BEILINSON-BLOCH-KATO CONJECTURE FOR
BEILINSON-BLOCH-KATO CONJECTURE FOR RANKIN-SELBERG MOTIVES II YICHAO TIAN
This is the second part of a report on my going project joint with Yifeng Liu, Liang Xiao, Wei Zhang and Xinwen Zhu. In the previous talk by Yifeng Liu, he explains the general framework of the problem and state a theorem on some new cases of Beilinson-Bloch-Kato conjecture on the relation between special values of L-functions and the Selmer groups attached to the Rankin-Selberg motives for GLn × GLn+1 over a CM field. In this talk, I will focus on the case n = 2, give a more precise statement of our main theorem in this case and outline the main ideas of the proof. 1. Statement of the main theorem Let F + be a totally real field, F0 be an imaginary quadratic field and F = F0 F + . Denote by c the complex conjugate of F/F + . For n = 2, 3, we consider a regular algebraic conjugate selfdual cuspidal automorphic (RACSDCA) representation Πn of GLn (AF ) such that the infinitesimal character of Πn,∞ is trivial. Let E be a number field where Π∞ n is defined. We fix a finite place λ of E, and put O = OEλ . Let $ be a uniformizor of O, and kλ = O/$. Let ` denote the characteristic of kλ . Then by the work of Clozel, Harris, Taylor, Shin et al., one has a continuous `-adic Galois representation ρn : ΓF := Gal(F /F ) → GLn (Eλ ) attached to Πn , which is unramified outside the finite set of finite places of F where Πn is not spherical. We take the geometric normalization for ρn so that ρcn ∼ = ρ∨ n (1 − n). Let ρ¯n : ΓF → GLn (kλ ) be the associated residue Galois representation, which is well-defined up to semi-simplification.1 We will impose the following assumptions for our main theorem: Assumption 1.1. We will impose the following assumptions on ρn (1) ` ≥ 11 is unramified in F . (2) ρ¯2 and ρ¯3 are absolutely irreducible. (3) If ` denotes the mod-` cyclotomoic character, the image of (¯ ρ2 , ρ¯3 , ` ) : ΓF → GL2 (kλ ) × GL3 (kλ ) × F× ` contains an element of the form 0 0 µ1 0 λ1 , 0 µ2 0 , ξ λ2 0 µ3 0 0 −1 −1 ±1 where λi , µj ∈ kλ× satisfying λ2 λ−1 1 = ξ, µ1 6= µ3 and λ2 λ1 µ3 µ1 6= ξ . (4) The image of ρ¯c2 ⊗ ρ¯c3 (2) contains a non-trivial scalar. 1Later on, we will assume that ρ¯ is absolutely irreducible so that there is no ambiguity. n 1
(5) ρ¯3 satisfies Caraiani-Scholze’s decomposed generic condition, i.e. there exists a rational prime p that splits completely in F such that ρ¯3 is unramified at every place v|p of F , and if α1 , · · · , αn are the eigenvalues of ρ¯3 (Frv ), then αi /αj ∈ / {1, p} for all i 6= j. Here, Frv ∈ ΓF denotes a geometric Frobenius element at v. (6) The spherical Hecke eigensystem for Π2 and Π3 are modulo λ-isolated. Our main theorem for n = 2 can be stated as follows. Theorem 1.2. Assume that L( 21 , Π2 × Π3 ) 6= 0. Then for a finite prime λ of E satisfying Assumption 1.1, then Hf1 (F, ρc2 ⊗ ρc3 (2)) = 0. Example 1.3. Let A and B are two modular elliptic curves over F + without complex multiplication. Assume that A and B are not isogenous to each other. We take E = Q and ρ2 to be the `-adic Galois representation over F on the Tate module of A, and ρ3 to be the symmetric square of the `-adic Galois representation on the Tate module of B. Then using a famous result of Serre, Assumptions 1.1 are verified for all but finitely many primes `. (m)
Put ρ = ρ2 ⊗ ρ3 (2). For every integer m ≥ 1, let ρ¯n denote the reduction modulo $m of (m) a ΓF -stable lattice in ρn with n = 2, 3. By Assumption 1.1(2), ρn is well-defined. We put (m) (m) ρ¯(m) := ρ¯2 ⊗ ρ¯3 (2). For a finite place v of F , let 1 Hsing (Fv , ρ¯(m) ) := H 1 (Iv , ρ¯(m) )Gal(kv /kv )
denote the singular part of the local Galois cohomology, where Iv is the inertia subgroup of Gal(F v /Fv ). We denote by locv : H 1 (F, ρ¯(m) ) → H 1 (Fv , ρ¯(m) ) the restriction map induced by a fixed embedding F ,→ F v , and by 1 ∂ : H 1 (Fv , ρ¯(m) ) → Hsing (Fv , ρ¯(m) )
