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 p∈ / S, let R denote the set of square-free positive integers divisible only by primes that lie outside S ∪ {p} and split in E/Q. For m ∈ 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 p∈ / 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 ∈ R, r ≥ 0, there exist classes 1 r ∗ cmpr ∈ H (E[mp ],Tπ (1 − n)) satisfying

r −1 E[`mp ] Pλ Frλ · cmpr if ` 6= p and `m ∈ R coresE[mpr ] c`mpr = cmpr if ` = p −1 where λ is the prime of E lying above `, Pλ(X) = det 1 − Frobλ X|Tπ(n) is the characteristic polynomial r of a geometric Frobenius at λ, and Frλ ∈ 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 conjugate 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 ∈ Gal E/E[1](µp∞ ) 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)\H0(A) is non-vanishing, for some φ ∈ π0. If it were, then by the conjecture of Getz–Wambach (see [GW14] and [PWZ19, Remark 1.7]) this would imply that the analogous period integral for Π0 would be non-vanishing and L(Π0, 1/2) 6= 0 – a contradiction to our assumption on the sign of the functional equation.

1.1. Outline of the paper. Let π0 be a cuspidal automorphic representation for the unitary group G0 of signature (1, 2n − 1) (defined in section 2.1), such that π0,∞ lies in the discrete series. Discrete series L-packets of G0 are parameterised by irreducible algebraic representations of GL2n /E; we let V0 denote the algebraic representation corresponding to the L-packet containing π0,∞. As explained in the previous section, we assume that π0,f admits a H0-linear model, i.e. we have Hom (π , ) 6= 0. H0(Af ) 0,f C In particular, this implies that V0 is self-dual and the central character of π0 is trivial. Let π denote a lift of π0 to the unitary similitude group G = GU(1, 2n − 1), and let Π = ω Π0 denote a weak base-change to GL1(AE) × GL2n(AE) as explained in Proposition 3.1. We assume that Π is cuspidal and, since the central character of π0 is trivial, we can choose π such that ω is equal to the trivial character (Lemma 2.10). Furthermore, π∞ lies in the discrete series L-packet corresponding to the trivial extension of V0 to an algebraic representation of GL1 /E × GL2n /E, which we denote by V (so in particular, V is also self-dual). 0 The representation Π decomposes as a restricted tensor product ⊗vΠv over the places of E, and for each (finite) prime λ where Πλ is unramified, we let L(Πλ, s) denote the standard local L-factor attached to Πλ (see Definition 2.9). We use similar notation for the representation Π0. Combining the results of several people (see Theorem 3.4), Chenevier and Harris have shown that there exists a semisimple continuous Galois representation ρπ : GE → GL2n(Qp) satisfying the following unramified local–global compatibility: for all but finitely many rational primes ` and for all λ|` −1 −s −1 −1 det 1 − Frobλ ` |ρπ = L(Πλ, s + 1/2 − n) = L(Π0,λ, s + 1/2 − n)

where Frobλ is an arithmetic Frobenius at λ and the last equality follows because ω = 1. We assume that this Galois representation appears in the (compactly-supported) ´etalecohomology of the Shimura variety 4 ANDREW GRAHAM AND SYED WAQAR ALI SHAH

ShG(K) associated with G (with appropriate level K ⊂ G(Af )), or equivalently, we assume that we have a “modular parameterisation”

2n−1 2n−1 ∨ ∗ H Sh , V (n) := lim H ShG(L) , V (n) π ⊗ ρ (1 − n). ´et G,Q −→ ´et Q f π L

where the limit is over all sufficiently small compact open subgroups L ⊂ G(Af ), and V denotes the p-adic sheaf associated with the representation V . In analogy with the Heegner point case, the Euler system will be obtained as the images of certain classes under this modular parameterisation. We also require that π satisfies a Siegel ordinarity condition at p. Briefly, this means that the representation πp contains a vector ϕp that is an eigenvector for the Hecke operator US := µ(τ)[JτJ], where p τ = 1 × ∈ GL ( ) × GL ( ) = G( ) 1 1 Qp 2n Qp Qp J is the parahoric subgroup associated with the Siegel parabolic subgroup (i.e. the parabolic subgroup × corresponding to the partition (n, n)), and µ: G(Qp) → Qp is the highest weight of V . We consider the following extended group Ge := G × T where T = U(1) is the torus controlling the variation in the anti-cyclotomic tower (see Lemma 7.6) – the group H = G (U(1, n − 1) × U(0, n)) then lifts to a subgroup of Ge via the map h1 det h2 (h1, h2) 7→ , . h2 det h1 In particular, the dimensions of the associated Shimura varieties1 are dim Sh = n−1 and dim Sh = 2n−1 H Ge respectively, and the inclusion H ,→ Ge provides us with a rich supply of codimension n cycles on the target ¯ variety, via Gysin morphisms. These cycles carry an action of both Ge (Af ) and Gal(E/E), and via Shimura reciprocity, the Galois action can be translated into an action of T(Af ). This will allow us to freely switch between the automorphic and Galois viewpoints. By considering the images under the cycle class map, we obtain classes in the degree 2n absolute étale cohomology of Sh . A consequence of the Siegel ordinarity condition is that we can modify these classes Ge so that they are universal norms in the anticyclotomic Zp-extension, and since this tower is a p-adic Lie extension of positive dimension, this will force our classes to be cohomologically trivial. We then obtain classes in the first group cohomology of the étalecohomology of ShG by applying the Abel–Jacobi map (an edge map in the Hochschild–Serre spectral sequence). The reason that these classes are universal norms will follow from the results of [Loe19], using the fact that the pair of subgroups in (1.2) give rise to a spherical variety at primes which split in the imaginary quadratic extension. This norm-compatibility will also imply the Euler system relations in the p-direction. We discuss these relations in section 6. For the convenience of the reader, we have provided the constructions of the Gysin morphisms and Abel–Jacobi maps for continuous étalecohomology in Appendix A. The key technique introduced in [LSZ17] is to rephrase the horizontal Euler system relations as an statement in local representation theory. The underlying idea is to consider the above construction for varying p p levels of the source and target Shimura varieties, and then package this data into a map of H(Af ) × Ge (Af ) representations. In section 5, we construct this map in an abstract setting using the language of cohomology functors. The ideas used here are essentially the same as those that appear in op.cit. but we hope that this formalism will be useful in proving Euler system relations for general pairs of reductive groups. We also note that these abstract cohomology functors appear in certain areas of representation theory, under the name “Mackey functors” (see [Dre73] for example), so these techniques can be considered as an application of Mackey theory. r r Recall that E[mp ] denotes the ring class field of E corresponding to the order Z + mp OE and let Dmpr ⊂ T(Af ) denote the compact open subgroup corresponding to this ring class field under the Artin reciprocity map. By composing the Gysin morphisms, the Abel–Jacobi map, the modular parameterisation

