BEILINSON–BLOCH–KATO CONJECTURE FOR

BEILINSON–BLOCH–KATO CONJECTURE FOR RANKIN–SELBERG MOTIVES YIFENG LIU

This is a report on the progress of a joint work by Yichao Tian, Liang Xiao, Wei Zhang, Xinwen Zhu, and the author, in which we study some new cases of Beilinson–Bloch–Kato conjecture on the relation between L-functions and Selmer groups, coming from automorphic motives of the Rankin–Selberg type GLn × GLn+1 . 1. Rankin–Selberg Selmer groups We fix a CM extension F/F + with c ∈ Gal(F/F + ) the complex conjugation. Let N ≥ 1 be an integer. We say that a representation Π of GLN (AF ) is relevant if • Π is irreducible and cuspidal automorphic; • Πc is isomorphic to Π∨ ; and • Π∞ is regular algebraic with the trivial infinitesimal character. We may associated to Π the following two kinds of objects. (1) For every number field E ⊆ C over which Π∞ is defined and every prime λ of E, there is a semisimple Galois representation ρΠ,Eλ : Gal(F /F ) → GLN (Eλ ), unique up to isomorphism, such that ρcΠ,Eλ ' ρ∨Π,Eλ (1 − N ); and that if Πv is spherical at a finite place v of F , then ρΠ,Eλ is unramified at v and the characteristic polynomial of a geometric Frobenius at v coincides with the Hecke polynomial of Πv . This is known by the works of Harris–Taylor, Shin, and Chenevier–Harris. (2) Let ΦΠ be the set of pairs (V, π), up to isomorphism, where V is a (nondegenerate) hermitian space over F (with respect to the involution c) of rank N and π is an irreducible cuspidal automorphic representation of U(V )(AF + ) such that for all but finitely many finite places v of F + at which πv is unramified, BC(πv ) ' Πv . We have the following proposition due to Arthur, Mok, and Kaletha–Minguez–Shin– White, which is part of Arthur’s classification of discrete automorphic representations. Proposition 1.1. For every member (V, π) of ΦΠ , we have (1) BC(πv ) ' Πv for every place v of F + that is either archimedean, split, or at which πv is unramified. In particular, πv must be a discrete series if v is archimedean. (2) π appears with multiplicity one in the discrete spectrum of U(V ). Date: May 13, 2018. 1

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YIFENG LIU

Now we move to the Rankin–Selberg case in our consideration. Let n ≥ 1 be an integer. Let Πn and Πn+1 be two relevant representations of GLn (AF ) and GLn+1 (AF ), ∞ respectively. We fix a number field E ⊆ C over which both Π∞ n and Πn+1 are defined. Conjecture 1.2 (Beilinson–Bloch–Kato). Let Πn , Πn+1 , and E be as above. Then for every prime λ of E, we have ords=1/2 L(s, Πn × Πn+1 ) = dimEλ H1f (F, ρcΠn ,Eλ ⊗ ρcΠn+1 ,Eλ (n)). 2. Main results The following conjecture is a special case of the global Gan–Gross–Prasad conjecture. Conjecture 2.1. Suppose L( 21 , Πn × Πn+1 ) 6= 0. Then there exists a unique pair of members (Vn , πn ) ∈ ΦΠn and (Vn+1 , πn+1 ) ∈ ΦΠn+1 such that Vn is totally positive definite; Vn+1 = Vn ⊕ F e with (e, e) = 1; and there exist fn ∈ πn and fn+1 ∈ πn+1 such that Z fn (g)fn+1 (g)dg 6= 0. U(Vn )(F + )\U(Vn )(AF + )