be the natural projection. The proof of Theorem 1.2 follows from the Kolyvagin’s strategy on the finiteness of ShafarevichTate groups for elliptic curves over Q. Similar ideas have been used by Bertonili-Darmon on their work on the anti-cyclotomic Iwasawa conjecture for elliptic curves over Q, and also by Wei Zhang on his proof of Kolyvagin conjecture. We shall construct a Kolyvagin system 1 (m) {Θ(v) )|m ≥ 1, v is m-admissible} m ∈ H (F, ρ (v)
indexed by certain (infinite set of) m-admissible places of F such that their images ∂locv (Θm ) ∈ 1 (F , ρ (m) ) are related to the Gan-Gross-Prasad period of Π × Π ; in particular, we show that Hsing v ¯ 2 3 (v)
∂locv (Θm ) 6= 0 for sufficiently large m ≥ 1 if L(1/2, Π2 × Π3 ) 6= 0. Once those properties are established, Theorem 1.2 follows easily from the general Kolyvagin’s machinery. 2. A Kolyvagin system and sketch of proofs (v)
In this section, we give more details on the construction of the cohomology classes Θm and their properties. We introduce first some notation. As explained in Yifeng Liu [Liu18], the global Gan-Gross-Prasad conjecture implies that there exists two unique pairs (V, π2 ) and (W, π3 ) where • V and W are totally positive definite Hermitian spaces over F/F + such that dimF (V ) = 2 and W = V ⊕ F u, where u ∈ W is a vector with (u, u) = 1, • π2 and π3 are respectively cuspidal automorphic representations on U (V )(AF + ) and U (W )(AF + ) such that the base change of πn to GLn (AF ) is Πn for n = 2, 3. 2
∞ Let KV ⊆ U (V )(A∞ F + ) and KW ⊆ U (W )(AF + ) be sufficiently small open compact subgroups such that (π2∞ )KV 6= 0, (π3∞ )KW 6= 0 and KV = KW ∩ U (V )(AF ), where we regard U (V ) as a subgroup of U (W ). Let Σ be the finite set of places of F such that KV and KW are hyperspecial at every Σ∪{v} Σ∪{v} Σ∪{v} finite place w ∈ / Σ. For a place v ∈ / Σ, let TvV = C (KV \U (V )(AF + )/KV , OEλ ) denote the spherical Hecke algebra away from v for U (V ). Similarly, one has TvW with V replaced by W , and we put Tv := TvV ⊗ TvW . Then the (modulo λ) Hecke eigensystem of π2 (resp. π3 ) determines a maximal ideal mπ2 ⊆ TvV (resp. mπ3 ⊆ TvW ). Let m denote the maximal ideal of Tv corresponding to π2 × π3 .
Definition 2.1. Let m ≥ 1 be an integer. A finite place v of F is called m-admissible if (1) the underlying rational prime p of v is unramified in F + and p mod ` avoid some explicit elements in F` (in particular, we assume p2 6≡ 1 mod `); (2) v ∈ / Σ and it is inert over some place v + of F + of degree 1, i.e. Fv++ ∼ = Qp2 , = Qp and Fv ∼ (3) if Frv ∈ ΓF denotes a geometric Frobenius α 1 0 (m) and ρ¯3 (Frv ) is conjugate to 0 2 0 p 0
(m)
element atv, then ρ¯2 (Frv ) is conjugate to 0 0 p2 0 with α ∈ / {1, p2 , p4 }. −1 4 0 α p
Remark 2.2. (a) Note that condition (3) above implies that 1 and p−2 appear as eigenvalues of ρ¯(m) (Frv ) with multiplicity one. (b) The Assumption 1.1(3) ensures that there exist infinitely many m-admisssible places for every integer m ≥ 1. (m) ) is free of rank one over 1 (F , ρ (c) An easy computation of Galois cohomology shows that Hsing v ¯ O/$m for an m-admissible place v. Fix an integer m ≥ 1 and an m-admissible place v. Note that V and W are quasi-split at v 2. Then there exists a unique nearby Hermitian space V (v) such that is isomorphic to V ⊗F A∞,v • V (v) ⊗F A∞,v F , F • V (v) ⊗Q R has signatures (1, 1) × (2, 0) × · · · × (2, 0). • V (v) ⊗F Fv is not quasi-split. Put W (v) = V (v) ⊕ F u. Let H (v) = ResF + /Q U (V (v) ) (resp. G(v) = ResF + /Q U (W (v) )) be the unitary group attached to V (v) (resp. W (v) ). We have thus the Shimura varieties X (v) (resp. Y (v) ) attached to H (v) (resp. G(v) ) defined over F of dimension 1 (resp. 2). The levels away from v of X (v) and Y (v) at v are the same as KV and KW , and their levels at v are defined by the stablizers of some non-self-dual lattices, hence they are non-hyperspecial at v. The diagonal embedding of unitary groups H (v) ,→ H (v) × G(v) induces an embedding of Shimura varieties ∆ : X (v) ,→ X (v) ×F Y (v) of codimenion 2. The arithmetic ´etale fundamental group H´e4t (X (v) ×F Y (v) , O/$m (2)) is equipped with an action by the spherical Hecke algebra Tv so that it makes sense to talk about the localization H´e4t (X (v) ×F Y (v) , O/$m (2))m . We consider now the cycle class map cl : Ch2 (X (v) ×F Y (v) ) → H´e4t (X (v) ×F Y (v) , O/$m (2))m . 2A Hermitian space for the unramified quadratic extension of a non-archimedean local filed K is called quasi-split
