otimes\mathbb{Q}_p \cong\mathbb{Q}_{p^6} L p = L ⊗ Q p ≅ Q p 6 B ⊗ k k p = End k p ( U p ) B \otimes_k k_p = \operatorname{End}_{k_p}(U_p) B ⊗ k k p = End k p ( U p ) U p U_p U p L p L_p L p Λ 0 ′ ⊂ U p \Lambda'_0 \subset U_p Λ 0 ′ ⊂ U p O L p \mathcal{O}_{L_p} O L p Λ ′ = Cor ( Λ 0 ′ ) \Lambda' = \operatorname{Cor}(\Lambda'_0) Λ ′ = Cor ( Λ 0 ′ ) L p / k p L_p/k_p L p / k p Λ 0 ′ \Lambda'_0 Λ 0 ′ W W W O k p → W \mathcal{O}_{k_p} \to W O k p → W W × W W \times W W × W Λ ′ ⊗ W \Lambda' \otimes W Λ ′ ⊗ W ( O K × O L ) ⊗ Z p (\mathcal{O}_K \times\mathcal{O}_L) \otimes\mathbb{Z}_p ( O K × O L ) ⊗ Z p ( O K × O L ) ⊗ Z p (\mathcal{O}_K \times\mathcal{O}_L) \otimes\mathbb{Z}_p ( O K × O L ) ⊗ Z p Λ ′ \Lambda' Λ ′ Λ ⊗ Z p = p j Cor ( Λ 0 ) \Lambda\otimes\mathbb{Z}_p = p^j \operatorname{Cor}(\Lambda_0) Λ ⊗ Z p = p j Cor ( Λ 0 ) U p U_p U p g Λ 0 g\Lambda_0 g Λ 0 g ∈ GL 2 ( k p ) = GL ( U p ) g \in\operatorname{GL}_2(k_p) = \operatorname{GL}(U_p) g ∈ GL 2 ( k p ) = GL ( U p ) G ( Q p ) G(\mathbb{Q}_p) G ( Q p ) G G G Res k / Q B × \operatorname{Res}_{k/\mathbb{Q}} B^\times Res k / Q B × B 1 B^1 B 1 Res k / Q G m \operatorname{Res}_{k/\mathbb{Q}} \mathbb{G}_m Res k / Q G m V V V G m G_{\mathrm{m}} G m Λ ′ = γ p ( Λ ⊗ Z p ) \Lambda'=\gamma_p(\Lambda\otimes\mathbb{Z}_p) Λ ′ = γ p ( Λ ⊗ Z p ) γ p ∈ G ( Q p ) \gamma_p\in G(\mathbb{Q}_p) γ p ∈ G ( Q p ) p − j p^{-j} p − j γ p \gamma_p γ p G G G c = [ h , ( γ p , 1 ) ] c=[h,(\gamma_p,1)] c = [ h , ( γ p , 1 )] H 1 ( X c , Z p ) = Λ ′ H_1(X_c,\mathbb{Z}_p)=\Lambda' H 1 ( X c , Z p ) = Λ ′
Reduction. Since X c X_c X c K × L K\times L K × L c c c S \mathcal{S} S t 0 t_0 t 0 t 0 t_0 t 0 μ \mu μ M ( X t 0 ) ⊗ Q M(X_{t_0})\otimes\mathbb{Q} M ( X t 0 ) ⊗ Q F F F K × L K\times L K × L Q p \mathbb{Q}_p Q p Q ‾ p \overline{\mathbb{Q}}_p Q p σ \sigma σ ( K × L ) ⊗ Q p ≅ Q p 2 × Q p 6 (K\times L)\otimes\mathbb{Q}_p\cong\mathbb{Q}_{p^2}\times\mathbb{Q}_{p^6} ( K × L ) ⊗ Q p ≅ Q p 2 × Q p 6 Q p \mathbb{Q}_p Q p Proposition 3.3 , t 0 t_0 t 0
Diagonal action. The elements of T ( Q ) T(\mathbb{Q}) T ( Q ) M ( X t 0 ) ⊗ Q M(X_{t_0})\otimes\mathbb{Q} M ( X t 0 ) ⊗ Q G F = { γ ∈ G ( Q ‾ p ) : γ F = F γ } G_F=\{\gamma\in G(\overline{\mathbb{Q}}_p):\gamma F=F\gamma\} G F = { γ ∈ G ( Q p ) : γ F = F γ } γ = γ 1 ⊗ γ 2 ⊗ γ 3 \gamma=\gamma_1\otimes\gamma_2\otimes\gamma_3 γ = γ 1 ⊗ γ 2 ⊗ γ 3 γ F γ − 1 \gamma F\gamma^{-1} γ F γ − 1 F F F γ = σ 2 ( γ ′ ) ⊗ γ ′ ⊗ σ ( γ ′ ) \gamma=\sigma^2(\gamma')\otimes\gamma'\otimes\sigma(\gamma') γ = σ 2 ( γ ′ ) ⊗ γ ′ ⊗ σ ( γ ′ ) γ ′ \gamma' γ ′ Δ \Delta Δ A σ 3 A\sigma^3 A σ 3 Mat 2 ( Q ‾ p ) \operatorname{Mat}_2(\overline{\mathbb{Q}}_p) Mat 2 ( Q p ) γ ′ A = A σ 3 ( γ ′ ) \gamma'A=A\sigma^3(\gamma') γ ′ A = A σ 3 ( γ ′ ) Δ \Delta Δ
( y p y ′ σ 3 ( y ′ ) σ 3 ( y ) ) \begin{pmatrix}
y & py' \\
\sigma^3(y') & \sigma^3(y)
\end{pmatrix} ( y σ 3 ( y ′ ) p y ′ σ 3 ( y ) ) with y , y ′ ∈ Q p 6 y,y'\in\mathbb{Q}_{p^6} y , y ′ ∈ Q p 6 Q p 3 \mathbb{Q}_{p^3} Q p 3 [27 Corollary 13.3.14] ). Let Δ T ⊆ Δ \Delta_T\subseteq\Delta Δ T ⊆ Δ Q p 3 \mathbb{Q}_{p^3} Q p 3 γ ′ \gamma' γ ′ γ ∈ T ( Q ) \gamma\in T(\mathbb{Q}) γ ∈ T ( Q ) T ( Q ) T(\mathbb{Q}) T ( Q ) γ \gamma γ γ ~ \widetilde{\gamma} γ γ ′ \gamma' γ ′ γ ~ ′ \widetilde{\gamma}' γ ′ γ ′ γ ~ ′ \gamma'\widetilde{\gamma}' γ ′ γ ′ γ ~ ′ γ ′ \widetilde{\gamma}'\gamma' γ ′ γ ′ ± 1 \pm1 ± 1 σ 2 \sigma^2 σ 2 σ \sigma σ G L 2 3 → G L ( M ⊗ Q ‾ p ) \mathrm{GL}_2^3\to\mathrm{GL}(M\otimes\overline{\mathbb{Q}}_p) GL 2 3 → GL ( M ⊗ Q p ) Lemma 3.1 ), so its cube is 1 and it is 1. As Δ \Delta Δ Δ T \Delta_T Δ T Q p 3 \mathbb{Q}_{p^3} Q p 3 dim Q p 3 Δ = 4 \dim_{\mathbb{Q}_{p^3}}\Delta=4 dim Q p 3 Δ = 4 Q ‾ p \overline{\mathbb{Q}}_p Q p T ( Q ) T(\mathbb{Q}) T ( Q ) M ⊗ Q ‾ p M\otimes\overline{\mathbb{Q}}_p M ⊗ Q p ( K × L ) ⊗ Q ‾ p ≅ Q ‾ p 8 (K\times L)\otimes\overline{\mathbb{Q}}_p\cong\overline{\mathbb{Q}}_p^8 ( K × L ) ⊗ Q p ≅ Q p 8 K × L K\times L K × L X t 0 X_{t_0} X t 0 § 2.2 . It lies in the tensor product of the Q ‾ p \overline{\mathbb{Q}}_p Q p σ 2 ( Δ T ) \sigma^2(\Delta_T) σ 2 ( Δ T ) Δ T \Delta_T Δ T σ ( Δ T ) \sigma(\Delta_T) σ ( Δ T ) Mat 2 ( Q ‾ p ) \operatorname{Mat}_2(\overline{\mathbb{Q}}_p) Mat 2 ( Q p ) Q ‾ p \overline{\mathbb{Q}}_p Q p Δ T \Delta_T Δ T Q ‾ p 8 \overline{\mathbb{Q}}_p^8 Q p 8 Q ‾ p × Q ‾ p \overline{\mathbb{Q}}_p\times\overline{\mathbb{Q}}_p Q p × Q p Δ T \Delta_T Δ T Q p 3 \mathbb{Q}_{p^3} Q p 3 Δ T ⊗ Q p 3 Q ‾ p \Delta_T\otimes_{\mathbb{Q}_{p^3}}\overline{\mathbb{Q}}_p Δ T ⊗ Q p 3 Q p Δ T \Delta_T Δ T Δ \Delta Δ Q p 6 \mathbb{Q}_{p^6} Q p 6 { diag ( y , σ 3 y ) : y ∈ Q p 6 } \{\operatorname{diag}(y,\sigma^3y):y\in\mathbb{Q}_{p^6}\} { diag ( y , σ 3 y ) : y ∈ Q p 6 } A σ 3 A\sigma^3 A σ 3 Δ × = A × Δ × \Delta^\times=A^\times\Delta^\times Δ × = A × Δ × A A A γ 0 ′ ∈ Δ × \gamma'_0\in\Delta^\times γ 0 ′ ∈ Δ × σ 2 ( γ 0 ′ ) ⊗ γ 0 ′ ⊗ σ ( γ 0 ′ ) \sigma^2(\gamma'_0)\otimes\gamma'_0\otimes\sigma(\gamma'_0) σ 2 ( γ 0 ′ ) ⊗ γ 0 ′ ⊗ σ ( γ 0 ′ ) T ( Q ) T(\mathbb{Q}) T ( Q ) T ( Q ) T(\mathbb{Q}) T ( Q ) Q \mathbb{Q} Q K × L K\times L K × L ( K × L ) ⊗ Q p (K\times L)\otimes\mathbb{Q}_p ( K × L ) ⊗ Q p ( O K × O L ) ⊗ Z p (O_K\times O_L)\otimes\mathbb{Z}_p ( O K × O L ) ⊗ Z p e s e_s e s Lemma 4.4 (ii) applied to the K K K
Lemma 6.2. In Proposition 6.1 , H 1 ( X c , Z ( p ) ) H_1(X_c,\mathbb{Z}_{(p)}) H 1 ( X c , Z ( p ) ) O K , ( p ) O_{K,(p)} O K , ( p ) X c X_c X c O K , ( p ) O_{K,(p)} O K , ( p )
Proof. By Proposition 6.1 (a), End ( X c ) ∩ K \operatorname{End}(X_c)\cap K End ( X c ) ∩ K p p p O K , ( p ) O_{K,(p)} O K , ( p ) ψ c \psi_c ψ c G ( W ) G(W) G ( W ) Λ ⊗ W \Lambda\otimes W Λ ⊗ W Λ ⊗ W \Lambda\otimes W Λ ⊗ W p p p Lemma 3.1 ), the ψ c \psi_c ψ c Λ ⊗ Z p \Lambda\otimes\mathbb{Z}_p Λ ⊗ Z p p j ′ Λ ⊗ Z p p^{j'}\Lambda\otimes\mathbb{Z}_p p j ′ Λ ⊗ Z p γ p \gamma_p γ p ψ c \psi_c ψ c Λ ′ = H 1 ( X c , Z p ) \Lambda' = H_1(X_c,\mathbb{Z}_p) Λ ′ = H 1 ( X c , Z p ) p j ′ ′ Λ ′ p^{j''}\Lambda' p j ′′ Λ ′ j ′ ′ ≤ 0 j''\leq0 j ′′ ≤ 0 ψ c \psi_c ψ c ψ c ( Λ ′ , Λ ′ ) ⊆ p j ′ ′ Z p \psi_c(\Lambda',\Lambda')\subseteq p^{j''}\mathbb{Z}_p ψ c ( Λ ′ , Λ ′ ) ⊆ p j ′′ Z p p j ′ ′ ψ c = p − j ′ ′ ψ c p^{j''}\psi_c=p^{-j''}\psi_c p j ′′ ψ c = p − j ′′ ψ c H 1 ( X c , Z ) H_1(X_c,\mathbb{Z}) H 1 ( X c , Z ) p p p ψ c \psi_c ψ c ℓ ≠ p \ell\neq p ℓ = p Λ ′ \Lambda' Λ ′ ψ c \psi_c ψ c K × L = Q [ T ( Q ) ] K\times L=\mathbb{Q}[T(\mathbb{Q})] K × L = Q [ T ( Q )] γ ∈ T ( Q ) \gamma\in T(\mathbb{Q}) γ ∈ T ( Q ) γ − 1 \gamma^{-1} γ − 1 K K K ψ c \psi_c ψ c Tr K / Q \operatorname{Tr}_{K/\mathbb{Q}} Tr K / Q p p p K K K O K , p O_{K,p} O K , p ψ c \psi_c ψ c Λ ′ \Lambda' Λ ′ p p p
A p p p
Markman’s classes
Let d ≥ 3 d \ge3 d ≥ 3 K d = Q ( − d ) K_d = \mathbb{Q}(\sqrt{-d}) K d = Q ( − d ) C C C X 3 = Pic 2 ( C ) X_3 = \operatorname{Pic}^{2}(C) X 3 = Pic 2 ( C ) Θ \Theta Θ X 3 ∨ X_3^\vee X 3 ∨ X 6 = X 3 × X 3 ∨ X_6 = X_3 \times X_3^\vee X 6 = X 3 × X 3 ∨ I 1 \mathcal{I}_1 I 1 I 2 \mathcal{I}_2 I 2 d + 1 d+1 d + 1 AJ ( C ) \operatorname{AJ}(C) AJ ( C ) − AJ ( C ) -\operatorname{AJ}(C) − AJ ( C ) Θ \Theta Θ Φ ( I 2 ⊠ I 1 ) \Phi(\mathcal{I}_2 \boxtimes\mathcal{I}_1) Φ ( I 2 ⊠ I 1 ) Φ : D b ( X 3 × X 3 ) → D b ( X 6 ) \Phi: D^b(X_3 \times X_3) \to D^b(X_6) Φ : D b ( X 3 × X 3 ) → D b ( X 6 ) [2 Section 1.3] , is up to shift a simple reflexive sheaf E \mathcal{E} E 8 d 8d 8 d X 6 X_6 X 6 [2 Theorem 1.4.1(2) and Proposition 9.2.2] . Write κ ( E ) = exp ( − c 1 ( E ) / 8 d ) ch ( E ) \kappa(\mathcal{E}) = \exp(-c_1(\mathcal{E})/8d)\operatorname{ch}(\mathcal{E}) κ ( E ) = exp ( − c 1 ( E ) /8 d ) ch ( E ) κ 3 ( E ) ∈ H 6 ( X 6 , Q ) \kappa_3(\mathcal{E}) \in H^6(X_6,\mathbb{Q}) κ 3 ( E ) ∈ H 6 ( X 6 , Q )