1Strictly speaking, the group H does not give rise to a Shimura variety but rather what we call a Shimura–Deligne variety (the axiom – typically denoted (SV3) – is not satisfied). In practice this does not affect the arguments in the paper, so the reader can safely pretend that it gives rise to a Shimura variety in the usual sense. We provide justifications for this viewpoint in Appendix B. ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS 5

above and passing to the “completion” (see Proposition 5.11) for varying levels, we construct a collection of maps 1×D r mp Lmpr ∨ 1 r ∗ H Ge (Af ) −−−→ πf ⊗ H (E[mp ], ρπ(1 − n))

C which satisfy a certain equivariance property under the action of Ge (Af ) × H(Af ). Here H(−) denotes the elements of the associated Hecke algebra with rational coefficients which are invariant under right-translation by elements in C (and the convolution product is with respect to the fixed Haar measure in §2.1). Let ϕ be an element of πf which is fixed by the Siegel parahoric level at p. We define our Euler system to be (m) cmpr ≈ (vϕ ◦ Lmpr )(φ ) (m) for suitable test data φ , where vϕ denotes the evaluation map at the the vector ϕ. Here, by approximately we mean up to some suitable volume factors which ensure that the classes cmpr land in the cohomology of a Galois stable lattice. To prove the tame norm relation at ` - m, we choose a character χ: Gal(E[mpr]/E) → × C , which can naturally be viewed as a character of T(Af ) via the Artin reciprocity map, and consider the χ χ-component Lmpr of the map Lmpr . It is enough to prove the tame norm relation locally at ` and, by applying Frobenius reciprocity, the map can naturally be viewed as an element χ −1 L r ∈ Hom (π , χ χ). mp ` H0(Q`) 0,` This reduces the tame norm relation to a calculation in local representation theory, which constitutes the main part of this paper. Once this relation is established, we then sum over all χ-components to obtain the full norm relation. We now describe the ingredients that go into the local computation described above, all of which take place in section 4 of the paper. If ` is a rational prime that splits in E/Q, then there exists a prime λ|` and ∼ ∼ an isomorphism π0,` = Π0,λ compatible with the identification G0(Q`) = GL2n(Q`). Furthermore, since π0,f admits a H0-linear model, we also have Hom (π , ) 6= 0. H0(Q`) 0,` C

We show (Proposition 4.3) that this implies that π0,` has a local Shalika model and as a consequence of this, we prove the following theorem:

× × Theorem 1.7. Let χ: Q` → C be an unramified finite-order character. Then (1) dim Hom (π , χ−1 χ) = 1, where χ−1 χ denotes the character of H ( ) given by sending C H0(Q`) 0,` 0 Q` a pair of matrices (h1, h2) to χ(det h2/ det h1). (2) There exists a compactly-supported locally constant function φ: G0(Q`) → Z (independent of χ) such that, for all z ∈ Hom (π , χ−1 χ) and all spherical vectors ϕ ∈ π H0(Q`) 0,` 0 0,` 2 `n (1.8) z(φ · ϕ ) = L(π ⊗ χ, 1/2)−1z(ϕ ). 0 ` − 1 0,` 0 The first part of this theorem is a result of Jacquet and Rallis [JR96] in the case that χ is trivial, and a result of Chen and Sun [CS20] in the general case. Since π0,` admits a local Shalika model, one obtains an explicit basis of the hom-space by considering an associated zeta integral, and the proof of part 2 then involves manipulating this zeta integral by applying several U`-operators to its input. By taking a suitable linear combination of these operators, we are able to simultaneously produce the relation appearing in (1.8) without losing the eventual integrality of the Euler system classes. Such a linear combination arises naturally from an inclusion-exclusion principle on flag varieties associated with GLm. 1.2. Notation and conventions. We fix the following notation and conventions throughout the paper: • An embedding Q ,→ C and an imaginary quadratic number field E ⊂ Q. • For any rational prime `, we fix an embedding Q ,→ Q` and an isomorphism ∼ Q` = C compatible with the fixed embeddings Q ,→ C, Q ,→ Q` . In particular, for each `, we have a choice of λ above `. • A rational prime p > 2 that splits in E. 6 ANDREW GRAHAM AND SYED WAQAR ALI SHAH