if it admits a self-dual lattice. 3
By Caraiani-Scholze’s result, one has H´eit ((X (v) ×F Y (v) )F , O/$m (2))m = 0 for i 6= 3. We get thus an Abel-Jacobi map 2 (v) AJ(v) ×F Y (v) ) → H 1 (F, H´e3t ((X (v) ×F Y (v) )F , O/$m )m (2)). m : Ch (X (v)
We denote by AJm ([X (v) ]) the image of the diagonal cycle [∆(X (v) )] ∈ Ch2 (X (v) ×F Y (v) ). Since the Hecke eigensystem of π2 and π3 are λ-isolated by Assumption 1.1(6), we see that (v) (v) H´e3t ((X (v) ×F Y (v) )F , O/$m (2))m ∼ = H´e1t (XF , O/$m (1))mπ2 ⊗ H´e2t (YF , O/$m (2))mπ3 (v)
is nothing but a finite copy of ρ¯(m) . The desired cohomology class Θm is then the projection of (v) AJm ([X (v) ]) onto one copy of H 1 (F, ρ¯(m) ) using certain Hecke operators. Remark 2.3. We should warn the reader on a technical subtlety. Strictly speaking, the Shimura varieties attached to H (v) and G(v) are of abelian type and we still do not have a satisfactory theory of integral models for such Shimura varieties at non-hyperspecial places. So we need to actually work with certain variants of the unitary groups H (v) and G(v) and their associated Shimura varieties. Let T0 = {x ∈ ResF/Q Gm,F |NF/F + (x) ∈ Gm,Q } e (v) = T0 × H (v) . Then it is shown by Rapoport, Smithling and Zhang [RSZ17] that the and put H e (v) , say X e (v) , has a nice moduli interpretation, and hence one can Shimura variety attached to H easily describe its integral model over OFv . Similarly, we have the Shimura variety Ye (v) attached e (v) := T0 × G(v) . We need to work with X e (v) and Ye (v) instead of X (v) and Y (v) . In this note, to G we ignore this technical subtlety. Theorem 2.4. Under the assumptions above, the following statements hold: (1) There exists a canonical isomorphism 1 Φ : Hsing (Fv , H´e3t ((X (v) ×F Y (v) )F , O/$m (2))m ) ∼
− → H 0 (ShKV (U (V )), O/$m )mπ2 ⊗ H 0 (ShKW (U (W )), O/$m )mπ3 , where ShKV (U (V )) denotes the discrete Shimura set U (V )(F )\U (V )(A∞ F )/KV , and similar meaning for ShKW (U (W )). (2) Let h•, •i denote the canonical bilinear pairing on H 0 (ShKV (U (V )), O/$m )mπ2 ⊗ H 0 (ShKW (U (W )), O/$m )mπ3 . Then there exists an explicit non-zero constant c such that for any f ∈ H 0 (ShKV (U (V )), O/$m )mπ2 ⊗ H 0 (ShKW (U (W )), O/$m )mπ3 , one has X (v) hΦ∂locv (AJ(v) ])), f i = c f (ι(x)), m ([X x∈ShKV (U (V ))
where c is some explicit non-zero constant in O, and ι : ShKV (U (V )) → ShKV (U (V )) × ShKV (U (V ))×ShKW (U (W )) is the diagonal embedding induced by the natural map U (V ) → U (W ). Note the right hand side of the formula in (2) is exactly the Gan-Gross-Prasad period of f , which is non-vanishing for some test vector f under the assumption that L( 12 , Π2 × Π3 ) 6= 0 by the global Gan-Gross-Prasad conjecture, proved recently by Wei Zhang []. Once this Theorem is established, Theorem 1.2 follows from Kolyvagin’s method of bounding Selmer groups using Kolyvagin system. We indicate how to prove Theorem 2.4. Statement (1) is reduced to establishing the following two isomorphisms 4
(v)
1 (F , H 1 (X (a) Φ2 : Hsing , O/$m (2))mπ2 ) ∼ = H 0 (ShKV (U (V )), O/$m )mπ2 . v ´ et Fv Γ (v) (b) Φ3 : H 2 (Y , O/$(1))mFv ∼ = H 0 (ShK (U (W )), O/$m )m π3
Fv
π3
W
Statement (2) will become clear when we understand how the isomorphism Φ in (1) is established. Part (a) is usually called the arithmetic level raising for U (V ). Note that the automorphic cuspidal representation π2 is hyperspecial at v + , while H (v) is not quasi-split at v + . Hence, if we consider the Eλ -coefficient (instead of O/$m -coefficient) ´etale cohomology of X (v) , the its localization at mπ2 should be 0. However, since the right hand side of Φ2 is non-zero, the existence of the isomorphism Φ2 shows that there exist an automorphic cuspidal representation π20 of H (v) , which should correspond to cuspidal automorphic representations of U (V ) with higher level at v via Jacquet-Langlands transfer for unitary groups, such that its spherical Hecke eigensystems away from v