By the works of Wei Zhang, Xue, and Beuzart-Plessis, Conjecture 2.1 is known when there is a finite place v of F of degree one over F + such that both Πn,v and Πn+1,v are supercuspidal (or under other rather technical local assumptions if one does not want any supercuspidality). Currently, Chaudouard and Zydor are working toward the full version of Conjecture 2.1. In what follows, we will admit Conjecture 2.1. From now on, we assume that F contains an imaginary quadratic field. Here is our main theorem. Theorem 2.2. Let Πn , Πn+1 , and E be as above such that L( 12 , Πn × Πn+1 ) 6= 0. Then for every admissible prime λ of E, we have H1f (F, ρcΠn ,Eλ ⊗ ρcΠn+1 ,Eλ (n)) = 0. The definition of admissible primes is very technical. Moreover, at this moment, we do not even know how to create any example when n ≥ 5 for which there exists admissible λ. Thus, instead of giving precise definition, we would like to state the following theorem, which is restrictive but more explicit. Theorem 2.3. Let A and B be two modular elliptic curves over F + that are not geometrically CM, and such that they are not geometrically isogenous to each other. Suppose that we are in one of the following cases: (1) n ≤ 3; (2) n = 4 and Sym4 B is automorphic; (3) n ≥ 5; both Symn−1 A and Symn B are automorphic. Then the nonvanishing of the central value, that is, L(n, Symn−1 AF × Symn BF ) 6= 0, implies H1f (F, Symn−1 V` (AF ) ⊗Q` Symn V` (BF )(1 − n)) = 0

BEILINSON–BLOCH–KATO CONJECTURE FOR RANKIN–SELBERG MOTIVES

3

for all but finitely many rational prime `. Here V` denotes the Q` -Tate module. Remark 2.4. The automorphy of Symn of a modular elliptic curve is known by Gelbart– Jacquet for n = 2, by Kim–Shahidi for n = 3, by Clozel–Thorne for n = 5, 7 with mild restriction on F + , and not known for other n ≥ 4. There are six main ingredients in the proof of Theorem 2.2 and Theorem 2.3. (1) Global Gan–Gross–Prasad conjecture, that is, Conjecture 2.1; (2) Vanishing of torsion cohomology, in the style of Caraiani–Scholze; (3) Arithmetic level raising via certain semistable model of unitary Shimura varieties; (4) Tate cycles on special fibers of unitary Shimura varieties; (5) Computation of Abel–Jacobi maps via nearby cycles; (6) Kolyvagin type argument for bounding Selmer groups. The third ingredient is the most involved one, which we will explain a little bit in the next section. 3. Semistable models and arithmetic level raising for unitary groups For simplicity, we assume F + 6= Q. We fix an embedding ι : F ,→ C. Let V be a totally positive definite hermitian space over F of rank N ≥ 2. Put G := U(V ). We fix a decomposable open compact subgroup K ⊆ G(A∞ F + ), a finite set Σ of places of F + such that for every v 6∈ Σ, v is finite and Kv is hyperspecial. Let ` be a rational Σ prime. Put T := Cc (K Σ \G(AΣ F ∞ )/K , Z` ) for the spherical Hecke algebra. Consider + a prime p of F inert in F away from Σ. In particular, Vp is a quasi-split hermitian Σ∪{p} space. Put Tp := Cc (K Σ∪{p} \G(AF ∞ )/K Σ∪{p} , Z` ) ⊆ T. Let W be the “nearby hermitian space” of V at p. Namely, it is the hermitian space over F , unique up to isomorphism, of signature (N − 1, 1) at the default place ι and isomorphic to V all places other than ι and p. Put H := U(W ). Regard K p as an open + ? compact subgroup of H(A∞,p F + ), and let Kp ⊆ H(Fp ) be a special maximal compact subgroup. We then have a Shimura variety associated to H and K p Kp? , which is smooth projective over F of dimension N − 1. Suppose that p is of degree one over Q1, then the Shimura variety has a natural semistable integral model2 X over OFp ' Zp2 . Put κ := OF /p ' Fp2 for the residue field. Proposition 3.1. The special fiber X ⊗OFp κ is a normal crossing divisor, which can be written as a union X1 ∪ X2 such that (1) X1 ⊗κ κ is a PN −1 -fibration over G(F + )\G(A∞ F + )/K; (2) X2 is smooth; 1This

condition is only for technical simplicity, which is not really necessary. fact, in order to get a reasonable integral model, we work with the Shimura variety associated to (ResF + /Q H) × T rather than ResF + /Q H, where T is certain torus contained in ResF/Q Gm . 2In