Let Γ Z = H 1 ( X 6 , Z ) = H 1 ( X 3 , Z ) ⊕ H 1 ( X 3 ∨ , Z ) \Gamma_{\mathbb{Z}} = H^1(X_6,\mathbb{Z}) = H^1(X_3,\mathbb{Z}) \oplus H^1(X_3^\vee,\mathbb{Z}) Γ Z = H 1 ( X 6 , Z ) = H 1 ( X 3 , Z ) ⊕ H 1 ( X 3 ∨ , Z ) ( ⋅ , ⋅ ) (\cdot,\cdot) ( ⋅ , ⋅ ) Γ Z \Gamma_{\mathbb{Z}} Γ Z [2 (1.2.2)] , and let Θ ♯ : H 1 ( X 3 ∨ , Z ) → H 1 ( X 3 , Z ) \Theta^\sharp: H^1(X_3^\vee,\mathbb{Z}) \to H^1(X_3,\mathbb{Z}) Θ ♯ : H 1 ( X 3 ∨ , Z ) → H 1 ( X 3 , Z ) Θ \Theta Θ [2 (2.4.3)] , an isomorphism because Θ \Theta Θ η \eta η Γ Q \Gamma_{\mathbb{Q}} Γ Q η 2 = − d \eta^2=-d η 2 = − d f = η / − d f=\eta/\sqrt{-d} f = η / − d [2 (2.4.1)] ), which, with the polarization ψ 6 ( x , y ) = ( η x , y ) \psi_6(x,y)=(\eta x,y) ψ 6 ( x , y ) = ( η x , y ) [2 (2.4.2)] , makes X 6 X_6 X 6 K d K_d K d [2 Section 1.2 and Proposition 2.4.4] whose Hermitian form H \mathcal{H} H ( 3 , 3 ) (3,3) ( 3 , 3 ) − 1 -1 − 1 [2 Lemmas 3.1.2 and 3.1.3] . On Γ K = Γ Z ⊗ K d \Gamma_K=\Gamma_{\mathbb{Z}}\otimes K_d Γ K = Γ Z ⊗ K d − d \sqrt{-d} − d Γ 1 , K = { ( − − d Θ ♯ y , y ) } \Gamma_{1,K}=\{(-\sqrt{-d}\Theta^\sharp y,y)\} Γ 1 , K = {( − − d Θ ♯ y , y )} − − d -\sqrt{-d} − − d Γ 2 , K = { ( − d Θ ♯ y , y ) } \Gamma_{2,K}=\{(\sqrt{-d}\Theta^\sharp y,y)\} Γ 2 , K = {( − d Θ ♯ y , y )} y ∈ H 1 ( X 3 ∨ , K d ) y\in H^1(X_3^\vee,K_d) y ∈ H 1 ( X 3 ∨ , K d ) [2 Section 1.2 and (2.4.6)] . Decomposing ( w , 0 ) (w,0) ( w , 0 ) ( 0 , y ) (0,y) ( 0 , y ) η ( w , 0 ) = ( 0 , − ( Θ ♯ ) − 1 w ) \eta(w,0)=(0,-(\Theta^\sharp)^{-1}w) η ( w , 0 ) = ( 0 , − ( Θ ♯ ) − 1 w ) η ( 0 , y ) = ( d Θ ♯ y , 0 ) \eta(0,y)=(d\Theta^\sharp y,0) η ( 0 , y ) = ( d Θ ♯ y , 0 ) η \eta η Γ Z \Gamma_{\mathbb{Z}} Γ Z ψ 6 \psi_6 ψ 6 d 6 d^6 d 6 Γ Z \Gamma_{\mathbb{Z}} Γ Z ⋀ i + j Γ K \bigwedge^{i+j}\Gamma_K ⋀ i + j Γ K K d K_d K d ( i , j ) (i,j) ( i , j ) ⋀ i Γ 1 , K ∧ ⋀ j Γ 2 , K \bigwedge^i\Gamma_{1,K}\wedge\bigwedge^j\Gamma_{2,K} ⋀ i Γ 1 , K ∧ ⋀ j Γ 2 , K Γ R \Gamma_{\mathbb{R}} Γ R η \eta η ψ 6 \psi_6 ψ 6 Γ 1 , K ⊗ K d C \Gamma_{1,K}\otimes_{K_d}\mathbb{C} Γ 1 , K ⊗ K d C H \mathcal{H} H D \mathcal{D} D SU H \operatorname{SU}_{\mathcal{H}} SU H ( Γ Q , H ) (\Gamma_{\mathbb{Q}},\mathcal{H}) ( Γ Q , H ) SU H ( R ) ≅ SU ( 3 , 3 ) \operatorname{SU}_{\mathcal{H}}(\mathbb{R})\cong\operatorname{SU}(3,3) SU H ( R ) ≅ SU ( 3 , 3 ) X 6 , t X_{6,t} X 6 , t t ∈ D t\in\mathcal{D} t ∈ D H 1 ( X 6 , t , Z ) = Γ Z H^1(X_{6,t},\mathbb{Z})=\Gamma_{\mathbb{Z}} H 1 ( X 6 , t , Z ) = Γ Z ( Γ Z , η , ψ 6 ) (\Gamma_{\mathbb{Z}},\eta,\psi_6) ( Γ Z , η , ψ 6 ) ⋀ ∙ Γ Z \bigwedge^\bullet\Gamma_{\mathbb{Z}} ⋀ ∙ Γ Z
Let Γ 1 = Γ 1 , K ∩ ( Γ Z ⊗ O K d ) \Gamma_1=\Gamma_{1,K}\cap(\Gamma_{\mathbb{Z}}\otimes O_{K_d}) Γ 1 = Γ 1 , K ∩ ( Γ Z ⊗ O K d ) Γ 2 = Γ ‾ 1 \Gamma_2=\overline{\Gamma}_1 Γ 2 = Γ 1 Z \mathbb{Z} Z ( y i ) (y_i) ( y i ) H 1 ( X 3 ∨ , Z ) H^1(X_3^\vee,\mathbb{Z}) H 1 ( X 3 ∨ , Z ) ( − − d Θ ♯ y i , y i ) (-\sqrt{-d}\Theta^\sharp y_i,y_i) ( − − d Θ ♯ y i , y i ) O K d O_{K_d} O K d Γ 1 \Gamma_1 Γ 1 Γ 1 , K \Gamma_{1,K} Γ 1 , K ω X 6 ∈ H 6 ( X 6 , O K d ) \omega_{X_6}\in H^6(X_6,O_{K_d}) ω X 6 ∈ H 6 ( X 6 , O K d ) ω ‾ X 6 \overline{\omega}_{X_6} ω X 6 p ∤ 2 d p\nmid2d p ∤ 2 d Γ Z ⊗ O K d , ( p ) = ( Γ 1 ⊕ Γ 2 ) ⊗ O K d O K d , ( p ) \Gamma_{\mathbb{Z}}\otimes O_{K_d,(p)}=(\Gamma_1\oplus\Gamma_2)\otimes_{O_{K_d}}O_{K_d,(p)} Γ Z ⊗ O K d , ( p ) = ( Γ 1 ⊕ Γ 2 ) ⊗ O K d O K d , ( p ) p p p { λ ω X 6 + λ ‾ ω ‾ X 6 : λ ∈ O K d , ( p ) } \{\lambda\omega_{X_6}+\overline{\lambda}\,\overline{\omega}_{X_6}:\lambda\in O_{K_d,(p)}\} { λ ω X 6 + λ ω X 6 : λ ∈ O K d , ( p ) }
Hypothesis 7.1 (M). For every even d > 3 d>3 d > 3 [2 Lemmas 9.3.6 and 9.3.11] and [2 Conjecture 7.3.9] , as proved for families of abelian varieties in [2 Section 7.4.2] , hold, and so do the other statements of [2 ] cited in this section.
In these statements the translates in I 1 \mathcal{I}_1 I 1 I 2 \mathcal{I}_2 I 2 X 3 X_3 X 3 d + 1 d+1 d + 1 C C C [2 Section 1.5 and Lemma 9.3.1] ; this changes neither I i \mathcal{I}_i I i κ ( E ) \kappa(\mathcal{E}) κ ( E ) Y 6 : X 6 → Y 6 \mathcal{Y}_6:X_6\to Y_6 Y 6 : X 6 → Y 6 ( d + 1 ) 2 (d+1)^2 ( d + 1 ) 2 [2 Section 1.5 and Lemma 9.3.3] . Lemma 9.3.6 of [2 ] states that Y 6 Y_6 Y 6 B \mathcal{B} B 8 d 8d 8 d μ 8 d \mu_{8d} μ 8 d B \mathcal{B} B 8 d 8d 8 d Y \mathcal{Y} Y Y 0 \mathcal{Y}_0 Y 0 ( Σ , 0 ) (\Sigma,0) ( Σ , 0 ) B \mathcal{B} B 8 d 8d 8 d Y 0 \mathcal{Y}_0 Y 0 μ 8 d \mu_{8d} μ 8 d B \mathcal{B} B Y \mathcal{Y} Y U U U 0 0 0 B \mathcal{B} B [2 Section 7.4.2] is flat over U U U μ 8 d \mu_{8d} μ 8 d B \mathcal{B} B B \mathcal{B} B μ 8 d \mu_{8d} μ 8 d [2 Remark 7.3.2] . This change does not affect B ⊗ 8 d \mathcal{B}^{\otimes8d} B ⊗ 8 d [2 Theorem 1.5.1] ; Hypothesis 7.1 also covers the other statements of [2 ] cited in this section, and part (b) of Theorem 1.3 is conditional on all of them.
The proof of Proposition 7.3 uses Chern classes of complexes of analytic sheaves, defined through the real-analytic resolutions of Atiyah and Hirzebruch. Let R \mathcal{R} R C ω \mathcal{C}^{\omega} C ω O R \mathcal{O}_{\mathcal{R}} O R J \mathcal{J} J J ⊗ C ω \mathcal{J} \otimes\mathcal{C}^{\omega} J ⊗ C ω C ω \mathcal{C}^{\omega} C ω [31 Proposition 2.9] , and every compact subset of R \mathcal{R} R J ⊗ C ω \mathcal{J} \otimes\mathcal{C}^{\omega} J ⊗ C ω C ω \mathcal{C}^{\omega} C ω [31 Proposition 2.6 and Section 4] . We write [ J ] t o p [\mathcal{J}]_{\mathrm{top}} [ J ] top K K K F \mathcal{F} F O R \mathcal{O}_{\mathcal{R}} O R [ F ] t o p = ∑ j ( − 1 ) j [ H j ( F ) ] t o p [\mathcal{F}]_{\mathrm{top}}=\sum_j(-1)^j[H^j(\mathcal{F})]_{\mathrm{top}} [ F ] top = ∑ j ( − 1 ) j [ H j ( F ) ] top R \mathcal{R} R K K K [31 ] are the usual ones [31 Section 1 and remark after Proposition 6.2] , [ J ] t o p [\mathcal{J}]_{\mathrm{top}} [ J ] top [31 Section 4] , and we write c ( F ) c(\mathcal{F}) c ( F ) ch ( F ) \operatorname{ch}(\mathcal{F}) ch ( F ) [ F ] t o p [\mathcal{F}]_{\mathrm{top}} [ F ] top Y Y Y cl : CH ∙ ( Y ) → H 2 ∙ ( Y , Z ) \operatorname{cl}\colon\operatorname{CH}^{\bullet}(Y) \to H^{2\bullet}(Y,\mathbb{Z}) cl : CH ∙ ( Y ) → H 2 ∙ ( Y , Z ) [32 4.15] . For algebraic coherent sheaves and complexes on a nonsingular projective Y Y Y c c c ch \operatorname{ch} ch
Lemma 7.2 (Chern classes in families). (a) Let R \mathcal{R} R [ ⋅ ] t o p [\cdot]_{\mathrm{top}} [ ⋅ ] top O R \mathcal{O}_{\mathcal{R}} O R
(b) Let R → Σ \mathcal{R} \to\Sigma R → Σ R t \mathcal{R}_t R t t ∈ Σ t \in\Sigma t ∈ Σ L i t ∗ L i_t^* L i t ∗ R t \mathcal{R}_t R t F \mathcal{F} F O R \mathcal{O}_{\mathcal{R}} O R t ↦ c ( L i t ∗ F ) t \mapsto c(L i_t^*\mathcal{F}) t ↦ c ( L i t ∗ F ) t ↦ ch ( L i t ∗ F ) t \mapsto\operatorname{ch}(L i_t^*\mathcal{F}) t ↦ ch ( L i t ∗ F ) Σ \Sigma Σ t ↦ ⨁ i H 2 i ( R t , Z ) t \mapsto\bigoplus_i H^{2i}(\mathcal{R}_t,\mathbb{Z}) t ↦ ⨁ i H 2 i ( R t , Z ) t ↦ ⨁ i H 2 i ( R t , Q ) t \mapsto\bigoplus_i H^{2i}(\mathcal{R}_t,\mathbb{Q}) t ↦ ⨁ i H 2 i ( R t , Q )
(c) Let Y Y Y F \mathcal{F} F O Y a n \mathcal{O}_{Y^{\mathrm{an}}} O Y an c ( F ) c(\mathcal{F}) c ( F ) cl \operatorname{cl} cl CH ∙ ( Y ) \operatorname{CH}^{\bullet}(Y) CH ∙ ( Y ) F \mathcal{F} F
(d) Let Y Y Y Z ⊆ Y Z \subseteq Y Z ⊆ Y dim Y − i \dim Y-i dim Y − i ch i ( O Z ) = cl ( [ Z ] dim Y − i ) \operatorname{ch}_i(\mathcal{O}_Z)=\operatorname{cl}([Z]_{\dim Y-i}) ch i ( O Z ) = cl ([ Z ] d i m Y − i ) [ Z ] dim Y − i [Z]_{\dim Y-i} [ Z ] d i m Y − i ( dim Y − i ) (\dim Y-i) ( dim Y − i ) Z Z Z [23 Tag 02QU] .