• AF will denote the adeles of a number field F and AF,f the finite adeles; if F = Q we simply write S S A and Af respectively. For any integer (or finite set of primes) S, we write A (resp. Af ) to mean the adeles (resp. finite adeles) away from S (i.e. at all primes not dividing S/not in S), and AS or ZS for the adeles at S. • For a ring R and a connected reductive group G over Q, we let φ is locally constant H (G( ),R) := φ: G( ) → R : Af Af and compactly-supported

denote the Hecke algebra, which carries an action of G(Af ) given by right-translation of the argument. We will omit the ring R from the notation when it is clear from the context. Additionally, if K ⊂ G(Af ) is a compact open subgroup, then H (K\G(Af )/K) will denote the subset of K- biinvariant functions. We also have similar notation for the Q`-points of G. • For a number field F , we let GF denote the absolute Galois group of F . Furthermore, the global Artin reciprocity map × ab ArtF : AF → GF is defined geometrically, i.e. it takes a uniformiser to the associated geometric Frobenius. • For a prime v of F , we let Frobv denote an arithmetic Frobenius in GF . If L/F is an abelian extension of number fields that is unramified at v, we will sometimes write Frv for the arithmetic Frobenius in Gal(L/F ) associated with the prime v. • For an imaginary quadratic number field E and an integer m, we let E[m] denote the ring class field × × of conductor m, i.e. the finite abelian extension with norm subgroup given by E · Obm where

Om = Z + mOE. • The smooth dual of a representation will be denoted by (−)∨ and the linear dual by (−)∗. • For a smooth quasi-projective scheme X over a characteristic zero field, and a number field F , we let CHM(X)F denote the category of relative Chow motives over X with an F -structure, as defined in [Tor19, Definition 2.4]. Let DMB,c(X) denote the category of constructible Beilinson motives, as

in [CD19, Deﬁnition 15.1.1]. Then, for any Chow motive (Y, e, n) ∈ CHM(X)Q, one can associate a constructible Beilinson motive eMX (Y )(n) as the mapping ﬁbre 1−e eMX (Y )(n) := Cone MX (Y )(n) −−→ MX (Y )(n) [−1]

where MX (Y ) is as in [CD19, §11.1.2]. This association assembles into a functor CHM(X)Q → DMB,c(X), so we can naturally view a Chow motive as a constructible Beilinson motive. If (Y, e, n) is equipped with an F -structure, then so is eMX (Y )(n) (i.e. one has a Q-linear homomorphism

F,→ EndDMB,c(X) (eMX (Y )(n)) ). • The motivic cohomology of X with values in a relative Chow motive F ∈ CHM(X)F is the F -vector space given by • Hmot (X, F ) := HomDMB,c(X) (1X , F [•]) where 1X denotes the trivial motive. This definition is compatible with the one arising from algebraic K-theory by [CD19, Corollary 14.2.14], and since the categories DMB,c(−) satisfy the six functor formalism, we can define pushforwards/pullbacks between these cohomology groups (see [Lem17, §4.1] and the references therein). • All étalecohomology groups refer to continuous étalecohomology in the sense of Jannsen ([Jan88]). A subscript c will denote the compactly supported version. • If H and G are locally profinite groups, and σ, π smooth representations of H and G respectively, then σ π will denote the tensor product representation of H × G. • Unless specified otherwise, all automorphic representations are assumed to be unitary. • Unless specified otherwise, all modules over a group ring will be left modules. n −1 • Let ` be a rational prime. For a positive integer n we write [n]` := (` − 1)(` − 1) . This definition extends to [n]`! := [n]` · [n − 1]` ··· [1]` and n [n] ! := ` m ` [m]`! · [(n − m)]`!

with the obvious convention that [0]`! = 1. ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS 7

1.3. Acknowledgements. We are greatly indebted to David Loeffler, Chris Skinner and Sarah Zerbes – a large amount of this paper owes its existence to the ideas introduced in [LSZ17] on constructing Euler systems in automorphic settings. We are also very grateful to Wei Zhang for suggesting this problem to us. The second named author would like to extend his gratitude to Barry Mazur for his constant guidance and to Sasha Petrov, Ziquan Yang and David Yang for many enlightening discussions. The authors would like to thank the anonymous referee for their feedback, which has led to significant improvements in the presentation of this article. Finally, we would like to thank the organizers of the conference “Automorphic p-adic L-functions and regulators” held in Lille, France in October 2019. This conference provided a pleasant opportunity for the authors to work together in person. The first named author was supported by the Engineering and Physical Sciences Research Council [EP/L015234/1], the EPSRC Centre for Doctoral Training in Geometry and Number Theory (The Lon- don School of Geometry and Number Theory), University College London and Imperial College London.

2. Preliminaries 2.1. The groups. In this section, we will define the algebraic groups and their associated Shimura data that will appear throughout the paper. All groups and morphisms will be defined over Q unless specified otherwise, and maps of algebraic groups will be defined on their R-points for Q-algebras R. For an arbitrary algebraic group K, we denote by TK its maximal abelian quotient, and ZK its centre. Let Jr,s denote the (r + s) × (r + s) diagonal matrix diag(1,..., 1, −1,..., −1) comprising of r copies of 1 and s copies of −1. We define U(p, q) (resp. GU(p, q)) to be the unitary (resp. unitary similitude) groups whose R-points for a Q-algebra R are given by t U(p, q)(R) := g ∈ GLp+q(E ⊗Q R): gJ¯ p,qg = Jp,q t GU(p, q)(R) := g ∈ GLp+q,E(E ⊗Q R): there exists λ ∈ Gm(R) s.t. gJ¯ p,qg = λJp,q . The group GU(p, q) comes with two morphisms

c: GU(p, q) → Gm det: GU(p, q) → ResE/QGm where c(g) = λ is the similitude factor and det(g) is the determinant. To ease notation, we will omit q if it is zero.