are exactly the same as those of π2 modulo $m . To show (a), we need to use the reduction of a certain integral model X (v) of X (v) over OFv (v) at v 3. It turns out that X (v) has semi-stable reduction, and its (geometric) special fiber XF is a p union X1 ∪ X2 , where • X1 is a P1 -bundle over the discrete Shimura set ShKV (U (V )), • X2 is a P1 -bundle over ShKVv Kv0 (U (V )), where KVv is the prime-to-v part of KV = KVv Kv , and Kv0 is another hyperspecial subgroup of U (V )(Fv++ ) such that Iwv = Kv0 ∩ Kv is an Iwahoric subgroup. The intersection X1 ∩ X2 is then exactly isomorphic to ShKVv Iwv (U (V )). Consider the composite map β
α
H 0 (X1 )mπ2 ⊕ H 0 (X2 )mπ2 − → H 0 (X1 ∩ X2 )mπ2 − → H 2 (X1 )mπ2 (1) ⊕ H 2 (X2 )mπ2 (1). Here, all the cohomology groups are ´etale cohomology with coefficients in O/$m (together with suitable Tate twists), α is given by natural restrictions and β is given by the Gysin maps. Now, statement (a) follows from the following facts: • Using the spectral sequence of nearby cycles, one can show that (v) 1 Coker(β ◦ α) ∼ (Fv , H´e1t (XF , O/$m (2))mπ2 ). = Hsing v
• On the other hand, one can give an explicit expression for β ◦ α in terms of Hecke operators at v of the group U (V ). Using this, one shows easily that the canonical map H 0 (ShK (U (V )))m ∼ = H 2 (X1 )m (1) → Coker(β ◦ α) V
π2
π2
is an isomorphism. To establish the isomorphism Φ3 in part (b), we use the reduction of an integral model4 Y (v) over OFv of Y (v) . The geometry for Y (v) is similar to X (v) . It has semi-stable reduction, and the (v) (geometric) special fiber YF is a union Y1 ∩ Y2 , where p
• Y1 is a fibration of P2 over ShKW (U (W )), • Y2 is smooth that intersect Y1 transversally, and Y1 ∩ Y2 is a fibration of Fermat curves of equation xp+1 + xp+1 + xp+1 = 0 over ShKW (U (W )). 1 2 3 Even the global geometry of Y2 is complicated, its basic/supersingular locus, denoted by Y2ss , is easy to describe: it is the union of Y1 ∩ Y2 and another stratum Z2 , which is a collection of P1 indexed ∞,v v K s (U (W )). Here, K v ⊆ U (W )(A + ) is the prime-to-v part of KW , and K s is a maximal by ShKW v W v F 3As in Remark 2.3, we need to actually work with a variant X e (v) of X (v) to get an integral model over OFv with
some moduli interpretations. 4The same remark as the previous footnote. See Remark 2.3. 5
special subgroup of U (W )(Fv++ ), the stablizer of a nearly self-lattice lattice in W ⊗F Fv . The dual graph attached to Y2ss is the quotient of the Bruhat-Tits tree for U (W )(FF++ ) by a discrete subgroup. Using the spectral sequence of nearby cycles for Y (v) , assertion (b) is reduced to computing the following composed Gysin-restriction map (called the intersection matrix of Y1 ∩ Y2 and Z2 ): Gysin
Restriction
H 0 (Y1 ∩ Y2 )mπ3 ⊕ H 0 (Z2 )mπ3 −−−−→ H 2 (Y2 )mπ3 (1) −−−−−−−→ H 2 (Y1 ∩ Y2 )mπ3 (1) ⊕ H 2 (Z2 )mπ3 (1). As in the case for X (v) , an explicit computation shows that the above composition map is nondegenerate under condition (3) of Definition 2.1. It follows that the Gysin map Gysin
H 0 (Y1 ∩ Y2 )mπ3 ⊕ H 0 (Z2 )mπ3 −−−−→ H 2 (Y2 )mπ3 (1) induces an isomorphism between H 0 (Y1 ∩ Y2 )mπ3 ⊕ H 0 (Z2 )mπ3 and the Galois invariant subgroup H 2 (Y2 )mπ3 (1)Gal(Fp /Fp2 ) , that is, all the Tate cycles in H 2 (Y2 )mπ3 (1) come from the cycle classes of basic/supersingular locus of Y2 . This is a variant of the generic Tate conjecture for the reduction of Shimura varieties at a bad place. The corresponding statement for the good reduction of unitary Shimura varieties was established recently in the joint work of Liang Xiao and Xinwen Zhu [XZ17+ ]. References [Liu18] Y. Liu, Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives I, this volume. [RSZ17] M. Rapoport, B. Smithling, W. Zhang, Arithmetic diagonal cycles on unitary Shimura varieties, arXiv:1710.06962v2. [XZ17+ ] L. Xiao and X. Zhu, Tate cycles on Shimura varieties via geometric Satake, arXiv:1707.07700. [Zha14b] W. Zhang, Fourier transform and the global Gan–Gross–Prasad conjecture for unitary groups, Ann. of Math. Vol. 180 (2014) Issue 3, 971-1049. Mathematics Institute, Endenicher Allee 60, Bonn, 53115, Germany