Proof. (a) At a point of R \mathcal{R} R C ω \mathcal{C}^{\omega} C ω O R \mathcal{O}_{\mathcal{R}} O R [23 Tags 00MB and 00MC] ). Locally free C ω \mathcal{C}^{\omega} C ω [31 Lemma 2.7] , so by the horseshoe lemma [ ⋅ ] t o p [\cdot]_{\mathrm{top}} [ ⋅ ] top C ω \mathcal{C}^{\omega} C ω C ω \mathcal{C}^{\omega} C ω [ ⋅ ] t o p [\cdot]_{\mathrm{top}} [ ⋅ ] top C ω \mathcal{C}^{\omega} C ω [ ⋅ ] t o p [\cdot]_{\mathrm{top}} [ ⋅ ] top O R \mathcal{O}_{\mathcal{R}} O R
(b) As R → Σ \mathcal{R} \to\Sigma R → Σ Σ \Sigma Σ Σ ′ \Sigma' Σ ′ H j ( F ) ⊗ C ω H^j(\mathcal{F}) \otimes\mathcal{C}^{\omega} H j ( F ) ⊗ C ω Σ ′ \Sigma' Σ ′ Σ ′ \Sigma' Σ ′ ∑ j ( − 1 ) j [ H j ( F ) ] t o p \sum_j(-1)^j[H^j(\mathcal{F})]_{\mathrm{top}} ∑ j ( − 1 ) j [ H j ( F ) ] top Σ ′ \Sigma' Σ ′ R t \mathcal{R}_t R t t ∈ Σ ′ t \in\Sigma' t ∈ Σ ′ R \mathcal{R} R Σ \Sigma Σ t t t R t \mathcal{R}_t R t C ω \mathcal{C}^{\omega} C ω C ω \mathcal{C}^{\omega} C ω O R \mathcal{O}_{\mathcal{R}} O R R t \mathcal{R}_t R t J ⊗ C ω \mathcal{J} \otimes\mathcal{C}^{\omega} J ⊗ C ω O R \mathcal{O}_{\mathcal{R}} O R J \mathcal{J} J R t \mathcal{R}_t R t Tor i O R ( J , O R t ) ⊗ C ω \operatorname{Tor}^{\mathcal{O}_{\mathcal{R}}}_i(\mathcal{J},\mathcal{O}_{\mathcal{R}_t}) \otimes\mathcal{C}^{\omega} Tor i O R ( J , O R t ) ⊗ C ω [ J ] t o p [\mathcal{J}]_{\mathrm{top}} [ J ] top R t \mathcal{R}_t R t [ L i t ∗ J ] t o p [L i_t^*\mathcal{J}]_{\mathrm{top}} [ L i t ∗ J ] top L i t ∗ L i_t^* L i t ∗ τ ≤ j − 1 F → τ ≤ j F → H j ( F ) [ − j ] \tau_{\leq j-1}\mathcal{F} \to\tau_{\leq j}\mathcal{F} \to H^j(\mathcal{F})[-j] τ ≤ j − 1 F → τ ≤ j F → H j ( F ) [ − j ] ∑ j ( − 1 ) j [ H j ( F ) ] t o p \sum_j(-1)^j[H^j(\mathcal{F})]_{\mathrm{top}} ∑ j ( − 1 ) j [ H j ( F ) ] top [ L i t ∗ F ] t o p [L i_t^*\mathcal{F}]_{\mathrm{top}} [ L i t ∗ F ] top
(c) For an algebraic vector bundle on Y Y Y cl \operatorname{cl} cl CH ∙ ( Y ) \operatorname{CH}^{\bullet}(Y) CH ∙ ( Y ) [33 Théorème 1] for the latter. For a line bundle with a rational section, the former is 1 1 1 cl \operatorname{cl} cl [32 4.13] and linearity, and the latter is 1 1 1 [33 Théorème 1] , applied to example 3° of [33 Section 2] on nonsingular quasi-projective varieties. The cohomology sheaves of F \mathcal{F} F [34 Exposé XII, Théorème 4.4] , each of which is quasi-isomorphic to a bounded complex of algebraic vector bundles (see e.g. [23 Tags 0FDC, 0F87 and 0F8E] ), and analytification is exact [34 Exposé XII, Proposition 1.3.1] . So by (a), [ F ] t o p [\mathcal{F}]_{\mathrm{top}} [ F ] top c ( F ) c(\mathcal{F}) c ( F ) cl \operatorname{cl} cl C H ∙ ( Y ) \mathrm{CH}^{\bullet}(Y) CH ∙ ( Y )
(d) For a closed subvariety Y ′ ⊆ Y Y' \subseteq Y Y ′ ⊆ Y i ′ i' i ′ Y ′ a n Y'^{\mathrm{an}} Y ′ an i ′ i' i ′ [34 Exposé XII, Propositions 2.1 and 2.4] , and ch ( O Y ′ ) \operatorname{ch}(\mathcal{O}_{Y'}) ch ( O Y ′ ) Y ′ a n Y'^{\mathrm{an}} Y ′ an [31 Proposition 5.4] plus terms of degree greater than 2 i ′ 2i' 2 i ′ [31 Propositions 3.4 and 6.2 and Lemma 5.6] . That class is cl ( Y ′ ) \operatorname{cl}(Y') cl ( Y ′ ) Y ′ ∖ Y s i n g ′ Y' \setminus Y'_{\mathrm{sing}} Y ′ ∖ Y sing ′ Y s i n g ′ Y'_{\mathrm{sing}} Y sing ′ Y ′ Y' Y ′ [32 proof of 4.5] , and this restriction is injective in degree 2 i ′ 2i' 2 i ′ [31 Lemma 5.3] . Now let [ Z ] dim Y − i ′ = ∑ Y ′ m Y ′ Y ′ [Z]_{\dim Y-i'} = \sum_{Y'}m_{Y'}Y' [ Z ] d i m Y − i ′ = ∑ Y ′ m Y ′ Y ′ [23 Tags 02MX and 02S9] and induction on the dimension, [ O Z ] − ∑ Y ′ m Y ′ [ O Y ′ ] [\mathcal{O}_Z]-\sum_{Y'}m_{Y'}[\mathcal{O}_{Y'}] [ O Z ] − ∑ Y ′ m Y ′ [ O Y ′ ] Y Y Y [ O Y ′ ′ ] [\mathcal{O}_{Y''}] [ O Y ′′ ] Y ′ ′ Y'' Y ′′ dim Y − i ′ \dim Y-i' dim Y − i ′ 2 i ′ 2i' 2 i ′ [ ⋅ ] t o p [\cdot]_{\mathrm{top}} [ ⋅ ] top Y Y Y ch i ′ ( O Z ) = cl ( [ Z ] dim Y − i ′ ) \operatorname{ch}_{i'}(\mathcal{O}_Z)=\operatorname{cl}([Z]_{\dim Y-i'}) ch i ′ ( O Z ) = cl ([ Z ] d i m Y − i ′ )
Proposition 7.3. Assume Hypothesis 7.1 and let d ≥ 3 d \ge3 d ≥ 3 On every member of the family of ( Γ Z , η , ψ 6 ) (\Gamma_Z,\eta,\psi_6) ( Γ Z , η , ψ 6 ) 2 ( 8 d ) 8 d κ 3 ( E ) ∈ ⋀ 6 Γ Q 2(8d)^{8d}\kappa_3(\mathcal{E}) \in\bigwedge^6\Gamma_{\mathbb{Q}} 2 ( 8 d ) 8 d κ 3 ( E ) ∈ ⋀ 6 Γ Q
Proof. Markman deduces the algebraicity with rational coefficients [2 proof of Theorem 1.5.1] ; the integral argument below is ours.
Near X 6 X_6 X 6 Markman states that ch ( B ) = κ ( B ) \operatorname{ch}(\mathcal{B})=\kappa(\mathcal{B}) ch ( B ) = κ ( B ) det B \det\mathcal{B} det B [2 Section 7.3, after Lemma 7.3.7] and that ϖ ∗ κ ( B ) = κ ( E ) \varpi^*\kappa(\mathcal{B})=\kappa(\mathcal{E}) ϖ ∗ κ ( B ) = κ ( E ) [2 Section 9.3, after Remark 9.3.7] ; his Chern character is defined through the Atiyah class, and for coherent sheaves such as E \mathcal{E} E [35 Section 1] ). He also states that the derived tensor power B ⊗ 8 d \mathcal{B}^{\otimes8d} B ⊗ 8 d ch ( B ⊗ 8 d ) = ch ( B ) 8 d \operatorname{ch}(\mathcal{B}^{\otimes8d})=\operatorname{ch}(\mathcal{B})^{8d} ch ( B ⊗ 8 d ) = ch ( B ) 8 d [2 (7.3.4) and Section 7.3, after Definition 7.3.6] , and we use, as he does there, that this is the rational Chern character of B ⊗ 8 d \mathcal{B}^{\otimes8d} B ⊗ 8 d κ 0 ( B ) = 8 d \kappa_0(\mathcal{B})=8d κ 0 ( B ) = 8 d κ 1 ( B ) = 0 \kappa_1(\mathcal{B})=0 κ 1 ( B ) = 0 c 1 ( B ⊗ 8 d ) = ch 1 ( B ⊗ 8 d ) = 0 c_1(\mathcal{B}^{\otimes8d})=\operatorname{ch}_1(\mathcal{B}^{\otimes8d})=0 c 1 ( B ⊗ 8 d ) = ch 1 ( B ⊗ 8 d ) = 0 ch 3 ( B ⊗ 8 d ) = ( 8 d ) 8 d κ 3 ( B ) \operatorname{ch}_3(\mathcal{B}^{\otimes8d})=(8d)^{8d}\kappa_3(\mathcal{B}) ch 3 ( B ⊗ 8 d ) = ( 8 d ) 8 d κ 3 ( B ) ch 3 = ( c 1 3 − 3 c 1 c 2 + 3 c 3 ) / 6 \operatorname{ch}_3=(c_1^3-3c_1c_2+3c_3)/6 ch 3 = ( c 1 3 − 3 c 1 c 2 + 3 c 3 ) /6 c 3 ( B ⊗ 8 d ) = 2 ( 8 d ) 8 d κ 3 ( B ) c_3(\mathcal{B}^{\otimes8d})=2(8d)^{8d}\kappa_3(\mathcal{B}) c 3 ( B ⊗ 8 d ) = 2 ( 8 d ) 8 d κ 3 ( B ) H 6 ( Y 6 , Z ) H^6(Y_6,\mathbb{Z}) H 6 ( Y 6 , Z ) ϖ ∗ c 3 ( B ⊗ 8 d ) = 2 ( 8 d ) 8 d κ 3 ( E ) \varpi^*c_3(\mathcal{B}^{\otimes8d})=2(8d)^{8d}\kappa_3(\mathcal{E}) ϖ ∗ c 3 ( B ⊗ 8 d ) = 2 ( 8 d ) 8 d κ 3 ( E )
Deformation. The group of translations defining ϖ \varpi ϖ Γ Z \Gamma_{\mathbb{Z}} Γ Z X 6 , t X_{6,t} X 6 , t t ∈ D t\in\mathcal{D} t ∈ D ϖ t : X 6 , t → Y 6 , t \varpi_t:X_{6,t}\to Y_{6,t} ϖ t : X 6 , t → Y 6 , t κ ( E ) \kappa(\mathcal{E}) κ ( E ) X 6 , t X_{6,t} X 6 , t [2 Theorem 1.4.1(3) and Corollary 1.3.2] , and ϖ t ∗ \varpi_t^* ϖ t ∗ ch ( B ) = κ ( B ) \operatorname{ch}(\mathcal{B})=\kappa(\mathcal{B}) ch ( B ) = κ ( B ) Y 6 , t Y_{6,t} Y 6 , t B \mathcal{B} B B U \mathcal{B}_{\mathcal{U}} B U Y 6 , t Y_{6,t} Y 6 , t U ⊆ D \mathcal{U}\subseteq\mathcal{D} U ⊆ D X 6 X_6 X 6 U \mathcal{U} U μ 8 d \mu_{8d} μ 8 d F = B U ⊗ 8 d \mathcal{F}=\mathcal{B}_{\mathcal{U}}^{\otimes8d} F = B U ⊗ 8 d Y 6 , t Y_{6,t} Y 6 , t B t ⊗ 8 d \mathcal{B}_t^{\otimes8d} B t ⊗ 8 d B t \mathcal{B}_t B t B U \mathcal{B}_{\mathcal{U}} B U B t ⊗ 8 d \mathcal{B}_t^{\otimes8d} B t ⊗ 8 d U \mathcal{U} U B ⊗ 8 d \mathcal{B}^{\otimes8d} B ⊗ 8 d U \mathcal{U} U Y 6 , t Y_{6,t} Y 6 , t ϖ t \varpi_t ϖ t [32 4.15] shows that 2 ( 8 d ) 8 d κ 3 ( E ) 2(8d)^{8d}\kappa_3(\mathcal{E}) 2 ( 8 d ) 8 d κ 3 ( E ) X 6 , t X_{6,t} X 6 , t t ∈ U t\in\mathcal{U} t ∈ U
Spreading out. Let A 0 A_0 A 0 G L ( Γ Z ) \mathrm{GL}(\Gamma_{\mathbb{Z}}) GL ( Γ Z ) [2 (3.1.1)] , the elements of S O + ( Γ Q , ( ⋅ , ⋅ ) ) \mathrm{SO}^+(\Gamma_{\mathbb{Q}},(\cdot,\cdot)) SO + ( Γ Q , ( ⋅ , ⋅ )) η \eta η 1 1 1 Γ 1 , K \Gamma_{1,K} Γ 1 , K Γ 2 , K \Gamma_{2,K} Γ 2 , K S O + \mathrm{SO}^+ SO + Spin ( Γ Q , ( ⋅ , ⋅ ) ) \operatorname{Spin}(\Gamma_{\mathbb{Q}},(\cdot,\cdot)) Spin ( Γ Q , ( ⋅ , ⋅ )) S O ( Γ Q , ( ⋅ , ⋅ ) ) \mathrm{SO}(\Gamma_{\mathbb{Q}},(\cdot,\cdot)) SO ( Γ Q , ( ⋅ , ⋅ )) κ ( E ) \kappa(\mathcal{E}) κ ( E ) [2 Corollary 1.3.2 and Lemma 3.1.1] . We show that A 0 A_0 A 0 S U H \mathrm{SU}_{\mathcal{H}} SU H ( ⋅ , ⋅ ) = 1 2 d Tr K / Q H (\cdot,\cdot)=\frac{1}{2d}\operatorname{Tr}_{K/\mathbb{Q}}\mathcal{H} ( ⋅ , ⋅ ) = 2 d 1 Tr K / Q H [2 (3.1.2)] , an element of G L ( Γ Q ) \mathrm{GL}(\Gamma_{\mathbb{Q}}) GL ( Γ Q ) η \eta η H \mathcal{H} H ( ⋅ , ⋅ ) (\cdot,\cdot) ( ⋅ , ⋅ ) K d K_d K d Γ Q \Gamma_{\mathbb{Q}} Γ Q Γ 1 , K \Gamma_{1,K} Γ 1 , K A 0 ⊆ A 1 = S U H ( Q ) ∩ G L ( Γ Z ) A_0\subseteq A_1=\mathrm{SU}_{\mathcal{H}}(\mathbb{Q})\cap\mathrm{GL}(\Gamma_{\mathbb{Z}}) A 0 ⊆ A 1 = SU H ( Q ) ∩ GL ( Γ Z ) A 0 A_0 A 0 A 1 A_1 A 1 Spin ( Γ Q , ( ⋅ , ⋅ ) ) \operatorname{Spin}(\Gamma_{\mathbb{Q}},(\cdot,\cdot)) Spin ( Γ Q , ( ⋅ , ⋅ )) Spin → SO \operatorname{Spin} \to\operatorname{SO} Spin → SO G ~ \widetilde{G} G SU H \operatorname{SU}_{H} SU H Spin ( Γ Q , ( ⋅ , ⋅ ) ) \operatorname{Spin}(\Gamma_{\mathbb{Q}},(\cdot,\cdot)) Spin ( Γ Q , ( ⋅ , ⋅ )) G ~ → SU H \widetilde{G} \to\operatorname{SU}_{H} G → SU H G ~ ( Q ) \widetilde{G}(\mathbb{Q}) G ( Q ) A 2 A_2 A 2 SU H ( Q ) \operatorname{SU}_{H}(\mathbb{Q}) SU H ( Q ) [26 Proposition 3.2] ). Then A 2 ∩ A 1 A_2 \cap A_1 A 2 ∩ A 1 A 1 A_1 A 1 A 0 A_0 A 0 A 0 A_0 A 0 A 1 A_1 A 1
Let A ⊆ A 0 A \subseteq A_0 A ⊆ A 0 [26 Proposition 3.5] ). Then M = A \ D \mathcal{M}=A\backslash\mathcal{D} M = A \ D [26 Proposition 3.1 and Theorem 3.12] ), and the polarized variation of Hodge structure Γ Z \Gamma_{\mathbb{Z}} Γ Z M \mathcal{M} M ( − 1 , 0 ) , ( 0 , − 1 ) (-1,0),(0,-1) ( − 1 , 0 ) , ( 0 , − 1 ) X → M \mathcal{X}\to\mathcal{M} X → M X t \mathcal{X}_t X t t ∈ M ( C ) t\in\mathcal{M}(\mathbb{C}) t ∈ M ( C ) Γ Z \Gamma_{\mathbb{Z}} Γ Z [26 Theorem 14.8] , and for projectivity [36 Section 2.1] and [23 Tag 0B45] ). The class 2 ( 8 d ) κ 3 ( E ) 2(8d)^{{}} \kappa_3(\mathcal{E}) 2 ( 8 d ) κ 3 ( E ) M \mathcal{M} M t ↦ H 6 ( X t , Q ) t\mapsto H^6(\mathcal{X}_t,\mathbb{Q}) t ↦ H 6 ( X t , Q ) M a l g ⊆ M ( C ) \mathcal{M}^{\mathrm{alg}}\subseteq\mathcal{M}(\mathbb{C}) M alg ⊆ M ( C )
For finitely many irreducible components H 1 , … , H m \mathcal{H}_1,\ldots,\mathcal{H}_m H 1 , … , H m X / M \mathcal{X}/\mathcal{M} X / M 3 3 3 M \mathcal{M} M [37 Section 1 and Theorem 5.1] ), and integers n 1 , … , n m n_1,\ldots,n_m n 1 , … , n m Q \mathcal{Q} Q H 1 × M ⋯ × M H m \mathcal{H}_1\times_{\mathcal{M}}\cdots\times_{\mathcal{M}}\mathcal{H}_m H 1 × M ⋯ × M H m Z i , s \mathcal{Z}_{i,s} Z i , s s ∈ Q s\in\mathcal{Q} s ∈ Q i i i Z i \mathcal{Z}_i Z i Lemma 7.2 (d), z ( s ) = ∑ i n i ch 3 ( O Z i , s ) z(s)=\sum_i n_i\operatorname{ch}_3(\mathcal{O}_{\mathcal{Z}_{i,s}}) z ( s ) = ∑ i n i ch 3 ( O Z i , s ) ∑ i n i [ Z i , s ] 3 \sum_i n_i[\mathcal{Z}_{i,s}]_3 ∑ i n i [ Z i , s ] 3 3 3 3 X t \mathcal{X}_t X t Z i , s \mathcal{Z}_{i,s} Z i , s s , s ′ ∈ Q s,s'\in\mathcal{Q} s , s ′ ∈ Q z ( s ′ ) z(s') z ( s ′ ) z ( s ) z(s) z ( s ) Q \mathcal{Q} Q Q \mathcal{Q} Q s s s s ′ s' s ′ [1 Chapter I, Lemma 5.20] , applied to the closure of the component in a projective space). Let Σ ~ \widetilde{\Sigma} Σ [23 Tag 0B8Y] ). The pull-back of X \mathcal{X} X Σ ~ \widetilde{\Sigma} Σ Σ ~ \widetilde{\Sigma} Σ Z ~ i \widetilde{\mathcal{Z}}_i Z i Z i \mathcal{Z}_i Z i Σ ~ \widetilde{\Sigma} Σ [34 Exposé XII, Propositions 3.1 and 3.2] . So a local coordinate of Σ ~ \widetilde{\Sigma} Σ O Z ~ i \mathcal{O}_{\widetilde{\mathcal{Z}}_i} O Z i O Z ~ i \mathcal{O}_{\widetilde{\mathcal{Z}}_i} O Z i O Z i , s \mathcal{O}_{\mathcal{Z}_{i,s}} O Z i , s s s s Q \mathcal{Q} Q Lemma 7.2 (b), z z z Σ ~ → Q \widetilde{\Sigma}\to\mathcal{Q} Σ → Q Σ ~ \widetilde{\Sigma} Σ z z z Q \mathcal{Q} Q Q \mathcal{Q} Q M \mathcal{M} M M a l g \mathcal{M}^{\mathrm{alg}} M alg M \mathcal{M} M U \mathcal{U} U [38 Section 4.2] .