Remark 2.1. These groups arise as the generic fibre of group schemes over Z by replacing E with OE in the definition. If ` is unramified in E, the pullback to Q` is unramified and the Z`-points form a hyperspecial maximal compact subgroup. Let n ≥ 1 be an integer. We define:

• G0 := U(1, 2n − 1). • G := GU(1, 2n − 1). • H0 := U(1, n − 1) × U(0, n). • H := GU(1, n − 1) ×c GU(0, n) where the product is ﬁbred over the similitude map.

We have natural inclusions ι: H0 ,→ G0 and ι: H ,→ G, both given by sending an element (h1, h2) to the 2n × 2n block matrix h1 ι(h1, h2) = . h2

Throughout, we will let T denote the algebraic torus U(1) = {x ∈ ResE/QGm,E :xx ¯ = 1} and we note that

• N : ResE/QGm,E → T is given by z 7→ z/z.

• ZG0 = T, ZH0 = T × T ∼ • ZG = GU(1), ZH = T × GU(1) ∼ ∼ • TH = T × GU(1) if n is odd, and TH = T × T × Gm if n is even. ∼ • ResE/Q Gm,E = GU(1). When n is odd, the quotient map is given explicitly by (n+1)/2 −1 H → TH, (h1, h2) 7→ (det h2/ det h1, c(h1) det(h1 )). Remark 2.2. This quotient in fact contains more information than we are interested in – to control the anticyclotomic variation of our Euler system we only need the fact that T is a quotient of H. 8 ANDREW GRAHAM AND SYED WAQAR ALI SHAH

We consider the group Ge = G × T. The group H extends to a subgroup of Ge via the map (h1, h2) 7→ (ι(h1, h2), ν(h1, h2)), where ν(h1, h2) = det h2/ det h1. We will need the following two lemmas later on: Lemma 2.3. The map

× N : AE,f → T(Af ) z 7→ z/z¯ is open and surjective. Proof. A continuous surjective homomorphism of σ-countable locally compact groups is open, and so it N suﬃces to prove that the map is surjective. We have an exact sequence 1 → Gm → GU(1) −−→ T → 1 of algebraic tori over Q, which gives an exact sequence × × 1 → Af → AE,f → T(Af ).

× 1 For each ﬁnite place `, the map (E ⊗Q`) → T(Q`) is surjective, since H (Q`, Gm) = 0 by Hilbert’s theorem 90. For ` - disc(E), both GU(1) and T have smooth models over Z` and the map extends to these models. 1 This induces a map GU(1)(F`) → T(F`). Since H (F`, Gm) = 0 by Lang’s lemma, this map is surjective and a lifting argument shows that GU(1)(Z`) → T(Z`) is also surjective. The claim for surjectivity on adelic points now follows.

Lemma 2.4. Consider the following groups

Z0 := ZG0 (A) and Z1 := ZG(A)G(Q) ∩ G0(A)

Then Z1 = Z0G0(Q).

Proof. The inclusion Z0G0(Q) ⊂ Z1 is obvious. Let zγ ∈ Z1 where z ∈ ZG(A) and γ ∈ G(Q). Then we have c(zγ) = zzc(γ) = 1 which implies that c(γ) is a norm locally everywhere. By the Hasse norm theorem, we must have that c(γ) × is a global norm, i.e. there exists an element ζ ∈ E = ZG(Q) such that c(γ) = c(ζ). Therefore, we can −1 replace z with zζ ∈ Z0 and γ with ζ γ ∈ G0(Q).

Notation. We ﬁx left Haar measures dg, dh and dt on G(Q`), H(Q`) and T(Q`) respectively. As these groups are unimodular [Ren10, p. 58], these are also right Haar measures. We normalise them so that the volume of any compact open subgroup lies in Q and the volume of any hyperspecial subgroup is 1 (when applicable). In particular, the products of these measures induce Haar measures on the Af -points of the corresponding groups, which we may also denote by the same letters.

Remark 2.5. If n is odd then the above groups are quasi-split at all finite places. Indeed, the groups H0 and H are automatically quasi-split at all finite places because there is only one unitary group (up to isomorphism) attached to Hermitian spaces of fixed odd dimension over a non-archimedean local field (see [Mor10, p. 152]). For G0 and G, we follow the argument in loc.cit., where the local cohomological invariant is calculated (the group is quasi-split when the invariant is 0). We note that these groups are automatically quasi-split at primes ` which do not divide the discriminant of E, because the group is unramified. If ` divides the discriminant of E, then the cohomological invariant is 0 if −1 is a norm in Q`. Otherwise, it is equal to 2n − 1 + n = 3n − 1 modulo 2. But this quantity is even, under the assumption that n is odd. Quasi-splitness at all finite places is used in the results invoked on weak base-change in section 3. However, we expect such results to hold without this hypothesis. 2.2. The Shimura–Deligne data. We now define the Shimura–Deligne data associated with H and G. Since the datum attached to the group H doesn’t satisfy all the usual axioms of a Shimura datum, we have extended the definition to that of a Shimura–Deligne datum (Appendix B). In this section, we will freely use notation and results from this appendix. ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS 9

: Let S = ResC/R Gm denote the Deligne torus and let hG and hH be the following algebraic homomorphisms:

hG : S → GR z 7→ (diag(z, z,¯ . . . , z¯))

hH : S → HR z 7→ (diag(z, z,¯ . . . , z¯), diag(¯z, . . . , z¯)) .