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YIFENG LIU

(3) X12 := X1 ∩ X2 is a Fermat hypersurface in X1 ; more precisely, X12 ⊗κ κ is a N −1 fibration over G(F + )\G(A∞ defined by the homogeF + )/K with fiber F ⊆ Pκ neous equation + · · · + xp+1 = 0. xp+1 1 N The special fiber of X has a natural notion of supersingular locus. To describe such locus on X2 , we need more notation. Let Kp0 ⊆ G(Fp+ ) be a special maximal subgroup such that Pp := Kp ∩ Kp0 is a Siegel parahoric subgroup as in the previous section. Proposition 3.2. We have (1) the supersingular locus of X1 is entire X1 ; (2) the supersingular locus of X2 is a union X12 ∪ Y2 in which Y2? ⊗κ κ is a fibration ? p 0 over G(F + )\G(A∞ F + )/K Kp , where Y2 denotes the normalization of Y2 , with each fiber being a smooth Deligne–Lusztig variety of dimension bN/2c; (3) An irreducible component of Y2 ⊗κ κ intersects with an irreducible component of X12 ⊗κ κ if and only if their parameters are in the image of the correspondence p + ∞ p 0 + ∞ G(F + )\G(A∞ F + )/K Pp → G(F )\G(AF + )/K Kp × G(F )\G(AF + )/K.

From now on, we suppose that N = 2r is even with r ≥ 2. Let ` be a rational prime different from p. Then the above proposition provides with the following diagram H´e2r−4 (X12 ⊗κ κ, Z` (r − 2)) t

H´e0t (Y2? ⊗κ κ, Z` ) i1

i2

+

t

H´e2r−2 (X2 ⊗κ κ, Z` (r − 1)) t β



H´e2r−2 (X12 ⊗κ κ, Z` (r − 1)) t 

α

H´e2rt (X2 ⊗κ κ, Z` (r)) j1

j2

s

*

H´e2rt (Y2? ⊗κ κ, Z` (r))

H´e2rt (X12 ⊗κ κ, Z` (r))

in which β, j1 , j2 are restriction maps and i1 , i2 , α are Gysin maps. Moreover, it is known that cup product with the hyperplane section in F induces canonical isomorphisms ∼

∼

H´e0t (X12 ⊗κ κ, Z` ) − → H´e2r−4 (X12 ⊗κ κ, Z` (r − 2)) − → H´e2rt (X12 ⊗κ κ, Z` (r)). t In particular, we have canonical isomorphisms 2r−4 C (G(F + )\G(A∞ (X12 ⊗κ κ, Z` (r − 2)) ' H´e2rt (X12 ⊗κ κ, Z` (r)); F + )/K, Z` ) ' H´ et p 0 0 ? 2r ? C (G(F + )\G(A∞ F + )/K Kp , Z` ) ' H´ et (Y2 ⊗κ κ, Z` ) ' H´ et (Y2 ⊗κ κ, Z` (r))

by Propositions 3.1 and 3.2. Thus, the above diagram induces an endomorphism Φ of the Z` -module + ∞ p 0 C (G(F + )\G(A∞ F + )/K, Z` ) ⊕ C (G(F )\G(AF + )/K Kp , Z` ).

BEILINSON–BLOCH–KATO CONJECTURE FOR RANKIN–SELBERG MOTIVES

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It is equivariant under the action of Tp on both sides. Proposition 3.3. Suppose ` - |G(OF + /p)|. Let m be a maximal ideal of T such that + ∞ C (G(F + )\G(A∞ F + )/K, Z` )m = C (G(F )\G(AF + )/K, Z` )mp

holds, where mp := m ∩ Tp . Write the Satake parameter of m at p as (α1 , . . . , αr ) with × αj ∈ F` . Then we have det(Φ ) = −(p + 1)p mp

r2

r  Y



(αi + αi−1 ) − (p + p−1 ) .