By Baire's theorem (see e.g. [23 Tag 0CQN] ), one of these closed subsets, M 1 \mathcal{M}_1 M 1 M 1 ∘ \mathcal{M}_1^\circ M 1 ∘ M ( C ) \mathcal{M}(\mathbb{C}) M ( C ) M 1 = M \mathcal{M}_1=\mathcal{M} M 1 = M M ( C ) \mathcal{M}(\mathbb{C}) M ( C ) D \mathcal{D} D M 1 ∘ \mathcal{M}_1^\circ M 1 ∘ M 1 ∘ \mathcal{M}_1^\circ M 1 ∘ M 1 \mathcal{M}_1 M 1 M 1 \mathcal{M}_1 M 1 M 1 ∘ \mathcal{M}_1^\circ M 1 ∘ M a l g = M ( C ) \mathcal{M}^{\mathrm{alg}}=\mathcal{M}(\mathbb{C}) M alg = M ( C ) X t \mathcal{X}_t X t
Lemma 7.4. Assume Hypothesis 7.1 . For even d > 3 d>3 d > 3 κ 3 ( E ) \kappa_3(\mathcal{E}) κ 3 ( E ) K d K_d K d ( 6 , 0 ) (6,0) ( 6 , 0 ) ( 0 , 6 ) (0,6) ( 0 , 6 ) ( 3 , 3 ) (3,3) ( 3 , 3 ) ( 6 , 0 ) ⊕ ( 0 , 6 ) (6,0)\oplus(0,6) ( 6 , 0 ) ⊕ ( 0 , 6 ) ± d + 1 2 d x \pm\frac{d+1}{2d}x ± 2 d d + 1 x x = 1 2 ( ω X 6 + ω ‾ X 6 ) x=\frac{1}{2}(\omega_{X_6}+\overline{\omega}_{X_6}) x = 2 1 ( ω X 6 + ω X 6 ) H 6 ( X 6 , Z ) H^6(X_6,\mathbb{Z}) H 6 ( X 6 , Z ) p p p K d K_d K d p ∤ 2 d ( d + 1 ) p\nmid2d(d+1) p ∤ 2 d ( d + 1 ) λ 6 ω X 6 + λ ‾ 6 ω ‾ X 6 \lambda_6\omega_{X_6}+\overline{\lambda}_6\overline{\omega}_{X_6} λ 6 ω X 6 + λ 6 ω X 6 λ 6 = ± d + 1 4 d ∈ O K d , ( p ) × \lambda_6=\pm\frac{d+1}{4d}\in\mathcal{O}_{K_d,(p)}^\times λ 6 = ± 4 d d + 1 ∈ O K d , ( p ) ×
Proof. By [2 Lemma 8.2.1] , ch ( I 1 ) = 1 + Θ − d 2 Θ 2 − d [ p t ] \operatorname{ch}(\mathcal{I}_1)=1+\Theta-\frac{d}{2}\Theta^2-d[\mathrm{pt}] ch ( I 1 ) = 1 + Θ − 2 d Θ 2 − d [ pt ] ch ( I 2 ) = ch ( I 1 ) \operatorname{ch}(\mathcal{I}_2)=\operatorname{ch}(\mathcal{I}_1) ch ( I 2 ) = ch ( I 1 ) − A J ( C ) -A_J(\mathbb{C}) − A J ( C ) Θ 2 / 2 \Theta^2/2 Θ 2 /2 A J ( C ) A_J(\mathbb{C}) A J ( C ) ρ 1 = exp ( − d Θ ) \rho_1=\exp(\sqrt{-d}\Theta) ρ 1 = exp ( − d Θ ) ρ 2 = ρ ‾ 1 \rho_2=\overline{\rho}_1 ρ 2 = ρ 1 Θ 3 = 6 [ p t ] \Theta^3=6[\mathrm{pt}] Θ 3 = 6 [ pt ] ch ( I 1 ) = α d ρ 1 + α ‾ d ρ 2 \operatorname{ch}(\mathcal{I}_1)=\alpha_d\rho_1+\overline{\alpha}_d\rho_2 ch ( I 1 ) = α d ρ 1 + α d ρ 2 α d = 1 2 ( 1 + 1 / − d ) \alpha_d=\frac{1}{2}(1+1/\sqrt{-d}) α d = 2 1 ( 1 + 1/ − d ) τ \tau τ H ∙ ( X 3 ) H^\bullet(X_3) H ∙ ( X 3 ) [2 (1.2.3)] , which exchanges ρ 1 \rho_1 ρ 1 ρ 2 \rho_2 ρ 2 ξ \xi ξ ϕ \phi ϕ [2 Proposition 6.4.1] , so that ch Φ ( I 2 ⊠ I 1 ) = ξ ( ch ( I 1 ) ⊗ τ ( ch ( I 1 ) ) ) \operatorname{ch}\Phi(\mathcal{I}_2\boxtimes\mathcal{I}_1)=\xi(\operatorname{ch}(\mathcal{I}_1)\otimes\tau(\operatorname{ch}(\mathcal{I}_1))) ch Φ ( I 2 ⊠ I 1 ) = ξ ( ch ( I 1 ) ⊗ τ ( ch ( I 1 ))) [2 proof of Lemma 8.3.1] . Put ξ ~ = exp ( − c 1 ( P ) / 2 ) ∪ ξ \widetilde{\xi}=\exp(-c_1(\mathcal{P})/2)\cup\xi ξ = exp ( − c 1 ( P ) /2 ) ∪ ξ P \mathcal{P} P X 6 X_6 X 6 [2 (6.1.4) and Proposition 6.1.2] , ξ ~ ( g c ⊗ g c ′ ) = g ˉ ( ξ ~ ( c ⊗ c ′ ) ) \widetilde{\xi}(gc\otimes gc')=\bar{g}(\widetilde{\xi}(c\otimes c')) ξ ( g c ⊗ g c ′ ) = g ˉ ( ξ ( c ⊗ c ′ )) g ∈ Spin ( Γ K ) g\in\operatorname{Spin}(\Gamma_K) g ∈ Spin ( Γ K ) g g g H ∙ ( X 3 , K d ) H^\bullet(X_3,K_d) H ∙ ( X 3 , K d ) g ˉ ∈ SO ( Γ K ) \bar{g}\in\operatorname{SO}(\Gamma_K) g ˉ ∈ SO ( Γ K ) H ∙ ( X 6 , K d ) = ⋀ ∙ Γ K H^\bullet(X_6,K_d)=\bigwedge^\bullet\Gamma_K H ∙ ( X 6 , K d ) = ⋀ ∙ Γ K κ \kappa κ κ ( Φ ( I 2 ⊠ I 1 ) ) = κ ( c ) \kappa(\Phi(\mathcal{I}_2\boxtimes\mathcal{I}_1))=\kappa(c) κ ( Φ ( I 2 ⊠ I 1 )) = κ ( c )
c = ξ ~ ( ch ( I 1 ) ⊗ τ ( ch ( I 1 ) ) ) = d + 1 4 d ( ξ ~ ( ρ 1 ⊗ ρ 1 ) + ξ ~ ( ρ 2 ⊗ ρ 2 ) ) + α d 2 ξ ~ ( ρ 1 ⊗ ρ 2 ) + α ‾ d 2 ξ ~ ( ρ 2 ⊗ ρ 1 ) , c=\widetilde{\xi}(\operatorname{ch}(\mathcal{I}_1)\otimes\tau(\operatorname{ch}(\mathcal{I}_1)))=\frac{d+1}{4d}\bigl(\widetilde{\xi}(\rho_1\otimes\rho_1)+\widetilde{\xi}(\rho_2\otimes\rho_2)\bigr)+\alpha_d^2\widetilde{\xi}(\rho_1\otimes\rho_2)+\overline{\alpha}_d^2\widetilde{\xi}(\rho_2\otimes\rho_1), c = ξ ( ch ( I 1 ) ⊗ τ ( ch ( I 1 ))) = 4 d d + 1 ( ξ ( ρ 1 ⊗ ρ 1 ) + ξ ( ρ 2 ⊗ ρ 2 ) ) + α d 2 ξ ( ρ 1 ⊗ ρ 2 ) + α d 2 ξ ( ρ 2 ⊗ ρ 1 ) , as α d α ‾ d = d + 1 4 d \alpha_d\overline{\alpha}_d=\frac{d+1}{4d} α d α d = 4 d d + 1 Spin ( Γ K ) \operatorname{Spin}(\Gamma_K) Spin ( Γ K ) K d ρ 1 K_d\rho_1 K d ρ 1 K d ρ 2 K_d\rho_2 K d ρ 2 ρ 1 ⊗ ρ 1 \rho_1\otimes\rho_1 ρ 1 ⊗ ρ 1 ρ 2 ⊗ ρ 2 \rho_2\otimes\rho_2 ρ 2 ⊗ ρ 2 Γ 1 , K \Gamma_{1,K} Γ 1 , K Γ 2 , K \Gamma_{2,K} Γ 2 , K ρ 1 ⊗ ρ 2 \rho_1\otimes\rho_2 ρ 1 ⊗ ρ 2 ρ 2 ⊗ ρ 1 \rho_2\otimes\rho_1 ρ 2 ⊗ ρ 1 [2 Section 6.4] . It contains an element that acts on Γ 1 , K \Gamma_{1,K} Γ 1 , K Γ 2 , K \Gamma_{2,K} Γ 2 , K 1 / 4 1/4 1/4 [2 Remark 2.4.3] , hence by 4 i − j 4^{i-j} 4 i − j K d K_d K d ( i , j ) (i,j) ( i , j ) ξ ~ \widetilde{\xi} ξ ξ ~ ( ρ 1 ⊗ ρ 1 ) \widetilde{\xi}(\rho_1\otimes\rho_1) ξ ( ρ 1 ⊗ ρ 1 ) ( 6 , 0 ) (6,0) ( 6 , 0 ) ξ ~ ( ρ 2 ⊗ ρ 2 ) \widetilde{\xi}(\rho_2\otimes\rho_2) ξ ( ρ 2 ⊗ ρ 2 ) ( 0 , 6 ) (0,6) ( 0 , 6 ) ( j , j ) (j,j) ( j , j ) c 1 ( c ) c_1(c) c 1 ( c ) ( 1 , 1 ) (1,1) ( 1 , 1 ) c c c ( j , j ) (j,j) ( j , j ) κ 3 ( c ) \kappa_3(c) κ 3 ( c ) ( 6 , 0 ) (6,0) ( 6 , 0 ) ( 0 , 6 ) (0,6) ( 0 , 6 ) ( 3 , 3 ) (3,3) ( 3 , 3 ) ( 6 , 0 ) (6,0) ( 6 , 0 ) d + 1 4 d ξ ~ ( ρ 1 ⊗ ρ 1 ) \frac{d+1}{4d}\widetilde{\xi}(\rho_1\otimes\rho_1) 4 d d + 1 ξ ( ρ 1 ⊗ ρ 1 ) κ \kappa κ ch Φ ( I 2 ⊠ I 1 ) \operatorname{ch}\Phi(\mathcal{I}_2\boxtimes\mathcal{I}_1) ch Φ ( I 2 ⊠ I 1 ) K d K_d K d
The element g = exp ( − d Θ ) g=\exp(\sqrt{-d}\Theta) g = exp ( − d Θ ) Spin ( Γ K ) \operatorname{Spin}(\Gamma_K) Spin ( Γ K ) H ∙ ( X 3 , K d ) H^\bullet(X_3,K_d) H ∙ ( X 3 , K d ) ρ 1 \rho_1 ρ 1 g ˉ ( w , y ) = ( w − − d Θ ♯ y , y ) \bar{g}(w,y)=(w-\sqrt{-d}\Theta^\sharp y,y) g ˉ ( w , y ) = ( w − − d Θ ♯ y , y ) [2 (2.4.4)] . So ξ ~ ( ρ 1 ⊗ ρ 1 ) = g ˉ ( ξ ~ ( 1 ⊗ 1 ) ) \widetilde{\xi}(\rho_1\otimes\rho_1)=\bar{g}(\widetilde{\xi}(1\otimes1)) ξ ( ρ 1 ⊗ ρ 1 ) = g ˉ ( ξ ( 1 ⊗ 1 )) [2 Section 1.3] , Φ \Phi Φ id × Φ P : D b ( X 6 ) → D b ( X 3 × X 3 ) \operatorname{id}\times\Phi_{\mathcal{P}}:D^b(X_6)\to D^b(X_3\times X_3) id × Φ P : D b ( X 6 ) → D b ( X 3 × X 3 ) X 3 × X 3 X_3\times X_3 X 3 × X 3 P \mathcal{P} P ( x , y ) ↦ ( x + y , y ) (x,y)\mapsto(x+y,y) ( x , y ) ↦ ( x + y , y ) X 3 × X 3 X_3\times X_3 X 3 × X 3 O X 3 × { 0 } \mathcal{O}_{X_3\times\{0\}} O X 3 × { 0 } O X 3 ⊠ P ∣ X 3 × { 0 } = O X 3 × X 3 \mathcal{O}_{X_3}\boxtimes\mathcal{P}|_{X_3\times\{0\}}=\mathcal{O}_{X_3\times X_3} O X 3 ⊠ P ∣ X 3 × { 0 } = O X 3 × X 3 X 3 × { 0 } X_3\times\{0\} X 3 × { 0 } [1 Chapter III, Section 1 and Summary 6.11] , X 3 X_3 X 3 O X 3 × X 3 \mathcal{O}_{X_3\times X_3} O X 3 × X 3 Φ ( O X 3 × X 3 ) = O X 3 × { 0 } \Phi(\mathcal{O}_{X_3\times X_3})=\mathcal{O}_{X_3\times\{0\}} Φ ( O X 3 × X 3 ) = O X 3 × { 0 } ξ ( 1 ⊗ 1 ) = ± [ X 3 × { 0 } ] \xi(1\otimes1)=\pm[X_3\times\{0\}] ξ ( 1 ⊗ 1 ) = ± [ X 3 × { 0 }] c 1 ( P ) c_1(\mathcal{P}) c 1 ( P ) ξ ~ ( 1 ⊗ 1 ) = ± y 1 ∧ ⋯ ∧ y 6 \widetilde{\xi}(1\otimes1)=\pm y_1\wedge\cdots\wedge y_6 ξ ( 1 ⊗ 1 ) = ± y 1 ∧ ⋯ ∧ y 6 Z \mathbb{Z} Z ( y i ) (y_i) ( y i ) H 1 ( X 3 , Z ) H^1(X_3,\mathbb{Z}) H 1 ( X 3 , Z ) ξ ~ ( ρ 1 ⊗ ρ 1 ) = ± ω X 6 \widetilde{\xi}(\rho_1\otimes\rho_1)=\pm\omega_{X_6} ξ ( ρ 1 ⊗ ρ 1 ) = ± ω X 6 κ 3 ( Φ ( I 2 ⊠ I 1 ) ) \kappa_3(\Phi(\mathcal{I}_2\boxtimes\mathcal{I}_1)) κ 3 ( Φ ( I 2 ⊠ I 1 )) ± d + 1 4 d ( ω X 6 + ω ‾ X 6 ) = ± d + 1 4 d x \pm\frac{d+1}{4d}(\omega_{X_6}+\overline{\omega}_{X_6})=\pm\frac{d+1}{4d}x ± 4 d d + 1 ( ω X 6 + ω X 6 ) = ± 4 d d + 1 x κ 3 \kappa_3 κ 3 E \mathcal{E} E Φ ( I 2 ⊠ I 1 ) \Phi(\mathcal{I}_2\boxtimes\mathcal{I}_1) Φ ( I 2 ⊠ I 1 ) [2 Theorem 1.4.1(2)] . Finally, expanding