6
This is the second part of a report on my going project joint with Yifeng Liu, Liang Xiao, Wei Zhang and Xinwen Zhu. In the previous talk by Yifeng Liu, he explains the general framework of the problem and state a theorem on some new cases of Beilinson-Bloch-Kato conjecture on the relation between special values of L-functions and the Selmer groups attached to the Rankin-Selberg motives for GLn × GLn+1 over a CM field. In this talk, I will focus on the case n = 2, give a more precise statement of our main theorem in this case and outline the main ideas of the proof. 1. Statement of the main theorem Let F + be a totally real field, F0 be an imaginary quadratic field and F = F0 F + . Denote by c the complex conjugate of F/F + . For n = 2, 3, we consider a regular algebraic conjugate selfdual cuspidal automorphic (RACSDCA) representation Πn of GLn (AF ) such that the infinitesimal character of Πn,∞ is trivial. Let E be a number field where Π∞ n is defined. We fix a finite place λ of E, and put O = OEλ . Let $ be a uniformizor of O, and kλ = O/$. Let ` denote the characteristic of kλ . Then by the work of Clozel, Harris, Taylor, Shin et al., one has a continuous `-adic Galois representation ρn : ΓF := Gal(F /F ) → GLn (Eλ ) attached to Πn , which is unramified outside the finite set of finite places of F where Πn is not spherical. We take the geometric normalization for ρn so that ρcn ∼ = ρ∨ n (1 − n). Let ρ¯n : ΓF → GLn (kλ ) be the associated residue Galois representation, which is well-defined up to semi-simplification.1 We will impose the following assumptions for our main theorem: Assumption 1.1. We will impose the following assumptions on ρn (1) ` ≥ 11 is unramified in F . (2) ρ¯2 and ρ¯3 are absolutely irreducible. (3) If ` denotes the mod-` cyclotomoic character, the image of (¯ ρ2 , ρ¯3 , ` ) : ΓF → GL2 (kλ ) × GL3 (kλ ) × F× ` contains an element of the form 0 0 µ1 0 λ1 , 0 µ2 0 , ξ λ2 0 µ3 0 0 −1 −1 ±1 where λi , µj ∈ kλ× satisfying λ2 λ−1 1 = ξ, µ1 6= µ3 and λ2 λ1 µ3 µ1 6= ξ . (4) The image of ρ¯c2 ⊗ ρ¯c3 (2) contains a non-trivial scalar. 1Later on, we will assume that ρ¯ is absolutely irreducible so that there is no ambiguity. n 1
(5) ρ¯3 satisfies Caraiani-Scholze’s decomposed generic condition, i.e. there exists a rational prime p that splits completely in F such that ρ¯3 is unramified at every place v|p of F , and if α1 , · · · , αn are the eigenvalues of ρ¯3 (Frv ), then αi /αj ∈ / {1, p} for all i 6= j. Here, Frv ∈ ΓF denotes a geometric Frobenius element at v. (6) The spherical Hecke eigensystem for Π2 and Π3 are modulo λ-isolated. Our main theorem for n = 2 can be stated as follows. Theorem 1.2. Assume that L( 21 , Π2 × Π3 ) 6= 0. Then for a finite prime λ of E satisfying Assumption 1.1, then Hf1 (F, ρc2 ⊗ ρc3 (2)) = 0. Example 1.3. Let A and B are two modular elliptic curves over F + without complex multiplication. Assume that A and B are not isogenous to each other. We take E = Q and ρ2 to be the `-adic Galois representation over F on the Tate module of A, and ρ3 to be the symmetric square of the `-adic Galois representation on the Tate module of B. Then using a famous result of Serre, Assumptions 1.1 are verified for all but finitely many primes `. (m)
Put ρ = ρ2 ⊗ ρ3 (2). For every integer m ≥ 1, let ρ¯n denote the reduction modulo $m of (m) a ΓF -stable lattice in ρn with n = 2, 3. By Assumption 1.1(2), ρn is well-defined. We put (m) (m) ρ¯(m) := ρ¯2 ⊗ ρ¯3 (2). For a finite place v of F , let 1 Hsing (Fv , ρ¯(m) ) := H 1 (Iv , ρ¯(m) )Gal(kv /kv )
denote the singular part of the local Galois cohomology, where Iv is the inertia subgroup of Gal(F v /Fv ). We denote by locv : H 1 (F, ρ¯(m) ) → H 1 (Fv , ρ¯(m) ) the restriction map induced by a fixed embedding F ,→ F v , and by 1 ∂ : H 1 (Fv , ρ¯(m) ) → Hsing (Fv , ρ¯(m) )
be the natural projection. The proof of Theorem 1.2 follows from the Kolyvagin’s strategy on the finiteness of ShafarevichTate groups for elliptic curves over Q. Similar ideas have been used by Bertonili-Darmon on their work on the anti-cyclotomic Iwasawa conjecture for elliptic curves over Q, and also by Wei Zhang on his proof of Kolyvagin conjecture. We shall construct a Kolyvagin system 1 (m) {Θ(v) )|m ≥ 1, v is m-admissible} m ∈ H (F, ρ (v)