i=1

H´ei t (X

Furthermore, if we have ⊗OFp Fp , Z` )mp = 0 for i 6= 2r − 1, and there is at most one j such that αj ∈ {p, p−1 }, then coker(Φmp ) is canonically a subquotient of H1sing (Fp , H´e2r−1 (X ⊗OFp Fp , Z` )mp ). t The proposition has the following immediate corollary, which reveals the level raising phenomenon. Corollary 3.4. Keep the notation as above, and suppose ` - |G(OF + /p)|. Let m be a maximal ideal of T such that + ∞ (1) C (G(F + )\G(A∞ F + )/K, Z` )m = C (G(F )\G(AF + )/K, Z` )mp holds; (2) H´ei t (X ⊗OFp Fp , Z` )mp = 0 for i 6= 2r − 1; (3) there is exactly one j such that αj ∈ {p, p−1 }. Then we have H´e2r−1 (X ⊗OFp Fp , Z` )mp 6= 0. t Example 3.5. When N = 4, the determinant of Φ is given by the Hecke operator 



−(p + 1) Tp,2 − p(p + 1)Tp,1 + p2 (p + 1)(p3 + 1) , where Tp,1 and Tp,2 are the spherical Hecke operators given by the matrices     

p 1 1 p

−1





  , 

   



p p p

−1

p−1

  , 

respectively. In particular, if we localize Φ at mp for m as in Proposition 3.3, then 



det(Φmp ) = −(p + 1)p4 (α1 + α1−1 ) − (p + p−1 )



(α2 + α2−1 ) − (p + p−1 ) ×

if the Satake parameter of m at p is (α1 , α2 ) for α1 , α2 ∈ F` . To prove Proposition 3.3, one key step is to show that the map (i1 , i2 ) : H´e2r−4 (X12 ⊗κ κ, Z` (r − 2)) ⊕ H´e0t (Y2? ⊗κ κ, Z` ) → H´e2r−2 (X2 ⊗κ κ, Z` (r − 1)) t t is an isomorphism after localization at mp . This is equivalent to Conjecture 3.6 introduced below assuming H´ei t (X ⊗OFp Fp , Z` )mp = 0 for i 6= 2r − 1, which appears very similar to the classical Ihara lemma. Let p be a prime of F + inert in F away from Σ. Suppose ` - |G(OF + /p)|. Then the K induced representation IndPpp F` is semisimple and contains a unique direct summand

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YIFENG LIU

σp satisfying: dim σp > 1 and its subspace of Iwahori fixed vectors has dimension one. We then have a natural projection map p + ∞ p Cc (G(F + )\G(A∞ F + )/K Pp , F` ) → Cc (G(F )\G(AF + )/K Kp , σp ).

Composing with the pullback map + ∞ p p 0 Cc (G(F + )\G(A∞ F + )/K Kp , F` ) → Cc (G(F )\G(AF + )/K Pp , F` ),

we obtain a map p 0 + ∞ p Ih := Cc (G(F + )\G(A∞ F + )/K Kp , F` ) → Cc (G(F )\G(AF + )/K Kp , σp )

which is equivariant under action Tp . Conjecture 3.6. Let m be a non-Eisenstein maximal ideal of T. Then for every p 6∈ Σ that is inert in F and satisfying ` - |G(OFp+ /p)|, the localized map Ihmp is injective, where mp := m ∩ Tp . We prove the following proposition, partially answering the above conjecture, which is enough for our proof of Proposition 3.3. Proposition 3.7. Conjecture 3.6 holds if we have (1) p is of degree one over Q3; + ∞ (2) C (G(F + )\G(A∞ F + )/K, Z` )m = C (G(F )\G(AF + )/K, Z` )mp ; i (3) H´et (X ⊗OFp Fp , Z` )mp = 0 for i 6= 2r − 1; and (4) there is at most one j such that αj ∈ {p, p−1 }. Department of Mathematics, Northwestern University, Evanston, IL 60208 E-mail address: [email protected]

3Again,

this condition is not really necessary.