Let XG (resp. XH) denote the G(R) (resp. H(R)) conjugacy class of homomorphisms containing hG (resp. hH). Then the pairs (G,XG) and (H,XH) both satisfy the axioms of a Shimura–Deligne datum as in Definition B.1 (axioms [Del79, 2.1.1.1-2.1.1.2]). The datum (G,XG) also satisfies 2.1.1.3 in loc.cit. so it is a Shimura datum in the usual sense. In particular, if K ⊂ G(Af ) is a sufficiently small compact open subgroup (Definition B.5), we let ShG(K) denote the associated Shimura–Deligne variety whose complex points equal ShG(K)(C) = G(Q)\[XG × G(Af )/K]

and similarly for H. The reflex field for the datum (G,XG) is the imaginary quadratic number field E fixed in the previous section for n > 1, and Q otherwise (see Example B.1). However, we will always consider the associated Shimura–Deligne varieties as schemes over E, unless stated otherwise. If the level K is small enough, the embedding ι induces a closed embedding on the associated Shimura–Deligne varieties. More precisely Definition 2.6. Let w equal the 2n × 2n block diagonal matrix 1 w := ι(1, −1) = . −1

We say that a compact open subgroup K ⊂ G(Af ) is H-small if there exists a compact open subgroup 0 0 −1 K ⊂ G(Af ), containing both K and wKw, such that for any g ∈ G(Af ), gK g ∩ G(Q) has no non-trivial stabilisers for its action on XG.

Proposition 2.7. Let K ⊂ G(Af ) be an H-small compact open subgroup. Then the natural map ι ShH(K ∩ H(Af )) −→ ShG(K) is a closed immersion. Proof. This follows the same argument in [LSZ17, Proposition 5.3.1]. Indeed, one has an injective map ι: ShH ,→ ShG on the infinite level Shimura–Deligne varieties. So it suffices to prove that for any u ∈ K − K ∩ H(Af ), one has ShH u ∩ ShH = ∅ as subsets of ShG. If Q and Qu are both in ShH, we must have that u0 = u · wu−1w fixes Q. By the conditions on K0, we have that u0 = 1. This implies that u is in the centraliser of w, which is precisely H(Af ).

We also consider the trivial Shimura–Deligne datum (T,XT), where XT is the singleton hT(z) =z/z ¯ (considered as a T( )-conjugacy class of homomorphisms → T) and the datum (G,X ) = (G,X ×X ). R S e Ge e G T Then the data associated with the groups H and Ge are clearly compatible with respect to the inclusion (ι, ν), and if U = K × C ⊂ Ge (Af ) is a sufficiently small compact open subgroup, we have a finite unramified map Sh (U ∩ H( )) → Sh (U) ∼ Sh (K) × Sh (C). H Af Ge = G T which is a closed embedding when K is H-small. The reflex field for (G,X ) is equal to the imaginary e Ge quadratic field E.

2.3. Dual groups, base-change, and L-factors. In this section, we describe the dual groups attached to the groups deﬁned in the previous section and several maps between them. Let WF denote the Weil group of a local or global ﬁeld F . In each of the following examples, we describe the connected component of the L-group. The Weil group WE will always act trivially on this component, so to describe the full L-group, it will be enough to specify how complex conjugation acts. i+1 As in [Ski12, §2.2], let Φm = (−1) δi,m+1−j i,j. The dual groups are as follows: 10 ANDREW GRAHAM AND SYED WAQAR ALI SHAH

• Gb = GL1(C) × GL2n(C) and complex conjugation acts by −1t −1 (λ; g) 7→ (λ · det g;Φ2n g Φ2n).

• Hb = GL1(C) × GLn(C) × GLn(C) and complex conjugation acts as −1t −1 −1t −1 (λ; h1, h2) 7→ (λ · det h1 · det h2;Φn h1 Φn, Φn h2 Φn)

• Gb 0 = GL2n(C) and Hb 0 = GLn(C) × GLn(C) and complex conjugation acts via the same formulae above, but omitting the GL1 factor. −1 • Tb = GL1(C) and complex conjugation acts by sending an element x to its inverse x . • Geb = Gb × Tb and complex conjugation acts diagonally. : We also note that we can choose a splitting such that the dual group of R GLm = ResE/Q GLm is equal to −1t −1 −1t −1 GLm(C) × GLm(C) and complex conjugation acts by (g1, g2) 7→ Φm g2 Φm, Φm g1 Φm . This implies that the natural diagonal embedding Gb 0 → R\GL2n extends to a map of L-groups, which we will denote by L L L L BC. Similarly, we have a map BC: G → (R GL1 ×R GL2n). We also have a natural map G → G0 given by projecting to the second component, and this is compatible with base-change in the sense that we have a commutative diagram: L BC L G (R GL1 ×R GL2n)

L BC L G0 R GL2n where the right-hand vertical arrow is projection to the second component. L L The map G → G0 corresponds on the automorphic side to restricting an automorphic representation of G to G0. It turns out that all irreducible constituents in this restriction lie in the same L-packet, and one can always lift a representation of G0 to one of G. We summarise this in the following proposition.