indexed by certain (infinite set of) m-admissible places of F such that their images ∂locv (Θm ) ∈ 1 (F , ρ (m) ) are related to the Gan-Gross-Prasad period of Π × Π ; in particular, we show that Hsing v ¯ 2 3 (v)
∂locv (Θm ) 6= 0 for sufficiently large m ≥ 1 if L(1/2, Π2 × Π3 ) 6= 0. Once those properties are established, Theorem 1.2 follows easily from the general Kolyvagin’s machinery. 2. A Kolyvagin system and sketch of proofs (v)
In this section, we give more details on the construction of the cohomology classes Θm and their properties. We introduce first some notation. As explained in Yifeng Liu [Liu18], the global Gan-Gross-Prasad conjecture implies that there exists two unique pairs (V, π2 ) and (W, π3 ) where • V and W are totally positive definite Hermitian spaces over F/F + such that dimF (V ) = 2 and W = V ⊕ F u, where u ∈ W is a vector with (u, u) = 1, • π2 and π3 are respectively cuspidal automorphic representations on U (V )(AF + ) and U (W )(AF + ) such that the base change of πn to GLn (AF ) is Πn for n = 2, 3. 2
∞ Let KV ⊆ U (V )(A∞ F + ) and KW ⊆ U (W )(AF + ) be sufficiently small open compact subgroups such that (π2∞ )KV 6= 0, (π3∞ )KW 6= 0 and KV = KW ∩ U (V )(AF ), where we regard U (V ) as a subgroup of U (W ). Let Σ be the finite set of places of F such that KV and KW are hyperspecial at every Σ∪{v} Σ∪{v} Σ∪{v} finite place w ∈ / Σ. For a place v ∈ / Σ, let TvV = C (KV \U (V )(AF + )/KV , OEλ ) denote the spherical Hecke algebra away from v for U (V ). Similarly, one has TvW with V replaced by W , and we put Tv := TvV ⊗ TvW . Then the (modulo λ) Hecke eigensystem of π2 (resp. π3 ) determines a maximal ideal mπ2 ⊆ TvV (resp. mπ3 ⊆ TvW ). Let m denote the maximal ideal of Tv corresponding to π2 × π3 .
Definition 2.1. Let m ≥ 1 be an integer. A finite place v of F is called m-admissible if (1) the underlying rational prime p of v is unramified in F + and p mod ` avoid some explicit elements in F` (in particular, we assume p2 6≡ 1 mod `); (2) v ∈ / Σ and it is inert over some place v + of F + of degree 1, i.e. Fv++ ∼ = Qp2 , = Qp and Fv ∼ (3) if Frv ∈ ΓF denotes a geometric Frobenius α 1 0 (m) and ρ¯3 (Frv ) is conjugate to 0 2 0 p 0
(m)
element atv, then ρ¯2 (Frv ) is conjugate to 0 0 p2 0 with α ∈ / {1, p2 , p4 }. −1 4 0 α p
Remark 2.2. (a) Note that condition (3) above implies that 1 and p−2 appear as eigenvalues of ρ¯(m) (Frv ) with multiplicity one. (b) The Assumption 1.1(3) ensures that there exist infinitely many m-admisssible places for every integer m ≥ 1. (m) ) is free of rank one over 1 (F , ρ (c) An easy computation of Galois cohomology shows that Hsing v ¯ O/$m for an m-admissible place v. Fix an integer m ≥ 1 and an m-admissible place v. Note that V and W are quasi-split at v 2. Then there exists a unique nearby Hermitian space V (v) such that is isomorphic to V ⊗F A∞,v • V (v) ⊗F A∞,v F , F • V (v) ⊗Q R has signatures (1, 1) × (2, 0) × · · · × (2, 0). • V (v) ⊗F Fv is not quasi-split. Put W (v) = V (v) ⊕ F u. Let H (v) = ResF + /Q U (V (v) ) (resp. G(v) = ResF + /Q U (W (v) )) be the unitary group attached to V (v) (resp. W (v) ). We have thus the Shimura varieties X (v) (resp. Y (v) ) attached to H (v) (resp. G(v) ) defined over F of dimension 1 (resp. 2). The levels away from v of X (v) and Y (v) at v are the same as KV and KW , and their levels at v are defined by the stablizers of some non-self-dual lattices, hence they are non-hyperspecial at v. The diagonal embedding of unitary groups H (v) ,→ H (v) × G(v) induces an embedding of Shimura varieties ∆ : X (v) ,→ X (v) ×F Y (v) of codimenion 2. The arithmetic ´etale fundamental group H´e4t (X (v) ×F Y (v) , O/$m (2)) is equipped with an action by the spherical Hecke algebra Tv so that it makes sense to talk about the localization H´e4t (X (v) ×F Y (v) , O/$m (2))m . We consider now the cycle class map cl : Ch2 (X (v) ×F Y (v) ) → H´e4t (X (v) ×F Y (v) , O/$m (2))m . 2A Hermitian space for the unramified quadratic extension of a non-archimedean local filed K is called quasi-split
