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Anticyclotomic Euler Systems for Unitary Groups

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Anticyclotomic Euler Systems for Unitary Groups ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS ANDREW GRAHAM AND SYED WAQAR ALI SHAH Abstract. Let n ≥ 1 be an odd integer. We construct an anticyclotomic Euler system for certain cuspidal automorphic representations of unitary groups with signature (1; 2n − 1). Contents 1. Introduction 1 2. Preliminaries 7 3. Cohomology of unitary Shimura varieties 13 4. Horizontal norm relations 15 5. Interlude: cohomology functors 25 6. Vertical norm relations at split primes 31 7. The anticyclotomic Euler system 41 Appendix A. Continuous ´etalecohomology 49 Appendix B. Shimura{Deligne varieties 55 Appendix C. Ancona's construction for Shimura{Deligne varieties 60 References 64 1. Introduction In [Kol89], Kolyvagin constructs an anticyclotomic Euler system for (modular) elliptic curves over Q which satisfy the so-called \Heegner hypothesis". One can view the classes in this construction as images under the modular parameterisation of certain divisors on modular curves arising from the embedding of reductive groups (1.1) ResE=Q Gm ,! GL2 where E is an imaginary quadratic number field. These divisors (Heegner points) are defined over ring class fields of E and satisfy trace compatibility relations as one varies the field of definition. In particular, Kolyvagin shows that if the bottom Heegner point is non-torsion, then the group of E-rational points has rank equal to one. Combining this so-called \Kolyvagin step" with the Gross{Zagier formula that relates the height of this point to the derivative of the L-function at the central value, one obtains instances of the Birch{ Swinnerton-Dyer conjecture in the analytic rank one case. The above construction has been generalised to higher weight modular forms and to situations where a more general hypothesis is placed on the modular form (see [Sch86], [Nek92], [BDP13] and [Zha01]). In this paper, we consider a possible generalisation of this setting; namely, we construct an anticyclotomic Euler system for the p-adic Galois representations attached to certain regular algebraic conjugate self-dual cuspidal automorphic representations of GL2n =E. More precisely, we consider the following \symmetric pair" (1.2) U(1; n − 1) × U(0; n) ,! U(1; 2n − 1): Note that both groups are outer forms of the groups appearing in (1.1) when n = 1. Let π0 be a cuspidal automorphic representation of U(1; 2n − 1) and let π denote a lift to the group of unitary similitudes. Under certain reasonable assumptions on π0 (for example, non-endoscopic and tempered with stable L-packet), there exists a cuspidal automorphic representation Π of GL2n(AE) × GL1(AE) which is locally isomorphic 1 2 ANDREW GRAHAM AND SYED WAQAR ALI SHAH to the base-change of π at all but finitely many primes. Let ρπ denote the Galois representation attached to Π, as constructed by Chenevier and Harris [CH13]. We impose the following assumptions. We require that the lift π of π0 as above is cohomological and the Galois-automorphic piece ρπ ⊗ πf appears in the middle degree cohomology of the Shimura variety for GU(1; 2n − 1) { see Assumption 3.5. If this assumption is satisfied, we say that ρπ ⊗ πf admits a \modular parameterisation". Our second assumption is an analogue of the Heegner hypothesis: we require that π0;f admits a H0-linear model, i.e. one has Hom (π ; ) 6= 0 H0(Af ) 0;f C where H0 is the subgroup U(1; n − 1) × U(0; n). Finally, we require that p is split in the quadratic extension E, and π0 (or equivalently, π) has a fixed eigenvector with respect to a certain Hecke operator associated with the Siegel parahoric level at the prime p. We refer to this assumption as \Siegel ordinarity". We now state our main result. Let S be a finite set of primes containing all primes where π0 is ramified, and all primes that ramify in E=Q. For p2 = S, let R denote the set of square-free positive integers divisible only by primes that lie outside S [ fpg and split in E=Q. For m 2 R and an integer r ≥ 0, we let E[mpr] denote the ring class field of E of conductor mpr. Theorem A. Let n ≥ 1 be an odd integer and p2 = S an odd prime that splits in E=Q. Let π0 be a cuspidal automorphic representation of U(1; 2n − 1) as above. We impose the following conditions: • ρπ ⊗ πf admits a \modular parameterisation" (see Assumption 3.5). • π0;f admits a H0-linear model. • π0 is \Siegel ordinary" (see Assumption 7.1). ∗ Let Tπ be a Galois stable lattice inside ρπ. Then there exists a split anticyclotomic Euler system for Tπ (1−n), i.e. for any collection of primes of E lying above primes in