Proposition 2.8. Let π be a cuspidal automorphic representation of G(A). Then all irreducible constituents

of π|G0 are cuspidal and lie in the same L-packet with L-parameter obtained by post-composing the L- L L parameter for π with the natural map G → G0. Conversely, if π0 is a cuspidal automorphic representation of G0(A), then π0 lifts to a cuspidal automorphic representation of G(A). Moreover, the same statements hold if we work over a local ﬁeld F , that is to say: if π is an irreducible admissible representation of G(F ), then all irreducible constituents of π|G0 lie in the same local L-packet L L (with parameter obtained by composing with GF → G0,F ), with cuspidal representations corresponding to each other, and every representation of G0(F ) lifts to one of G(F ). If F = R and π0 lies in the discrete series, then we can lift π0 to a discrete series representation of G(R).

If ωπ0 denotes the central character of a representation π0, and ω is a (unitary) character extending ωπ0 to ZG(Q)\ZG(A), then there exists a lift π of π0 as above with central character ω. A similar statement holds for local fields. Proof. See Proposition 1.8.1 (and the discussion preceeding it) in [HL04] and Theorem 1.1.1 in [LS18]. The last part follows from the techniques used in [LS18, Theorem 5.2.2] and Lemma 2.4. We will frequently need to talk about local L-factors attached to unramified representations of the groups defined in §2.1, so for completeness, we recall the definition here. We follow §1.11, §1.14 and §5.1 Example (b) in [BR94].

Definition 2.9. Let ` be a prime which is unramified in E/Q and for which G0,Q` is quasi-split. For a positive integer m, let Stdm denote the standard representation of GLm. L (a) Let F be a finite extension of Q`. For G ∈ {GL2n, GL1 × GL2n}, let rstd : GF → GL2n(C) denote the following representation: L Std2n(g) if G = GL2n with GF = GL2n(C) × WF rstd(g, x) = L (Std1 Std2n)(g) if G = GL1 × GL2n with GF = GL1(C) × GL2n(C) × WF

Let q denote the order of the residue field of F and Frob an arithmetic Frobenius in WF . For any unramified representation σ of G (F ), with corresponding (unramified) Langlands parameter ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS 11

L ϕσ : WF → GF , the standard local L-factor is deﬁned to be

−s −1 −1 L(σ, s) := det 1 − q (rstd ◦ ϕσ)(Frob )

where s ∈ C. ∼ (b) If ` splits in E/Q, then we have an isomorphism G0,Q` = GL2n,Q` , and for any unramified representation σ of G0(F ), we define the local L-factor of σ to be as in part (a), viewing σ as a representation ∼ of GL2n(F ) = G0(F ). We have a similar definition for representations of G(F ).

2.4. Discrete series representations. One has an identification GE = GL1,E × GL2n,E given by sending an element g to (c(g); g0), where g0 is the matrix obtained by projecting its entries to the first component of the identification ∼ E ⊗Q E −→ E ⊕ E x ⊗ λ 7→ (λx, λx).

1+2n We let Gm ⊂ GE denote the standard torus with respect to this identiﬁcation. Then any algebraic character of this torus is of the form

1+2n Gm → Gm

c0 Y ci (t0; t1, . . . , t2n) 7→ t0 · ti i 1+2n where c = (c0; c1, . . . , c2n) ∈ Z . Such a character is called dominant if c1 ≥ · · · ≥ c2n and these characters classify all irreducible algebraic representations of GE. Indeed, any such representation corresponds to a representation with highest weight given by c. There is a similar description for G0 by omitting the ﬁrst Gm-factor. The L-packets of discrete series representations of G(R) are described by the above algebraic representations. More precisely, following [Clo91, §3.3] (the extension to similitude groups is immediate), if π∞ lies in L × the discrete series, the Langlands parameter ϕ∞ : WR → G associated with π∞, restricted to WC = C , is equivalent to a parameter of the form

p p + 2n−1 p + 2n−1 z 7→ (z/z¯) 0 ;(z/z¯) 1 2 ,..., (z/z¯) 2n 2

with pi ∈ Z and p1 > ··· > p2n. Then π∞ corresponds to an irreducible algebraic representation of GE with highest weight c = (c0; c1, . . . , c2n) satisfying ci = pi + (i − 1). We have a similar description for discrete series L-packets for G0(R).

Lemma 2.10. Let π0 be a cuspidal automorphic representation of G0(A) such that π0,∞ lies in the discrete series L-packet corresponding to the highest weight c = (c1, . . . , c2n). Suppose that the central character of π0 is trivial. Then there exists a cuspidal automorphic representation π of G(A) such that: • π has trivial central character. • π∞ lies in the discrete series L-packet corresponding to the weight (0; c1, . . . , c2n).

Proof. By looking at the inﬁnitesimal character of π0,∞, we must have c1 + ··· + c2n = 0 because the central character is trivial. By Proposition 2.8, there exists a cuspidal automorphic representation π of G(A) lifting π0 such that π∞ lies in the discrete series. Furthermore, we can arrange it so that the central character of π is trivial. In particular, if (c0; c1, . . . , c2n) is the weight corresponding to the L-packet of π∞, then since π has trivial central character, we must have

2n X c0 + ci = 0. i=1

This implies that c0 = 0.

We call π in the above lemma a unitary lift of π0. We are interested in algebraic representations V of 0 GE with highest weight c = (0; c1, . . . , c2n) that are self dual: in this case, we must have c = −c , where 0 c = (0; c2n, . . . , c1). We have the following “branching law”: 12 ANDREW GRAHAM AND SYED WAQAR ALI SHAH

Lemma 2.11. Let V be a self-dual algebraic representation of GE with highest weight c = (0; c1, . . . , c2n) as ∗ ∗ above and let ι V denote its restriction to an algebraic representation of HE. Then ι V contains the trivial representation as a direct summand, with multiplicity one. The same statement holds for representations of Ge E of the form Ve := V 1, with respect to the embedding (ι, ν). Proof. This is well-known and follows from Proposition 6.3.1 in [GRG14] for example. These self-dual representations will correspond to the coeﬃcient sheaves that we will consider throughout the paper. More precisely we make the following deﬁnition.