ω X 6 = ⋀ i ( y i − − d Θ ♯ y i ) \omega_{X_6}=\bigwedge_i(y_i-\sqrt{-d}\Theta^\sharp y_i) ω X 6 = i ⋀ ( y i − − d Θ ♯ y i ) shows that x x x y i y_i y i Θ ♯ y i \Theta^\sharp y_i Θ ♯ y i ( − d ) k (-d)^k ( − d ) k y 1 ∧ ⋯ ∧ y 6 y_1\wedge\cdots\wedge y_6 y 1 ∧ ⋯ ∧ y 6 x x x λ 6 = ∓ d + 1 4 d \lambda_6=\mp\frac{d+1}{4d} λ 6 = ∓ 4 d d + 1 p p p p ∤ 2 d ( d + 1 ) p\nmid2d(d+1) p ∤ 2 d ( d + 1 )
The derivation uses [2 Proposition 6.1.2] . An exact computation through the cohomological Fourier–Mukai transform of Orlov’s equivalence, which does not use it, gives the same types and the same content d + 1 2 d \frac{d+1}{2d} 2 d d + 1 d = 4 , 6 , 8 , 10 , 12 , 28 d=4,6,8,10,12,28 d = 4 , 6 , 8 , 10 , 12 , 28
The Weil cycle on the CM fourfold
Proposition 7.5. Assume Hypothesis 7.1. Let X c X_c X c K = Q ( − d 0 ) K=\mathbb{Q}(\sqrt{-d_0}) K = Q ( − d 0 ) d 0 d_0 d 0 ω = ⋀ 4 \omega=\bigwedge^4 ω = ⋀ 4 O K , ( p ) \mathcal{O}_{K,(p)} O K , ( p ) K K K H 1 ( X c , Z ( p ) ) ⊗ O K H^1(X_c,\mathbb{Z}_{(p)})\otimes\mathcal{O}_K H 1 ( X c , Z ( p ) ) ⊗ O K Z Z Z X c X_c X c λ ω + λ ‾ ω ‾ \lambda\omega+\overline{\lambda}\overline{\omega} λω + λ ω λ ∈ O K , ( p ) × \lambda\in\mathcal{O}_{K,(p)}^\times λ ∈ O K , ( p ) × K K K ( i , 4 − i ) (i,4-i) ( i , 4 − i )
Proof. Put d = 4 d 0 d=4d_0 d = 4 d 0 d ≥ 4 d\ge4 d ≥ 4 K d = K K_d=K K d = K p p p K K K p ∤ 2 d p\nmid2d p ∤ 2 d p ∤ d + 1 p\nmid d+1 p ∤ d + 1 − d ≡ 1 -d\equiv1 − d ≡ 1 p p p − d 0 -d_0 − d 0 p p p p p p K K K ψ 6 \psi_6 ψ 6 X 6 X_6 X 6 d 6 d^6 d 6 Γ Z \Gamma_{\mathbb{Z}} Γ Z Γ Z \Gamma_{\mathbb{Z}} Γ Z p p p
Schoen’s degeneration at $p. We follow Schoen’s proof of [39 Proposition 10] . Let Λ c = H 1 ( X c , Z ) \Lambda_c=H_1(X_c,\mathbb{Z}) Λ c = H 1 ( X c , Z ) Tr K / Q ( ⋅ / 2 − d ) \operatorname{Tr}_{K/\mathbb{Q}}(\cdot/2\sqrt{-d}) Tr K / Q ( ⋅ /2 − d ) p p p p p p [2 (2.4.2) and (3.1.2)] , ψ 6 = Tr K d / Q ( H / 2 − d ) \psi_6=\operatorname{Tr}_{K_d/\mathbb{Q}}(H/2\sqrt{-d}) ψ 6 = Tr K d / Q ( H /2 − d ) H H H Λ c \Lambda_c Λ c Γ Z \Gamma_{\mathbb{Z}} Γ Z X 6 , t X_{6,t} X 6 , t x ↦ ψ 6 ( x , ⋅ ) x\mapsto\psi_6(x,\mathord{\cdot}) x ↦ ψ 6 ( x , ⋅ ) Γ Z \Gamma_{\mathbb{Z}} Γ Z H 1 ( X 6 , t , Z ) = Hom ( Γ Z , Z ) H_1(X_{6,t},\mathbb{Z})=\operatorname{Hom}(\Gamma_Z,\mathbb{Z}) H 1 ( X 6 , t , Z ) = Hom ( Γ Z , Z ) d 6 d^6 d 6 p p p ψ 6 \psi_6 ψ 6 ψ 6 \psi_6 ψ 6 η \eta η ψ 6 \psi_6 ψ 6 ( ⋅ , ⋅ ) (\cdot,\cdot) ( ⋅ , ⋅ ) [2 Section 2.4] , this map carries − η -\eta − η X 6 , t X_{6,t} X 6 , t η \eta η − d \sqrt{-d} − d Γ Q \Gamma_{\mathbb{Q}} Γ Q − η -\eta − η − H ‾ -\overline{\mathcal{H}} − H ψ 6 \psi_6 ψ 6 H \mathcal{H} H ( 3 , 3 ) (3,3) ( 3 , 3 ) − 1 -1 − 1 K K K K K K ( i , j ) (i,j) ( i , j ) ( j , i ) (j,i) ( j , i ) K K K η \eta η K K K Γ Q ′ \Gamma'_{\mathbb{Q}} Γ Q ′ ( 1 , 1 ) (1,1) ( 1 , 1 ) Λ c ⊗ Q ⊕ Γ Q ′ \Lambda_c\otimes\mathbb{Q}\oplus\Gamma'_{\mathbb{Q}} Λ c ⊗ Q ⊕ Γ Q ′ Γ Q = Γ Z ⊗ Q \Gamma_{\mathbb{Q}}=\Gamma_Z\otimes\mathbb{Q} Γ Q = Γ Z ⊗ Q [39 Section 7] . Since det Λ c \det\Lambda_c det Λ c det Γ Z \det\Gamma_Z det Γ Z p p p K p K_p K p det Γ Q ′ \det\Gamma'_{\mathbb{Q}} det Γ Q ′ p p p Γ Q ′ \Gamma'_{\mathbb{Q}} Γ Q ′ O K O_K O K Γ Z ′ \Gamma'_Z Γ Z ′ p p p p p p X 2 X_2 X 2 K K K Ψ p : ( Λ c ⊕ Γ Z ′ ) ⊗ Z p → Γ Z ⊗ Z p \Psi_p:(\Lambda_c\oplus\Gamma'_Z)\otimes\mathbb{Z}_p\to\Gamma_Z\otimes\mathbb{Z}_p Ψ p : ( Λ c ⊕ Γ Z ′ ) ⊗ Z p → Γ Z ⊗ Z p Ψ 0 : V = Λ c ⊗ Q ⊕ Γ Q ′ → Γ Q \Psi_0:V=\Lambda_c\otimes\mathbb{Q}\oplus\Gamma'_{\mathbb{Q}}\to\Gamma_{\mathbb{Q}} Ψ 0 : V = Λ c ⊗ Q ⊕ Γ Q ′ → Γ Q g p = Ψ 0 − 1 Ψ p g_p=\Psi_0^{-1}\Psi_p g p = Ψ 0 − 1 Ψ p Υ \Upsilon Υ ( Λ c ⊕ Γ Z ′ ) ⊗ Z p (\Lambda_c\oplus\Gamma'_Z)\otimes\mathbb{Z}_p ( Λ c ⊕ Γ Z ′ ) ⊗ Z p U V ( Q p ) U_V(\mathbb{Q}_p) U V ( Q p ) g ∈ g p Υ ∩ U V ( Q ) g\in g_p\Upsilon\cap U_V(\mathbb{Q}) g ∈ g p Υ ∩ U V ( Q ) Ψ = Ψ 0 g \Psi=\Psi_0g Ψ = Ψ 0 g Ψ ( ( Λ c ⊕ Γ Z ′ ) ⊗ Z ( p ) ) = Γ Z ⊗ Z ( p ) \Psi((\Lambda_c\oplus\Gamma'_Z)\otimes\mathbb{Z}_{(p)})=\Gamma_Z\otimes\mathbb{Z}_{(p)} Ψ (( Λ c ⊕ Γ Z ′ ) ⊗ Z ( p ) ) = Γ Z ⊗ Z ( p ) N Ψ N\Psi N Ψ Λ c ⊕ Γ Z ′ \Lambda_c\oplus\Gamma'_Z Λ c ⊕ Γ Z ′ Γ Z \Gamma_Z Γ Z N N N p p p f : X c × X 2 → X 6 ′ f:X_c\times X_2\to X'_6 f : X c × X 2 → X 6 ′ p p p K K K Ψ \Psi Ψ ψ 6 \psi_6 ψ 6 N Ψ N\Psi N Ψ N 2 N^2 N 2 X c × X 2 X_c\times X_2 X c × X 2 X c × X 2 X_c\times X_2 X c × X 2 Ψ \Psi Ψ η \eta η ψ 6 \psi_6 ψ 6 f f f X 6 ′ X'_6 X 6 ′ ( Γ Z , η , ψ 6 ) (\Gamma_Z,\eta,\psi_6) ( Γ Z , η , ψ 6 ) z z z X 6 ′ X'_6 X 6 ′ 2 ( 8 d ) 8 d κ 3 ( E ) 2(8d)^{8d}\kappa_3(\mathcal{E}) 2 ( 8 d ) 8 d κ 3 ( E ) λ 6 ′ ω X 6 ′ + λ 6 ′ ‾ ω ‾ X 6 ′ \lambda'_6\omega_{X'_6}+\overline{\lambda'_6}\overline{\omega}_{X'_6} λ 6 ′ ω X 6 ′ + λ 6 ′ ω X 6 ′ λ 6 ′ = 2 ( 8 d ) 8 d λ 6 ∈ O K , ( p ) \lambda'_6=2(8d)^{8d}\lambda_6\in O_{K,(p)} λ 6 ′ = 2 ( 8 d ) 8 d λ 6 ∈ O K , ( p ) 2 ( 8 d ) 8 d 2(8d)^{8d} 2 ( 8 d ) 8 d p p p ω X 6 \omega_{X_6} ω X 6 ( Γ Z , η ) (\Gamma_Z,\eta) ( Γ Z , η ) ω X 6 ′ \omega_{X'_6} ω X 6 ′
Pull-back and push-forward. The pull-back f ∗ f^* f ∗ H 1 ( ⋅ , Z ( p ) ) H^1(\cdot,\mathbb{Z}_{(p)}) H 1 ( ⋅ , Z ( p ) ) O K , ( p ) O_{K,(p)} O K , ( p ) Γ 1 ⊗ Z ( p ) \Gamma_1\otimes\mathbb{Z}_{(p)} Γ 1 ⊗ Z ( p ) η \eta η − d \sqrt{-d} − d ι \iota ι H 1 ( X c × X 2 , Z ( p ) ) ⊗ O K H^1(X_c\times X_2,\mathbb{Z}_{(p)})\otimes O_K H 1 ( X c × X 2 , Z ( p ) ) ⊗ O K X c X_c X c X 2 X_2 X 2 f ∗ ω X 6 ′ = det K ( f ∗ ) ⋅ ω ⊗ ω ′ f^*\omega_{X'_6}=\det_K(f^*)\cdot\omega\otimes\omega' f ∗ ω X 6 ′ = det K ( f ∗ ) ⋅ ω ⊗ ω ′ ω ′ \omega' ω ′ ω \omega ω X 2 X_2 X 2 det K ( f ∗ ) ∈ O K , ( p ) × \det_K(f^*)\in O^\times_{K,(p)} det K ( f ∗ ) ∈ O K , ( p ) × ω ′ + ω ‾ ′ ∈ H 2 ( X 2 , Z ( p ) ) \omega'+\overline{\omega}'\in H^2(X_2,\mathbb{Z}_{(p)}) ω ′ + ω ′ ∈ H 2 ( X 2 , Z ( p ) ) ( 1 , 1 ) (1,1) ( 1 , 1 ) N ′ ( ω ′ + ω ‾ ′ ) N'(\omega'+\overline{\omega}') N ′ ( ω ′ + ω ′ ) N ′ N' N ′ p p p ( 1 , 1 ) (1,1) ( 1 , 1 ) [38 Theorem 2] ). Put Z = pr X c ∗ ( f ∗ z ⋅ pr X 2 ∗ ( N ′ ( ω ′ + ω ‾ ′ ) ) ) Z=\operatorname{pr}_{X_c*}(f^*z\cdot\operatorname{pr}_{X_2}^*(N'(\omega'+\overline{\omega}'))) Z = pr X c ∗ ( f ∗ z ⋅ pr X 2 ∗ ( N ′ ( ω ′ + ω ′ ))) K K K ( i , j ) (i,j) ( i , j ) X c × X 2 X_c\times X_2 X c × X 2 ω ′ + ω ‾ ′ \omega'+\overline{\omega}' ω ′ + ω ′ ( 2 , 0 ) (2,0) ( 2 , 0 ) ( 0 , 2 ) (0,2) ( 0 , 2 ) ( 4 , 0 ) (4,0) ( 4 , 0 ) X c X_c X c ( i , j ) (i,j) ( i , j ) ( 6 , 0 ) (6,0) ( 6 , 0 ) ( 5 , 1 ) (5,1) ( 5 , 1 ) ( 4 , 2 ) (4,2) ( 4 , 2 ) ( 6 , 0 ) (6,0) ( 6 , 0 ) λ ω \lambda\omega λω λ = N ′ λ 6 ′ det K ( f ∗ ) ∫ X 2 ω ′ ∧ ω ‾ ′ \lambda=N'\lambda'_6\det_K(f^*)\int_{X_2}\omega'\wedge\overline{\omega}' λ = N ′ λ 6 ′ det K ( f ∗ ) ∫ X 2 ω ′ ∧ ω ′ ω ′ ∧ ω ‾ ′ \omega'\wedge\overline{\omega}' ω ′ ∧ ω ′ H 4 ( X 2 , O K , ( p ) ) H^4(X_2,O_{K,(p)}) H 4 ( X 2 , O K , ( p ) ) Γ Z ′ \Gamma'_Z Γ Z ′ p p p ( 0 , 4 ) (0,4) ( 0 , 4 )
Specialization