R, and for m 2 R, r ≥ 0, there exist classes 1 r ∗ cmpr 2 H (E[mp ];Tπ (1 − n)) satisfying r −1 E[`mp ] Pλ Frλ · cmpr if ` 6= p and `m 2 R coresE[mpr ] c`mpr = cmpr if ` = p −1 where λ is the prime of E lying above `, Pλ(X) = det 1 − Frobλ XjTπ(n) is the characteristic polynomial r of a geometric Frobenius at λ, and Frλ 2 Gal (E[mp ]=E) denotes the arithmetic Frobenius at λ. Remark 1.3. Evidently, we can take cmpr = 0 for all m; r, and the above theorem is vacuous. However, the Euler system classes we construct arise from special cycles on Shimura varieties, and the non-vanishing of these classes is expected to be related to the behaviour of the L-function attached to π0 (more precisely, the non-vanishing of the derivative of the L-function at its central value). Proving such a relation constitutes an explicit reciprocity law, and we will pursue this in future work. The above definition of a split anticyclotomic Euler system originates from the forthcoming work of Jetchev, Nekov´aˇrand Skinner, in which a general machinery for bounding Selmer groups attached to conju- gate self-dual representations is developed. An interesting feature of this work is that one only needs norm relations for primes that split in the CM extension, rather than a \full" anticyclotomic Euler system. As a consequence of this, we expect to obtain the following corollary: Corollary. Let π0 be as in Theorem A, and suppose that the Galois representation ρπ satisfies the following hypotheses: • The primes for which ρπ is ramified lie above primes that split in E=Q. • The representation ρπ is absolutely irreducible. • There exists an element σ 2 Gal E=E[1](µp1 ) such that rank ρπ=(σ − 1)ρπ = 1: E[1] 1 Then, if c := coresE c1 is non-torsion, the Bloch{Kato Selmer group Hf (E; ρπ(n)) is one-dimensional. Remark 1.4. The existence of such an element σ in the above corollary is expected to follow if the image of the Galois representation ρπ : Gal(Q=Q) ! GL2n(Qp) is sufficiently large. In particular, this will exclude automorphic representations of \CM type". ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS 3 Remark 1.5. In this paper, we have chosen to focus on the case where E is an imaginary quadratic number field { however, we expect the results to also hold for general CM fields. Moreover, it should be possible to construct similar classes for certain inner forms of the groups appearing in (1.2). Another direction worth exploring is the case of inert primes which is closer in spirit to Kolyvagin's original construction { these questions will also be investigated further in future work. Remark 1.6. In its simplest form the Heegner hypothesis says that the conductor of the elliptic curve is divisible only by primes that split in the imaginary quadratic extension. By the Tunnell{Saito theorem, × this implies that the automorphic representation attached to the elliptic curve admits a AE;f -linear model ([YZZ13, Theorem 1.3]). To generalise this, we start with a cuspidal automorphic representation τ0 of GL2n(A) that is of \symplectic type" (i.e. it is the automorphic base-change of a cuspidal automorphic representation σ0 of GSpin2n+1(A)) and we assume that σ0 has trivial central character. Choose an imaginary quadratic extension E=Q such that every prime that divides the conductor of τ0 splits in E=Q, and let Π0 denote the base-change of τ0 to GL2n(AE), which is generically cuspidal (for example, if τ0 is not isomorphic to its quadratic twist [AC89, Chapter 3]). Since τ0 is of symplectic type and σ0 has trivial central character, ∼ c ∼ _ we have Π0 = Π0 = Π0 and Π0 admits a local Shalika model at all primes that lie above primes that split in E=Q. In fact, this implies that Π0 admits a global Shalika model if we additionally assume that Π0,λ is a discrete series representation of GL2n(Eλ) = GL2n(Q`) (see the discussion following Proposition 6.3 in [PWZ19]). By our conditions on the ramification of τ0, we expect that Π0 descends to a cuspidal automorphic representation π0 of the unitary group G0 = U(1; 2n − 1) that is quasi-split at all finite places (with the additional assumption that n is odd), and that π0 admits a H0(Af )-linear model. Indeed, for the primes where Π0 is ramified, there is nothing to check, and for the primes where Π0 is unramified, one should be able to check this explicitly via Satake parameters (similar to the calculations in [Sak08, §4]). If we further assume that the sign of the functional equation of the (completed) L-function of Π0 is −1, then it is not expected that π0 is globally distinguished, in the sense that the following period integral Z φ(h)dh H0(Q)nH0(A) is non-vanishing, for some φ 2 π0.
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