0 Deﬁnition 2.12. Let c be a dominant weight of GE satisfying c = −c (so the associated representation V = Vc is self dual). We consider the following algebraic representation of Ge E given by Ve := V 1. The ? restriction (ι, ν) Ve contains the trivial representation of HE with multiplicity one, by Lemma 2.11. We denote the inclusion of this factor by br.

2.5. Coefficient sheaves. Let p be a prime that splits in E/Q and let P denote the prime of E lying above p which is fixed by the embedding E,→ Qp (so EP is identified with Qp). In this section we briefly describe how to associate motivic and P-adic sheaves to algebraic representations of G. We follow [Pin92], [Anc15], [Tor19] and [LSZ17] closely (although note that the conventions are slightly different in the latter to that of the first three – we follow conventions in the first three references). In particular, this will mean that the multiplier character of G is sent to Tate motive under the functor AncG. Since the group H does not give rise to a Shimura datum in the usual sense, we have provided justifications for the results in this section in Appendix C. Let Tn be a finite-dimensional representation of a (sufficiently small) compact open subgroup K ⊂ G(Af ) n with coefficients in Z/p Z. Since Tn is finite, there exists a finite-index normal subgroup L ⊂ K that acts trivially on Tn. We let µG,K,n(Tn) be the locally constant étalesheaf of abelian groups on ShG(K) corresponding to (ShG(L) × Tn) /Γ → ShG(K) where Γ := K/L acts via (x, t) · h = (xh, h−1t). This construction is independent of the choice of L. ¯ If x0 = Spec(k) is a point on ShG(K) and x = Spec(k) is the associated geometric point obtained from fixing a separable closure of k, then we can explicitly describe the action of Gal(k/k¯ ) on the stalk ¯ (µG,K,n(Tn))x. Indeed, lift x to a geometric point xe on ShG(L). Then for σ ∈ Gal(k/k), there exists a unique element ψ(σ) ∈ Γ such that ψ(σ) σ · xe = xe . Since the Galois action commutes with that of Γ, the map ψ defines a homomorphism ψ : Gal(k/k¯ ) → Γ.

The stalk (µG,K,n(Tn))x is isomorphic to Tn with the Galois action given by ψ. The functors µG,K,n are compatible as n varies: if we let T be a continuous representation of K with coefficients in Zp, then we let TK = µG,K (T ) denote the corresponding sheaf on ShG(K). By considering the category of lisse sheaves on ShG(K) with coefficients in Zp up to isogeny, this construction extends to finite-dimensional continuous representations of G(Qp) with coefficients in Qp. In particular, any algebraic representation of GQp gives rise to a continuous representation of G(Qp) (with coefficients in Qp). One obtains an additive tensor functor µ : Rep (G) → Et(Sh´ (K)) G,K Qp G Qp from the category of algebraic representations of G over Qp to the category of lisse sheaves on ShG(K) with coefficients in Qp. In [Anc15] (see also [Tor19, §10]), Ancona shows that this functor µG,K fits into a commutative diagram (up to natural isomorphism)

AncG,K RepE(G) CHM (ShG(K))E

−⊗EP r´et

µG,K Rep (G) Et´ (ShG(K)) Qp Qp

where CHM(−)E denotes the category of relative Chow motives (over E) and r´et denotes the P-adic reali- sation of a motive. The functor AncG,K is additive and preserves duals and tensor products. ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS 13

Remark 2.13. Ancona’s construction applies to general PEL Shimura–Deligne varieties (see Appendix C). In particular, since ShH is also PEL, all of the constructions in this section hold for the group H. Furthermore, these constructions extend to representations of Ge of the form V 1 (note that Ge does not give rise to a PEL datum).

2.6. Equivariant coeﬃcient sheaves. If we wish to study the sheaves VK = µG,K (V ) and AncG,K (V ) as K varies, we need to keep track of the action of G( f ). More precisely, let Et´ (ShG) (resp. CHM (ShG) ) A Qp E denote the category of equivariant ´etalesheaves (resp. relative Chow motives) – that is to say – a collection ∗ ∼ −1 of sheaves VK and isomorphisms σ VK = VL for any σ ∈ G(Af ) such that σ Lσ ⊂ K. By [LSZ17, Proposition 6.2.4], the functors µG,K and AncG,K are compatible as K varies, so give rise to functors:

• AncG : RepE(G) → CHM (ShG)E • µG : Rep (G) → Et´ (ShG) Qp Qp which are compatible under P-adic realisations. Given such an equivariant sheaf V , one can deﬁne H• (Sh , ) := lim H• (Sh (K), ) G V −→ G VK K which is a smooth admissible G(Af )-representation. We will sometimes drop the level from the subscript when we talk about the cohomology of these equivariant sheaves at ﬁnite level. Remark 2.14. As in the previous section, the above results also apply to the group H and representations of Ge of the form V 1.