Choose β ∈ O K ∩ End ( X c ) \beta\in O_K\cap\operatorname{End}(X_c) β ∈ O K ∩ End ( X c ) ( O K / p ) × ≅ F p 2 × (O_K/p)^\times\cong\mathbb{F}_{p^2}^\times ( O K / p ) × ≅ F p 2 × End ( X c ) ∩ K \operatorname{End}(X_c)\cap K End ( X c ) ∩ K p p p K K K ( i , 4 − i ) (i,4-i) ( i , 4 − i ) H 4 ( X c ) H^4(X_c) H 4 ( X c ) [ β ] ∗ [\beta]^* [ β ] ∗ β i β ‾ 4 − i ≡ ζ i ( p + 1 − i ) ( m o d p ) \beta^i\overline{\beta}^{\,4-i}\equiv\zeta^{i(p+1-i)}\pmod p β i β 4 − i ≡ ζ i ( p + 1 − i ) ( mod p ) ζ ≡ β p + 1 ( m o d p ) \zeta\equiv\beta^{p+1}\pmod p ζ ≡ β p + 1 ( mod p ) 0 ≤ i < j ≤ 4 0\le i<j\le4 0 ≤ i < j ≤ 4 β i β ‾ 4 − i = β j β ‾ 4 − j \beta^i\overline{\beta}^{\,4-i}=\beta^j\overline{\beta}^{\,4-j} β i β 4 − i = β j β 4 − j ( p + 1 ) ( j − i ) (p+1)(j-i) ( p + 1 ) ( j − i ) p > 5 p>5 p > 5 Q Q Q β ‾ 4 \overline{\beta}^{\,4} β 4 β 4 \beta^4 β 4 β i β ‾ 4 − i \beta^i\overline{\beta}^{\,4-i} β i β 4 − i i = 1 , 2 , 3 i=1,2,3 i = 1 , 2 , 3 O K , ( p ) O_{K,(p)} O K , ( p ) β i β ‾ 4 − i \beta^i\overline{\beta}^{\,4-i} β i β 4 − i β 4 − i β ‾ i \beta^{4-i}\overline{\beta}^{\,i} β 4 − i β i Z ( p ) \mathbb{Z}_{(p)} Z ( p ) P = Q ( [ β ] ∗ ) P=Q([\beta]^*) P = Q ([ β ] ∗ ) P P P Z ( p ) \mathbb{Z}_{(p)} Z ( p ) [ β ] ∗ [\beta]^* [ β ] ∗ Proposition 6.1 (d) and Lemma 4.4 (iii).
Proposition 7.6. Assume Hypothesis 7.1. In the situation of Proposition 6.1 , L ( X t 0 ) ⊆ Alg 2 ( X t 0 ) \mathcal{L}(X_{t_0})\subseteq\operatorname{Alg}^2(X_{t_0}) L ( X t 0 ) ⊆ Alg 2 ( X t 0 )
Proof. The fourfold X c X_c X c Q ‾ \overline{\mathbb{Q}} Q Z Z Z C \mathbb{C} C Q ‾ \overline{\mathbb{Q}} Q [40 ] ), which has the same class, and we replace Z Z Z O O O p p p X c X_c X c Z Z Z β \beta β X / O \mathcal{X}/O X / O X c X_c X c X t 0 X_{t_0} X t 0 Proposition 6.1 (c)); β \beta β X \mathcal{X} X O O O Z 1 = P Z Z_1=PZ Z 1 = P Z Z 2 = [ β ] ∗ Z 1 Z_2=[\beta]^*Z_1 Z 2 = [ β ] ∗ Z 1 Z ( p ) \mathbb{Z}_{(p)} Z ( p ) λ ω + λ ‾ ω ‾ \lambda\omega+\overline{\lambda}\overline{\omega} λω + λ ω β 4 λ ω + β ‾ 4 λ ‾ ω ‾ \beta^4\lambda\omega+\overline{\beta}^4\overline{\lambda}\overline{\omega} β 4 λω + β 4 λ ω ω ∧ ω = 0 \omega\wedge\omega=0 ω ∧ ω = 0
deg ( Z i ⋅ Z j ) = u Tr K / Q ( λ ( i ) λ ( j ) ‾ ) \deg(Z_i\cdot Z_j)=u\operatorname{Tr}_{K/\mathbb{Q}}\left(\lambda_{(i)}\overline{\lambda_{(j)}}\right) deg ( Z i ⋅ Z j ) = u Tr K / Q ( λ ( i ) λ ( j ) ) with λ ( 1 ) = λ \lambda_{(1)}=\lambda λ ( 1 ) = λ λ ( 2 ) = β 4 λ \lambda_{(2)}=\beta^4\lambda λ ( 2 ) = β 4 λ u = ∫ ω ∧ ω ‾ u=\int\omega\wedge\overline{\omega} u = ∫ ω ∧ ω p p p K K K H 1 ( X c , Z ( p ) ) ⊗ O K H^1(X_c,\mathbb{Z}_{(p)})\otimes O_K H 1 ( X c , Z ( p ) ) ⊗ O K ω ∧ ω ‾ \omega\wedge\overline{\omega} ω ∧ ω H 8 ( X c , O K , ( p ) ) H^8(X_c,O_{K,(p)}) H 8 ( X c , O K , ( p ) ) u ∈ Z ( p ) u\in\mathbb{Z}_{(p)} u ∈ Z ( p ) β 4 ≢ β ‾ 4 ( m o d p ) \beta^4\not\equiv\overline{\beta}^4\pmod p β 4 ≡ β 4 ( mod p ) ζ 4 ( p − 1 ) = 1 \zeta^{4(p-1)}=1 ζ 4 ( p − 1 ) = 1 ( p + 1 ) ∣ 4 (p+1)\mid4 ( p + 1 ) ∣ 4 p p p
By (R5) and (R4), the specialized cycles sp Z i \operatorname{sp}Z_i sp Z i X t 0 X_{t_0} X t 0 cl ( sp Z i ) ∈ T 4 ( X t 0 ) \operatorname{cl}(\operatorname{sp}Z_i)\in T^4(X_{t_0}) cl ( sp Z i ) ∈ T 4 ( X t 0 ) cl ( sp Z 1 ) = P cl ( sp Z ) \operatorname{cl}(\operatorname{sp}Z_1)=P\operatorname{cl}(\operatorname{sp}Z) cl ( sp Z 1 ) = P cl ( sp Z ) L \mathcal{L} L cl ( sp Z 2 ) = [ β ] ∗ cl ( sp Z 1 ) \operatorname{cl}(\operatorname{sp}Z_2)=[\beta]^*\operatorname{cl}(\operatorname{sp}Z_1) cl ( sp Z 2 ) = [ β ] ∗ cl ( sp Z 1 ) ∫ cl ( sp Z i ) cl ( sp Z j ) = deg ( sp Z i ⋅ sp Z j ) = deg ( Z i ⋅ Z j ) \int\operatorname{cl}(\operatorname{sp}Z_i)\operatorname{cl}(\operatorname{sp}Z_j)=\deg(\operatorname{sp}Z_i\cdot\operatorname{sp}Z_j)=\deg(Z_i\cdot Z_j) ∫ cl ( sp Z i ) cl ( sp Z j ) = deg ( sp Z i ⋅ sp Z j ) = deg ( Z i ⋅ Z j ) T 4 T^4 T 4 T 8 = Z p [ p t ] T^8=\mathbb{Z}_p[\mathrm{pt}] T 8 = Z p [ pt ] L \mathcal{L} L Z p \mathbb{Z}_p Z p L ′ \mathcal{L}' L ′ cl ( sp Z 1 ) , cl ( sp Z 2 ) \operatorname{cl}(\operatorname{sp}Z_1),\operatorname{cl}(\operatorname{sp}Z_2) cl ( sp Z 1 ) , cl ( sp Z 2 ) disc L ′ = [ L : L ′ ] 2 disc L \operatorname{disc}\mathcal{L}'=[\mathcal{L}:\mathcal{L}']^2\operatorname{disc}\mathcal{L} disc L ′ = [ L : L ′ ] 2 disc L L ′ = L \mathcal{L}'=\mathcal{L} L ′ = L
No p p p p p p
Proof of the main theorem
Theorem 1.3 is proved at one supersingular point of a Mumford family and carried to every abelian fourfold with p p p G p \mathcal{G}_p G p p p p
Lemma 8.1. Let X , X ′ X,X' X , X ′ F ‾ p \overline{\mathbb{F}}_p F p X [ p ∞ ] ≅ X ′ [ p ∞ ] X[p^\infty]\cong X'[p^\infty] X [ p ∞ ] ≅ X ′ [ p ∞ ] f : X ′ → X f:X'\to X f : X ′ → X p p p f ∗ f^* f ∗ T 2 i ( X ) T^{2i}(X) T 2 i ( X ) Alg i ( X ) \operatorname{Alg}^i(X) Alg i ( X ) S ( X ) S(X) S ( X ) T 2 i ( X ′ ) T^{2i}(X') T 2 i ( X ′ ) Alg i ( X ′ ) \operatorname{Alg}^i(X') Alg i ( X ′ ) S ( X ′ ) S(X') S ( X ′ )
Proof. By (R3), Hom ( X ′ , X ) ⊗ Z p = Hom ( X ′ [ p ∞ ] , X [ p ∞ ] ) \operatorname{Hom}(X',X)\otimes\mathbb{Z}_p=\operatorname{Hom}(X'[p^\infty],X[p^\infty]) Hom ( X ′ , X ) ⊗ Z p = Hom ( X ′ [ p ∞ ] , X [ p ∞ ]) f 0 f_0 f 0 p p p f ∈ Hom ( X ′ , X ) f\in\operatorname{Hom}(X',X) f ∈ Hom ( X ′ , X ) f − f 0 ∈ p Hom ( X ′ [ p ∞ ] , X [ p ∞ ] ) f-f_0\in p\operatorname{Hom}(X'[p^\infty],X[p^\infty]) f − f 0 ∈ p Hom ( X ′ [ p ∞ ] , X [ p ∞ ]) f [ p ∞ ] ∈ f 0 ( 1 + p End ( X ′ [ p ∞ ] ) ) f[p^\infty]\in f_0(1+p\operatorname{End}(X'[p^\infty])) f [ p ∞ ] ∈ f 0 ( 1 + p End ( X ′ [ p ∞ ])) f f f p p p X X X X ′ X' X ′ p p p X 0 ′ X'_0 X 0 ′ ker f \ker f ker f X ′ X' X ′ f [ p ∞ ] f[p^\infty] f [ p ∞ ] X 0 ′ [ p ∞ ] ⊆ X ′ [ p ∞ ] X'_0[p^\infty]\subseteq X'[p^\infty] X 0 ′ [ p ∞ ] ⊆ X ′ [ p ∞ ] 2 dim X 0 ′ 2\dim X'_0 2 dim X 0 ′ f [ p ∞ ] f[p^\infty] f [ p ∞ ] X 0 ′ = 0 X'_0=0 X 0 ′ = 0 ker f \ker f ker f p p p X ′ [ p ∞ ] X'[p^\infty] X ′ [ p ∞ ] f [ p ∞ ] f[p^\infty] f [ p ∞ ] f ∗ f^* f ∗ M M M T 2 i T^{2i} T 2 i T 2 ( X ) T^2(X) T 2 ( X ) T 2 ( X ′ ) T^2(X') T 2 ( X ′ ) S ( X ) S(X) S ( X ) S ( X ′ ) S(X') S ( X ′ ) f ∗ f ∗ = f ∗ f ∗ = deg f ∈ Z p × f_*f^*=f^*f_*=\deg f\in\mathbb{Z}_p^\times f ∗ f ∗ = f ∗ f ∗ = deg f ∈ Z p × f ∗ Alg i ( X ) = Alg i ( X ′ ) f^*\operatorname{Alg}^i(X)=\operatorname{Alg}^i(X') f ∗ Alg i ( X ) = Alg i ( X ′ )
Lemma 8.2. For fixed ( k , B , Λ ) (k,B,\Lambda) ( k , B , Λ ) ( h 3 ) (h3) ( h 3 ) ( h 4 ) (h4) ( h 4 ) p p p p ≥ 5 p\geq5 p ≥ 5 ( k , B , Λ , K p ) (k,B,\Lambda,K^p) ( k , B , Λ , K p ) ( h 1 ) (h1) ( h 1 ) ( h 4 ) (h4) ( h 4 )
Proof. Choose integers N 1 , N 0 N_1,N_0 N 1 , N 0 Ξ 3 + N 1 Ξ + N 0 \Xi^3+N_1\Xi+N_0 Ξ 3 + N 1 Ξ + N 0 p p p F p \mathbb{F}_p F p N 1 N_1 N 1 p p p N 1 < 0 N_1<0 N 1 < 0 − 4 N 1 3 − 27 N 0 2 > 0 -4N_1^3-27N_0^2>0 − 4 N 1 3 − 27 N 0 2 > 0 Ξ 3 + N 1 Ξ + N 0 \Xi^3+N_1\Xi+N_0 Ξ 3 + N 1 Ξ + N 0 p p p p p p k = Q [ Ξ ] / ( Ξ 3 + N 1 Ξ + N 0 ) k=\mathbb{Q}[\Xi]/(\Xi^3+N_1\Xi+N_0) k = Q [ Ξ ] / ( Ξ 3 + N 1 Ξ + N 0 ) B B B k k k Cor k / Q B \operatorname{Cor}_{k/\mathbb{Q}}B Cor k / Q B 1 2 + 1 2 = 0 \frac{1}{2}+\frac{1}{2}=0 2 1 + 2 1 = 0 ∞ \infty ∞ 0 0 0 Cor k / Q B ≅ Mat 8 ( Q ) \operatorname{Cor}_{k/\mathbb{Q}}B\cong\operatorname{Mat}_8(\mathbb{Q}) Cor k / Q B ≅ Mat 8 ( Q ) [14 Section 2.1.2] ), and B B B p p p G Q p G_{\mathbb{Q}_p} G Q p [26 Section 16] ). Concretely, take Λ p = Cor ( Λ 0 ) \Lambda_p=\operatorname{Cor}(\Lambda_0) Λ p = Cor ( Λ 0 ) O k p O_{k_p} O k p Λ 0 ⊂ U p \Lambda_0\subset U_p Λ 0 ⊂ U p Λ p \Lambda_p Λ p Λ ⊂ V \Lambda\subset V Λ ⊂ V ψ \psi ψ Λ \Lambda Λ K p K^p K p O B ⊂ B \mathcal{O}_B \subset B O B ⊂ B Nm : B → Cor k / Q B = End ( V ) \operatorname{Nm}: B \to\operatorname{Cor}_{k/\mathbb{Q}} B = \operatorname{End}(V) Nm : B → Cor k / Q B = End ( V ) Q \mathbb{Q} Q p p p p p p k k k B B B p p p O B ⊗ Z p = End O k p ( Λ 0 ) \mathcal{O}_B \otimes\mathbb{Z}_p = \operatorname{End}_{\mathcal{O}_{k_p}}(\Lambda_0) O B ⊗ Z p = End O k p ( Λ 