if it admits a self-dual lattice. 3
By Caraiani-Scholze’s result, one has H´eit ((X (v) ×F Y (v) )F , O/$m (2))m = 0 for i 6= 3. We get thus an Abel-Jacobi map 2 (v) AJ(v) ×F Y (v) ) → H 1 (F, H´e3t ((X (v) ×F Y (v) )F , O/$m )m (2)). m : Ch (X (v)
We denote by AJm ([X (v) ]) the image of the diagonal cycle [∆(X (v) )] ∈ Ch2 (X (v) ×F Y (v) ). Since the Hecke eigensystem of π2 and π3 are λ-isolated by Assumption 1.1(6), we see that (v) (v) H´e3t ((X (v) ×F Y (v) )F , O/$m (2))m ∼ = H´e1t (XF , O/$m (1))mπ2 ⊗ H´e2t (YF , O/$m (2))mπ3 (v)
is nothing but a finite copy of ρ¯(m) . The desired cohomology class Θm is then the projection of (v) AJm ([X (v) ]) onto one copy of H 1 (F, ρ¯(m) ) using certain Hecke operators. Remark 2.3. We should warn the reader on a technical subtlety. Strictly speaking, the Shimura varieties attached to H (v) and G(v) are of abelian type and we still do not have a satisfactory theory of integral models for such Shimura varieties at non-hyperspecial places. So we need to actually work with certain variants of the unitary groups H (v) and G(v) and their associated Shimura varieties. Let T0 = {x ∈ ResF/Q Gm,F |NF/F + (x) ∈ Gm,Q } e (v) = T0 × H (v) . Then it is shown by Rapoport, Smithling and Zhang [RSZ17] that the and put H e (v) , say X e (v) , has a nice moduli interpretation, and hence one can Shimura variety attached to H easily describe its integral model over OFv . Similarly, we have the Shimura variety Ye (v) attached e (v) := T0 × G(v) . We need to work with X e (v) and Ye (v) instead of X (v) and Y (v) . In this note, to G we ignore this technical subtlety. Theorem 2.4. Under the assumptions above, the following statements hold: (1) There exists a canonical isomorphism 1 Φ : Hsing (Fv , H´e3t ((X (v) ×F Y (v) )F , O/$m (2))m ) ∼
− → H 0 (ShKV (U (V )), O/$m )mπ2 ⊗ H 0 (ShKW (U (W )), O/$m )mπ3 , where ShKV (U (V )) denotes the discrete Shimura set U (V )(F )\U (V )(A∞ F )/KV , and similar meaning for ShKW (U (W )). (2) Let h•, •i denote the canonical bilinear pairing on H 0 (ShKV (U (V )), O/$m )mπ2 ⊗ H 0 (ShKW (U (W )), O/$m )mπ3 . Then there exists an explicit non-zero constant c such that for any f ∈ H 0 (ShKV (U (V )), O/$m )mπ2 ⊗ H 0 (ShKW (U (W )), O/$m )mπ3 , one has X (v) hΦ∂locv (AJ(v) ])), f i = c f (ι(x)), m ([X x∈ShKV (U (V ))
where c is some explicit non-zero constant in O, and ι : ShKV (U (V )) → ShKV (U (V )) × ShKV (U (V ))×ShKW (U (W )) is the diagonal embedding induced by the natural map U (V ) → U (W ). Note the right hand side of the formula in (2) is exactly the Gan-Gross-Prasad period of f , which is non-vanishing for some test vector f under the assumption that L( 12 , Π2 × Π3 ) 6= 0 by the global Gan-Gross-Prasad conjecture, proved recently by Wei Zhang []. Once this Theorem is established, Theorem 1.2 follows from Kolyvagin’s method of bounding Selmer groups using Kolyvagin system. We indicate how to prove Theorem 2.4. Statement (1) is reduced to establishing the following two isomorphisms 4
(v)
1 (F , H 1 (X (a) Φ2 : Hsing , O/$m (2))mπ2 ) ∼ = H 0 (ShKV (U (V )), O/$m )mπ2 . v ´ et Fv Γ (v) (b) Φ3 : H 2 (Y , O/$(1))mFv ∼ = H 0 (ShK (U (W )), O/$m )m π3
Fv
π3
W
Statement (2) will become clear when we understand how the isomorphism Φ in (1) is established. Part (a) is usually called the arithmetic level raising for U (V ). Note that the automorphic cuspidal representation π2 is hyperspecial at v + , while H (v) is not quasi-split at v + . Hence, if we consider the Eλ -coefficient (instead of O/$m -coefficient) ´etale cohomology of X (v) , the its localization at mπ2 should be 0. However, since the right hand side of Φ2 is non-zero, the existence of the isomorphism Φ2 shows that there exist an automorphic cuspidal representation π20 of H (v) , which should correspond to cuspidal automorphic representations of U (V ) with higher level at v via Jacquet-Langlands transfer for unitary groups, such that its spherical Hecke eigensystems away from v are exactly the same as those of π2 modulo $m . To show (a), we need to use the reduction of a certain integral model X (v) of X (v) over OFv (v) at v 3. It turns out that X (v) has semi-stable reduction, and its (geometric) special fiber XF is a p union X1 ∪ X2 , where • X1 is a P1 -bundle over the discrete Shimura set ShKV (U (V )), • X2 is a P1 -bundle over ShKVv Kv0 (U (V )), where KVv is the prime-to-v part of KV = KVv Kv , and Kv0 is another hyperspecial subgroup of U (V )(Fv++ ) such that Iwv = Kv0 ∩ Kv is an Iwahoric subgroup. The intersection X1 ∩ X2 is then exactly isomorphic to ShKVv Iwv (U (V )). Consider the composite map β