0 ) O k p \mathcal{O}_{k_p} O k p Λ 0 ⊂ U p \Lambda_0 \subset U_p Λ 0 ⊂ U p Nm ( O B ⊗ Z p ) \operatorname{Nm}(\mathcal{O}_B \otimes\mathbb{Z}_p) Nm ( O B ⊗ Z p ) Λ ⊗ Z p \Lambda\otimes\mathbb{Z}_p Λ ⊗ Z p Nm \operatorname{Nm} Nm Z \mathbb{Z} Z O B \mathcal{O}_B O B End ( V ) \operatorname{End}(V) End ( V ) W W W Nm \operatorname{Nm} Nm ( x 1 , x 2 , x 3 ) ↦ x 1 ⊗ x 2 ⊗ x 3 (x_1,x_2,x_3) \mapsto x_1 \otimes x_2 \otimes x_3 ( x 1 , x 2 , x 3 ) ↦ x 1 ⊗ x 2 ⊗ x 3 Mat 2 ( W ) 3 \operatorname{Mat}_2(W)^3 Mat 2 ( W ) 3 O B ⊗ Z p \mathcal{O}_B \otimes\mathbb{Z}_p O B ⊗ Z p Nm \operatorname{Nm} Nm x 1 ⊗ x 2 ⊗ x 3 x_1 \otimes x_2 \otimes x_3 x 1 ⊗ x 2 ⊗ x 3 ( x 1 , 0 , 0 ) (x_1,0,0) ( x 1 , 0 , 0 ) ( 0 , x 2 , 0 ) (0,x_2,0) ( 0 , x 2 , 0 ) ( 0 , 0 , x 3 ) (0,0,x_3) ( 0 , 0 , x 3 ) W W W Nm ( O B ⊗ Z p ) \operatorname{Nm}(\mathcal{O}_B \otimes\mathbb{Z}_p) Nm ( O B ⊗ Z p ) End W ( Cor ( Λ 0 ) ⊗ W ) \operatorname{End}_W(\operatorname{Cor}(\Lambda_0) \otimes W) End W ( Cor ( Λ 0 ) ⊗ W ) W W W Z p \mathbb{Z}_p Z p Z p \mathbb{Z}_p Z p End Z p ( Cor Λ 0 ) ≅ Mat 8 ( Z p ) \operatorname{End}_{\mathbb{Z}_p}(\operatorname{Cor}\Lambda_0) \cong\operatorname{Mat}_8(\mathbb{Z}_p) End Z p ( Cor Λ 0 ) ≅ Mat 8 ( Z p ) End Z p ( Cor Λ 0 ) \operatorname{End}_{\mathbb{Z}_p}(\operatorname{Cor}\Lambda_0) End Z p ( Cor Λ 0 ) p j Cor ( Λ 0 ) p^j\operatorname{Cor}(\Lambda_0) p j Cor ( Λ 0 ) j ∈ Z j \in\mathbb{Z} j ∈ Z
Theorem 8.3. Assume (h1)–(h4), and let t ∈ S ( F ‾ p ) t \in\mathcal{S}(\overline{\mathbb{F}}_p) t ∈ S ( F p ) X = A t X=A_t X = A t X [ p ∞ ] ≅ G p X[p^\infty] \cong\mathcal{G}_p X [ p ∞ ] ≅ G p Theorem 5.2 applies to X X X z ∈ Alg 2 ( X ) z \in\operatorname{Alg}^2(X) z ∈ Alg 2 ( X ) S ( X ) S(X) S ( X ) U ∗ z U^*z U ∗ z U ∈ End ( X ) U \in\operatorname{End}(X) U ∈ End ( X ) T 4 ( X ) T^4(X) T 4 ( X ) z z z Q ‾ \overline{\mathbb{Q}} Q p p p Alg 2 ( X ) = T 4 ( X ) \operatorname{Alg}^2(X)=T^4(X) Alg 2 ( X ) = T 4 ( X ) H c r i s 2 ( X ) ⊗ H c r i s 2 ( X ) H^2_{\mathrm{cris}}(X)\otimes H^2_{\mathrm{cris}}(X) H cris 2 ( X ) ⊗ H cris 2 ( X ) T 4 ( X × X ) T^4(X\times X) T 4 ( X × X )
Proof. The first assertion is Proposition 3.3. For the rest, let t 0 t_0 t 0 L ( X t 0 ) ⊂ Alg 2 ( X t 0 ) L(X_{t_0}) \subset\operatorname{Alg}^2(X_{t_0}) L ( X t 0 ) ⊂ Alg 2 ( X t 0 ) Alg 2 \operatorname{Alg}^2 Alg 2 S S S End ( X t 0 ) ∗ \operatorname{End}(X_{t_0})^* End ( X t 0 ) ∗ Alg 2 ( X t 0 ) = T 4 ( X t 0 ) \operatorname{Alg}^2(X_{t_0})=T^4(X_{t_0}) Alg 2 ( X t 0 ) = T 4 ( X t 0 ) z ∈ L ∖ p L z \in\mathcal{L}\setminus p\mathcal{L} z ∈ L ∖ p L Theorem 5.2 . On X t 0 × X t 0 X_{t_0}\times X_{t_0} X t 0 × X t 0 ( 2 , 2 ) (2,2) ( 2 , 2 ) U 1 U_1 U 1 j U 1 jU_1 j U 1 j = 0 , … , 4 j=0,\ldots,4 j = 0 , … , 4 ( i , 4 − i ) (i,4-i) ( i , 4 − i ) ( U 1 , U 2 ) ∗ x (U_1,U_2)^*x ( U 1 , U 2 ) ∗ x j i j^i j i 0 , … , 4 0,\ldots,4 0 , … , 4 p ≥ 5 p\geq5 p ≥ 5 T 22 T_{22} T 22 t t t A t [ p ∞ ] ≅ X t 0 [ p ∞ ] A_t[p^\infty]\cong X_{t_0}[p^\infty] A t [ p ∞ ] ≅ X t 0 [ p ∞ ] A t A_t A t A t × A t A_t\times A_t A t × A t
Proof of Theorem 1.3. By (R1), M ( X ) M(X) M ( X ) T 4 ( X ) T^4(X) T 4 ( X ) S ( X ) S(X) S ( X ) ( M ( X ) , F ) (M(X),F) ( M ( X ) , F ) p p p t 0 t_0 t 0 X t 0 [ p ∞ ] ≅ G p ≅ X [ p ∞ ] X_{t_0}[p^\infty]\cong\mathcal{G}_p\cong X[p^\infty] X t 0 [ p ∞ ] ≅ G p ≅ X [ p ∞ ] f : X → X t 0 f:X\to X_{t_0} f : X → X t 0 p p p Alg 2 ( X ) = f ∗ Alg 2 ( X t 0 ) = f ∗ T 4 ( X t 0 ) = T 4 ( X ) \operatorname{Alg}^2(X)=f^*\operatorname{Alg}^2(X_{t_0})=f^*T^4(X_{t_0})=T^4(X) Alg 2 ( X ) = f ∗ Alg 2 ( X t 0 ) = f ∗ T 4 ( X t 0 ) = T 4 ( X ) f × f f\times f f × f
At primes that are not inert in k k k T 4 / S T^4/S T 4 / S
Proposition 8.4 (computer-assisted). Let p ≥ 5 p\geq5 p ≥ 5 k k k B B B k k k p p p A ⊗ Z p = p j Cor ( Λ 0 ) A\otimes\mathbb{Z}_p=p^j\operatorname{Cor}(\Lambda_0) A ⊗ Z p = p j Cor ( Λ 0 ) O k p ⊗ Z p \mathcal{O}_{k_p}\otimes\mathbb{Z}_p O k p ⊗ Z p Λ 0 ⊂ U p \Lambda_0\subset U_p Λ 0 ⊂ U p t ∈ S ( F ‾ p ) t\in\mathcal{S}(\overline{\mathbb{F}}_p) t ∈ S ( F p ) X = A t X=A_t X = A t T 4 ( X ) = S ( X ) T^4(X)=S(X) T 4 ( X ) = S ( X ) H c r i s 2 ( X ) ⊗ H c r i s 2 ( X ) H^2_{\mathrm{cris}}(X)\otimes H^2_{\mathrm{cris}}(X) H cris 2 ( X ) ⊗ H cris 2 ( X ) T 4 ( X × X ) T^4(X\times X) T 4 ( X × X ) X × X X\times X X × X
Proof. Lemma 3.1 holds, with the same proof, for p p p k k k k p k_p k p Q p \mathbb{Q}_p Q p Λ i \Lambda_i Λ i O k p → W \mathcal{O}_{k_p}\to W O k p → W σ \sigma σ k k k p p p p p p k k k σ \sigma σ b i ∈ GL 2 ( W ) b_i\in\operatorname{GL}_2(W) b i ∈ GL 2 ( W ) r = 1 r=1 r = 1 ( M , F ) ≅ ( W 2 , A σ ) ⊗ ( W 4 , σ ) (M,F)\cong(W^2,A\sigma)\otimes(W^4,\sigma) ( M , F ) ≅ ( W 2 , A σ ) ⊗ ( W 4 , σ ) E 0 4 E_0^4 E 0 4 p = p 1 p 2 p=p_1p_2 p = p 1 p 2 p 1 \mathfrak{p}_1 p 1 p 2 \mathfrak{p}_2 p 2 p 1 \mathfrak{p}_1 p 1 ( W 2 ⊗ W 2 , ( A ⊗ 1 ) σ ′ ) (W^2 \otimes W^2,(A \otimes1)\sigma') ( W 2 ⊗ W 2 , ( A ⊗ 1 ) σ ′ ) σ ′ ( x ⊗ y ) = σ y ⊗ σ x \sigma'(x \otimes y)=\sigma y \otimes\sigma x σ ′ ( x ⊗ y ) = σ y ⊗ σ x r = 2 r=2 r = 2 ( M , F ) (M,F) ( M , F ) § 2.2 , and the exact computation of Appendix § 10 gives T 4 = S T^4=S T 4 = S T 4 ( X × X ) T^4(X \times X) T 4 ( X × X ) S ( X × X ) S(X \times X) S ( X × X ) □ \square □
Remark 8.5. By (5) , G p \mathcal{G}_p G p p p p G p ≅ E 0 [ p ∞ ] × H \mathcal{G}_p \cong E_0[p^\infty] \times\mathcal{H} G p ≅ E 0 [ p ∞ ] × H H \mathcal{H} H y y y F 3 y = V 3 y F^3y=V^3y F 3 y = V 3 y a a a
Remark 8.6. For a very general complex fiber X C X_{\mathbb{C}} X C H 4 ( X C × X C , Q ) H^4(X_{\mathbb{C}} \times X_{\mathbb{C}},\mathbb{Q}) H 4 ( X C × X C , Q ) [6 ] . Modulo that span they form a space of dimension 2 in the Künneth ( 2 , 2 ) (2,2) ( 2 , 2 ) Q \mathbb{Q} Q SL 2 \operatorname{SL}_2 SL 2 [6 ] , which by Mumford’s construction acts on H 1 H^1 H 1 U 1 ⊗ U 2 ⊗ U 3 U_1 \otimes U_2 \otimes U_3 U 1 ⊗ U 2 ⊗ U 3 § 1 ), and ⋀ 2 ( U 1 ⊗ U 2 ⊗ U 3 ) \bigwedge^2(U_1 \otimes U_2 \otimes U_3) ⋀ 2 ( U 1 ⊗ U 2 ⊗ U 3 ) Sym 2 U i ⊗ Sym 2 U j \operatorname{Sym}^2U_i \otimes\operatorname{Sym}^2U_j Sym 2 U i ⊗ Sym 2 U j ( 2 , 2 ) (2,2) ( 2 , 2 ) H 4 H^4 H 4 N δ N\delta N δ δ \delta δ p ∤ N p\nmid N p ∤ N δ t \delta_t δ t t t t p p p δ t \delta_t δ t δ \delta δ δ \delta δ δ t \delta_t δ t Proposition 8.4 show that, under Hypothesis 7.1, no such δ t \delta_t δ t ( 2 , 2 ) (2,2) ( 2 , 2 ) T 4 ( X t × X t ) T^4(X_t \times X_t) T 4 ( X t × X t ) δ \delta δ
Other supersingular Dieudonné modules
A monomial supersingular Dieudonné module of height 8 and dimension 4 is ( M w , F ) (M_w,F) ( M w , F ) w w w M w M_w M w M v M_v M v v v v w w w M v M_v M v e i e_i e i i ∈ Z / ∣ v ∣ i \in\mathbb{Z}/|v| i ∈ Z /∣ v ∣ F e i = p a ( i ) e i + 1 Fe_i=p^{a(i)}e_{i+1} F e i = p a ( i ) e i + 1 a ( 0 ) , a ( 1 ) , … a(0),a(1),\ldots a ( 0 ) , a ( 1 ) , … v v v E 0 4 E_0^4 E 0 4 G p \mathcal{G}_p G p M w M_w M w E 0 4 E_0^4 E 0 4 v v v 2 m 2m 2 m ( e i ′ ) (e'_i) ( e i ′ ) 2 m 2m 2 m M ( E 0 m ) M(E_0^m) M ( E 0 m ) e i ↦ p c i e i ′ e_i \mapsto p^{c_i}e'_i e i ↦ p c i e i ′ c i + 1 = c i + 1 − ( − 1 ) i 2 − a ( i ) c_{i+1}=c_i+\frac{1-(-1)^i}{2}-a(i) c i + 1 = c i + 2 1 − ( − 1 ) i − a ( i ) v v v m m m F F F V V V M w M_w M w M ( E 0 4 ) M(E_0^4) M ( E 0 4 ) F F F V V V M ( E 0 4 / H ) M(E_0^4/H) M ( E 0 4 / H ) H ⊂ E 0 4 [ p ∞ ] H\subset E_0^4[p^\infty] H ⊂ E 0 4 [ p ∞ ] [17 Corollary 3.16] ). § 2.2 applies to it.
Different cycle decompositions can give isomorphic modules. Let v v v M M M v v ⋯ v vv\cdots v v v ⋯ v m ∣ v ∣ m|v| m ∣ v ∣ e i e_i e i i ∈ Z / m ∣ v ∣ i \in\mathbb{Z}/m|v| i ∈ Z / m ∣ v ∣ φ v = p − ∣ v ∣ / 2 F ∣ v ∣ \varphi_v=p^{-|v|/2}F^{|v|} φ v = p − ∣ v ∣/2 F ∣ v ∣ σ ∣ v ∣ \sigma^{|v|} σ ∣ v ∣ M 0 = ⨁ 0 ≤ j < m W e j ∣ v ∣ M_0=\bigoplus_{0\leq j<m}W e_{j|v|} M 0 = ⨁ 0 ≤ j < m W e j ∣ v ∣ M 0 M_0 M 0 e 1 ′ , … , e m ′ e'_1,\ldots,e'_m e 1 ′ , … , e m ′ φ v \varphi_v φ v p − ( a ( 0 ) + ⋯ + a ( i − 1 ) ) F i e j ′ p^{-(a(0)+\cdots+a(i-1))}F^ie'_j p − ( a ( 0 ) + ⋯ + a ( i − 1 )) F i e j ′ 0 ≤ i < ∣ v ∣ 0\leq i<|v| 0 ≤ i < ∣ v ∣ 1 ≤ j ≤ m 1\leq j\leq m 1 ≤ j ≤ m F F F M v ⊕ m M_v^{\oplus m} M v ⊕ m p p p p p p ker F = im V \ker F = \operatorname{im} V ker F = im V p p p p p p [18 Section 9] ). So the twenty cycle decompositions form fourteen isomorphism classes.
For a supersingular abelian fourfold X X X F ‾ p \overline{\mathbb{F}}_p F p P \mathcal{P} P X × X ∨ X \times X^\vee X × X ∨
S ^ ( X ) = Z p -span of 1 24 pr X ∗ ( pr X ∨ ∗ y ⋅ c 1 ( P ) 4 ) , y ∈ S ( X ∨ ) . (7) \widehat{S}(X)=\mathbb{Z}_p\text{-span of }\frac{1}{24}\operatorname{pr}_{X*}\left(\operatorname{pr}_{X^\vee}^{*}y\cdot c_1(\mathcal{P})^4\right),\qquad y\in S(X^\vee). \tag*{(7)} S ( X ) = Z p -span of 24 1 pr X ∗ ( pr X ∨ ∗ y ⋅ c 1 ( P ) 4 ) , y ∈ S ( X ∨ ) . (7) This is the degree-4 part of the cohomological Fourier transform of S ( X ∨ ) S(X^\vee) S ( X ∨ ) [41 ] ).
Lemma 9.2. For p ≥ 5 p \ge5 p ≥ 5 S ^ ( X ) ⊆ Alg 2 ( X ) \widehat{S}(X) \subseteq\operatorname{Alg}^2(X) S ( X ) ⊆ Alg 2 ( X )
Proof. By (R3), S ( X ∨ ) ⊆ Alg 2 ( X ∨ ) S(X^\vee) \subseteq\operatorname{Alg}^2(X^\vee) S ( X ∨ ) ⊆ Alg 2 ( X ∨ ) c 1 ( P ) c_1(\mathcal{P}) c 1 ( P ) Z p \mathbb{Z}_p Z p □ \square □
The lattice S ^ ( X ) \widehat{S}(X) S ( X ) ( M ( X ) , F ) (M(X),F) ( M ( X ) , F ) ( e i ) (e_i) ( e i ) M ( X ) M(X) M ( X ) c 1 ( P ) = ∑ i e i ⊗ e i c_1(\mathcal{P})=\sum_i e_i\otimes e^i c 1 ( P ) = ∑ i e i ⊗ e i e i ∈ M ( X ∨ ) e^i\in M(X^\vee) e i ∈ M ( X ∨ ) P \mathcal{P} P X × 0 X\times0 X × 0 0 × X ∨ 0\times X^\vee 0 × X ∨ c 1 ( P ) c_1(\mathcal{P}) c 1 ( P ) ( 2 , 0 ) (2,0) ( 2 , 0 ) ( 0 , 2 ) (0,2) ( 0 , 2 ) ∫ c 1 ( P ) 8 \int c_1(\mathcal{P})^8 ∫ c 1 ( P ) 8 ± 8 ! \pm8! ± 8 ! W W W e i e^i e i M ( X ∨ ) M(X^\vee) M ( X ∨ ) X × X ∨ X\times X^\vee X × X ∨ deg ( c 1 ( P ) 8 ) = 8 ! χ ( P ) = ± 8 ! \deg(c_1(\mathcal{P})^8)=8!\chi(\mathcal{P})=\pm8! deg ( c 1 ( P ) 8 ) = 8 ! χ ( P ) = ± 8 ! χ ( P ) 2 \chi(\mathcal{P})^2 χ ( P ) 2 X × X ∨ → X ∨ × X ∨ ∨ X\times X^\vee\to X^\vee\times X^{\vee\vee} X × X ∨ → X ∨ × X ∨∨ P \mathcal{P} P ( x , x ∨ ) ↦ ( x ∨ , x ) (x,x^\vee)\mapsto(x^\vee,x) ( x , x ∨ ) ↦ ( x ∨ , x ) x x x X ∨ ∨ X^{\vee\vee} X ∨∨ [1 Chapter I, Theorems 11.1 and 8.9] ). So this determinant is a unit, the e i e^i e i M ( X ∨ ) M(X^\vee) M ( X ∨ ) c 1 ( P ) n / n ! = ∑ ∣ I ∣ = n ± e I ⊗ e I c_1(\mathcal{P})^n/n!=\sum_{|I|=n}\pm e_I\otimes e^I c 1 ( P ) n / n ! = ∑ ∣ I ∣ = n ± e I ⊗ e I n n n F ( y ) = pr X ∗ ( pr X ∨ ∗ y ⋅ c 1 ( P ) 8 − n / ( 8 − n ) ! ) \mathcal{F}(y)=\operatorname{pr}_{X*}(\operatorname{pr}_{X^\vee}^{*}y\cdot c_1(\mathcal{P})^{8-n}/(8-n)!) F ( y ) = pr X ∗ ( pr X ∨ ∗ y ⋅ c 1 ( P ) 8 − n / ( 8 − n )!) e J e_J e J ∣ J ∣ = n |J|=n ∣ J ∣ = n e J c e_{J^c} e J c J c J^c J c J J J ⋀ n M ( X ∨ ) → ⋀ 8 − n M ( X ) \bigwedge^n M(X^\vee)\to\bigwedge^{8-n}M(X) ⋀ n M ( X ∨ ) → ⋀ 8 − n M ( X ) c 1 ( P ) c_1(\mathcal{P}) c 1 ( P ) F c 1 ( P ) = p c 1 ( P ) \mathcal{F}c_1(\mathcal{P})=pc_1(\mathcal{P}) F c 1 ( P ) = p c 1 ( P ) ∫ F x = p 4 ∫ x \int\mathcal{F}x=p^4\int x ∫ F x = p 4 ∫ x X ∨ X^\vee X ∨ F F ~ ( y ) = p 4 − n F ( F y ) \mathcal{F}\widetilde{\mathcal{F}}(y)=p^{4-n}\mathcal{F}(\mathcal{F}y) F F ( y ) = p 4 − n F ( F y ) F \mathcal{F} F T n ( X ∨ ) T^n(X^\vee) T n ( X ∨ ) T 8 − n ( X ) T^{8-n}(X) T 8 − n ( X ) ( e i ) (e_i) ( e i ) n = 2 n=2 n = 2 T 2 ( X ∨ ) ⊗ W T^2(X^\vee)\otimes W T 2 ( X ∨ ) ⊗ W ( e J ) (e^J) ( e J ) T 6 ( X ) T^6(X) T 6 ( X ) e J c e_{J^c} e J c S ^ ( X ) ⊗ W \widehat{S}(X)\otimes W S ( X ) ⊗ W S ( X ∨ ) ⊗ W S(X^\vee)\otimes W S ( X ∨ ) ⊗ W e J e_J e J ∣ J ∣ = 4 |J|=4 ∣ J ∣ = 4 ν T ( J ′ ) + ν T ( J ′ ′ ) \nu_T(J')+\nu_T(J'') ν T ( J ′ ) + ν T ( J ′′ ) J ′ J' J ′ J ′ ′ J'' J ′′ J ′ ∩ J ′ ′ = J J'\cap J''=J J ′ ∩ J ′′ = J J ′ ∪ J ′ ′ J'\cup J'' J ′ ∪ J ′′ ν T \nu_T ν T T 6 ( X ) T^6(X) T 6 ( X )
Proposition 9.3 (computer-assisted). Let p ≥ 5 p \ge5 p ≥ 5 [ T 4 : S ] [T^4:S] [ T 4 : S ] [ T 4 : S + S ^ ] [T^4:S+\widehat{S}] [ T 4 : S + S ] Table 1 . Consequently the integral crystalline Tate conjecture holds in codimension 2, without Hypothesis 7.1, for every abelian fourfold over F ‾ p \overline{\mathbb{F}}_p F p S + S ^ = T 4 S+\widehat{S}=T^4 S + S = T 4 01 + 000111 01+000111 01 + 000111 Theorem 1.3 ). For the remaining four modules the index [ T 4 : S + S ^ ] [T^4:S+\widehat{S}] [ T 4 : S + S ] S + S ^ S+\widehat{S} S + S T 4 T^4 T 4 S + S ^ = T 4 S+\widehat{S}=T^4 S + S = T 4 T 4 T^4 T 4 p p p G p \mathcal{G}_p G p
Table 1. The twenty cycle decompositions of monomial supersingular Dieudonné modules of height 8 and dimension 4, named by their words. They form fourteen isomorphism classes: the five decompositions of the first row give isomorphic modules, as do 01 + 01 + 0011 and 0011 + 0101, and 0011 + 0011 and 00110011. “Self-dual” means isomorphic to the dual module, whose words are the complementary words, with 0 and 1 exchanged; it is necessary for the existence of a polarization of degree prime to p p p X [ p ∞ ] ≅ X ∨ [ p ∞ ] X[p^\infty] \cong X^\vee[p^\infty] X [ p ∞ ] ≅ X ∨ [ p ∞ ] M ( X ∨ ) M(X^\vee) M ( X ∨ ) [17 Definition 3.6 and Proposition 3.23] . A row naming two modules that are not self-dual lists a module and its dual, which are not isomorphic but have the same a a a [ T 4 : S + S ^ ] [T^4:S+\widehat{S}] [ T 4 : S + S ] [ T 4 : S ] [T^4:S] [ T 4 : S ] p p p p ≥ 5 p \ge5 p ≥ 5
Proof. By Section 2.2 and the description of S ^ \widehat{S} S f f f ⋀ 4 \bigwedge^4 ⋀ 4 4 × 4 4\times4 4 × 4 f ∗ f^* f ∗ 4 ! 4! 4 ! S ⊆ Alg 2 S\subseteq\operatorname{Alg}^2 S ⊆ Alg 2 T 4 T^4 T 4 S S S □ \square □
Remark 9.4. For each of the four open modules, the exact computation shows that one lattice of integral Weil classes of an inert imaginary quadratic action of signature ( 2 , 2 ) (2,2) ( 2 , 2 ) T 4 T^4 T 4 S S S a a a S + S ^ S+\widehat{S} S + S T 4 T^4 T 4 p p p p p p p p p p 26 p^{26} p 26
Appendix A. The computations
All computations are exact, with integers or, for the check of Lemma 7.4, in Q ( − d ) \mathbb{Q}(\sqrt{-d}) Q ( − d ) § 2.2 , every lattice in Propositions 4.3, 5.1 and 9.3 and in the proof of Proposition 8.4 is monomial over W W W ⋀ N M \bigwedge^N M ⋀ N M End ( M ) \operatorname{End}(M) End ( M ) ⋀ 4 \bigwedge^4 ⋀ 4 a ( s ) a(s) a ( s ) p p p p ≥ 5 p \ge5 p ≥ 5 4 ! 4! 4 !
Proposition 4.3. For the model (4.1), the program lists the two π \pi π ν T ( J ) \nu_T(J) ν T ( J ) J J J ν S ( J ) \nu_S(J) ν S ( J ) J J J T 6 T^6 T 6 F F F ∣ J ∩ { s 1 = 1 } ∣ \lvert J \cap\{s_1 = 1\}\rvert ∣ J ∩ { s 1 = 1 }∣
Proposition 5.1. The program computes the exponents of End ( M , F ) ⊗ W \operatorname{End}(M,F) \otimes W End ( M , F ) ⊗ W D ( U 1 , … , U 4 ) ( e J ) D(U_1,\ldots,U_4)(e_J) D ( U 1 , … , U 4 ) ( e J ) ( U 1 , U 2 ) (U_1,U_2) ( U 1 , U 2 ) ( 2 , 2 ) (2,2) ( 2 , 2 ) ⋀ 4 ( M ⊕ M ) \bigwedge^4(M \oplus M) ⋀ 4 ( M ⊕ M )
Check of Lemma 7.4. For d = 4 , 6 , 8 , 10 , 12 , 28 d = 4,6,8,10,12,28 d = 4 , 6 , 8 , 10 , 12 , 28 Q ( − d ) \mathbb{Q}(\sqrt{-d}) Q ( − d ) ch ( I 1 ) ⊠ ch ( I 1 ) \operatorname{ch}(\mathcal{I}_1) \boxtimes\operatorname{ch}(\mathcal{I}_1) ch ( I 1 ) ⊠ ch ( I 1 ) H ∙ ( X 3 × X 3 ) H^\bullet(X_3 \times X_3) H ∙ ( X 3 × X 3 ) H ∙ ( X 3 × X 3 ∨ ) H^\bullet(X_3 \times X_3^\vee) H ∙ ( X 3 × X 3 ∨ ) exp ( − c 1 / rk ) \exp(-c_1/\operatorname{rk}) exp ( − c 1 / rk ) Γ 1 \Gamma_1 Γ 1 ρ 1 ⊠ ρ 2 \rho_1 \boxtimes\rho_2 ρ 1 ⊠ ρ 2 κ 3 \kappa_3 κ 3 K d K_d K d
Proposition 8.4. The same computations for the two models of the proof give [ T 4 : S ] = 1 [T^4:S] = 1 [ T 4 : S ] = 1 ( 2 , 2 ) (2,2) ( 2 , 2 ) T 4 T^4 T 4
Proposition 9.3. For each of the twenty cycle decompositions the program computes T 4 T^4 T 4 S S S S ^ \widehat{S} S 4 × 4 4 \times4 4 × 4 T 4 T^4 T 4 S + S ^ = T 4 S + \widehat{S} = T^4 S + S = T 4 T 4 T^4 T 4 E 0 4 E_0^4 E 0 4 G p \mathcal{G}_p G p ( 2 , 2 ) (2,2) ( 2 , 2 ) S S S S + S ^ S + \widehat{S} S + S