α
H 0 (X1 )mπ2 ⊕ H 0 (X2 )mπ2 − → H 0 (X1 ∩ X2 )mπ2 − → H 2 (X1 )mπ2 (1) ⊕ H 2 (X2 )mπ2 (1). Here, all the cohomology groups are ´etale cohomology with coefficients in O/$m (together with suitable Tate twists), α is given by natural restrictions and β is given by the Gysin maps. Now, statement (a) follows from the following facts: • Using the spectral sequence of nearby cycles, one can show that (v) 1 Coker(β ◦ α) ∼ (Fv , H´e1t (XF , O/$m (2))mπ2 ). = Hsing v
• On the other hand, one can give an explicit expression for β ◦ α in terms of Hecke operators at v of the group U (V ). Using this, one shows easily that the canonical map H 0 (ShK (U (V )))m ∼ = H 2 (X1 )m (1) → Coker(β ◦ α) V
π2
π2
is an isomorphism. To establish the isomorphism Φ3 in part (b), we use the reduction of an integral model4 Y (v) over OFv of Y (v) . The geometry for Y (v) is similar to X (v) . It has semi-stable reduction, and the (v) (geometric) special fiber YF is a union Y1 ∩ Y2 , where p
• Y1 is a fibration of P2 over ShKW (U (W )), • Y2 is smooth that intersect Y1 transversally, and Y1 ∩ Y2 is a fibration of Fermat curves of equation xp+1 + xp+1 + xp+1 = 0 over ShKW (U (W )). 1 2 3 Even the global geometry of Y2 is complicated, its basic/supersingular locus, denoted by Y2ss , is easy to describe: it is the union of Y1 ∩ Y2 and another stratum Z2 , which is a collection of P1 indexed ∞,v v K s (U (W )). Here, K v ⊆ U (W )(A + ) is the prime-to-v part of KW , and K s is a maximal by ShKW v W v F 3As in Remark 2.3, we need to actually work with a variant X e (v) of X (v) to get an integral model over OFv with
some moduli interpretations. 4The same remark as the previous footnote. See Remark 2.3. 5
special subgroup of U (W )(Fv++ ), the stablizer of a nearly self-lattice lattice in W ⊗F Fv . The dual graph attached to Y2ss is the quotient of the Bruhat-Tits tree for U (W )(FF++ ) by a discrete subgroup. Using the spectral sequence of nearby cycles for Y (v) , assertion (b) is reduced to computing the following composed Gysin-restriction map (called the intersection matrix of Y1 ∩ Y2 and Z2 ): Gysin
Restriction
H 0 (Y1 ∩ Y2 )mπ3 ⊕ H 0 (Z2 )mπ3 −−−−→ H 2 (Y2 )mπ3 (1) −−−−−−−→ H 2 (Y1 ∩ Y2 )mπ3 (1) ⊕ H 2 (Z2 )mπ3 (1). As in the case for X (v) , an explicit computation shows that the above composition map is nondegenerate under condition (3) of Definition 2.1. It follows that the Gysin map Gysin
H 0 (Y1 ∩ Y2 )mπ3 ⊕ H 0 (Z2 )mπ3 −−−−→ H 2 (Y2 )mπ3 (1) induces an isomorphism between H 0 (Y1 ∩ Y2 )mπ3 ⊕ H 0 (Z2 )mπ3 and the Galois invariant subgroup H 2 (Y2 )mπ3 (1)Gal(Fp /Fp2 ) , that is, all the Tate cycles in H 2 (Y2 )mπ3 (1) come from the cycle classes of basic/supersingular locus of Y2 . This is a variant of the generic Tate conjecture for the reduction of Shimura varieties at a bad place. The corresponding statement for the good reduction of unitary Shimura varieties was established recently in the joint work of Liang Xiao and Xinwen Zhu [XZ17+ ]. References [Liu18] Y. Liu, Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives I, this volume. [RSZ17] M. Rapoport, B. Smithling, W. Zhang, Arithmetic diagonal cycles on unitary Shimura varieties, arXiv:1710.06962v2. [XZ17+ ] L. Xiao and X. Zhu, Tate cycles on Shimura varieties via geometric Satake, arXiv:1707.07700. [Zha14b] W. Zhang, Fourier transform and the global Gan–Gross–Prasad conjecture for unitary groups, Ann. of Math. Vol. 180 (2014) Issue 3, 971-1049. Mathematics Institute, Endenicher Allee 60, Bonn, 53115, Germany
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