Let RepE(Ge )0 denote the full subcategory of RepE(Ge ) consisting of representations of the form V 1. We have the following compatibility with base-change. Proposition 2.15. One has a commutative diagram (up to natural isomorphism):

Anc Rep (G) Gf CHM Sh E e 0 Ge E

(ι,ν)∗ (ι,ν)∗

AncH RepE(H) CHM (ShH)E Proof. By [Tor19, Theorem 9.7] the theorem holds for the pair of groups (G, H) since the morphism of associated PEL data is “admissible” (Deﬁnition 9.1 in op.cit.). The result immediately extends to the

category RepE(Ge )0. We will apply this proposition to the representations Ve defined in Definition 2.12. Indeed, since (ι, ν)?Ve contains the trivial representation with multiplicity one, we have a map of relative Chow motives br: E → (ι, ν)? Anc (V ). Ge e This “branching law” will allow us to construct classes in the motivic cohomology of Sh with coefficients Ge in Anc (V ) ⊗ E(n), where E(−1) denotes the Lefschetz motive. Ge e 3. Cohomology of unitary Shimura varieties In this section we recall the construction of Galois representations attached to cuspidal automorphic representations for the group G = GU(1, 2n − 1). To apply the results of [Mor10], we assume that n is odd throughout this section. Let π0 be a cuspidal automorphic representation of G0(A) such that π0,∞ lies in the discrete series, and let π denote a lift to G(A) (see Proposition 2.8). If π0 has trivial central character, then we take π to be a lift with trivial central character. Assume that π is cohomological, i.e. there exists: K • A sufficiently small level K ⊂ G(Af ) such that πf 6= {0}. • An irreducible algebraic representation V of GE and an integer i ∈ Z such that i 0 H (gC,K∞; π∞ ⊗ VC) 6= {0} 0 ◦ where g = Lie(G(R)) and K∞ = K∞ZG(R) , with K∞ a maximal compact subgroup of G(R). 14 ANDREW GRAHAM AND SYED WAQAR ALI SHAH

We say that a representation Π = ψ Π of GL ( ) × GL ( ) is θ-stable if Πc ∼ Π∨ and ψ = ψcωc , 0 1 AE 2n AE 0 = 0 Π0 where ωΠ0 is the central character of Π0. Recall that our choice of Hermitian pairing implies that the group G is quasi-split at all ﬁnite places. Then we have the following result due to Morel:

Proposition 3.1. Let S be the set of places of Q containing ∞, all primes which ramify in E and all places where K is not hyperspecial (so π is unramiﬁed outside S).

(1) There exists an irreducible admissible representation Π of ResE/Q GE(A) such that for all but ﬁnitely many primes ∼ (3.2) ϕΠ` = BC (ϕπ` )

where ϕΠ` and ϕπ` are the local Langlands parameters of Π` and π` respectively, and BC is the local base-change morphism defined in section 2.3. We say Π is a weak base-change of π. (2) If the representation Π above is cuspidal then any weak base-change of π is a θ-stable regular algebraic cuspidal automorphic representation and (3.2) holds for all `∈ / S. Furthermore, the infinitesimal θ ∗ θ character of Π∞ is the same as that of (V ⊗V ) , where V is the representation with highest weight (c0 + ··· + c2n; −c2n,..., −c1). (3) Suppose that Π is cuspidal and π has trivial central character (for example, if π0,f admits a H0-linear ∼ model). Then under the isomorphism ResE/Q GE(AQ) = GL1(AE) × GL2n(AE), the representation Π is of the form 1 Π0 for some conjugate self-dual regular algebraic cuspidal automorphic representation Π0 of GL2n(AE) satisfying ϕΠ0,` = BC ϕπ0,` for all `∈ / S. Proof. Most of the proposition follows from Corollary 8.5.3, Remark 8.5.4 and the proof of Lemma 8.5.6 in [Mor10]. The reason that (3.2) holds for all `∈ / S follows from the fact that G is quasi-split at all finite places. Moreover, if π0,f admits a H0-linear model then the assertion that Π is of the form 1 Π0 follows from the commutative diagram in section 2.3 and the fact that the central character of πf is trivial. Indeed, if we write Π = ψ Π0 then it is enough to show that ψ` = 1 for all primes `∈ / S, and this follows from a direct computation involving Satake parameters. For the assertion about the infinitesimal character see the proofs of [Mor10, Lemma 8.5.6] and [Ski12, Theorem 9]. Remark 3.3. If the highest weight of the representation V is regular then the above proposition follows from Theorem A in [Ski12] (which also builds on the work of Morel). However we are only interested in the case where Π is cuspidal which does not require this assumption. From now on we assume that any weak base change of π is cuspidal. In this case, thanks to the work of many people (see for example [CH13], [HT01], [Mor10], [Shi11] and [Ski12]), there exists a semisimple Galois representation attached to π. Theorem 3.4. Let π be as above and assume that the central character of π is trivial. Suppose that a weak base-change Π = 1 Π0 of π is cuspidal. Then there exists a continuous semisimple Galois representation ∼ ρπ : GE → GL(Wπ) = GL2n(Qp) such that

(1) ρπ is conjugate self-dual (up to a twist), i.e. one has an isomorphism c ∼ ∗ ρπ = ρπ(1 − 2n) c where ρπ denotes the representation given by conjugating the argument by complex conjugation in GQ (recall that we have ﬁxed an embedding Q ,→ C).

(2) For all `∈ / S ∪ {p} and λ|`, the representation ρπ|Eλ is unramified and satisfies −s −1 −s −1 Pλ(Nm λ ) := det 1 − Frobλ (Nm λ) | ρπ = L (Πλ, s + (1 − 2n)/2) . In particular, if ` splits in E/Q and λ is the prime lying above ` corresponding to the fixed embedding −n −1 Q ,→ Q`, then Pλ(` ) = L(π`, 1/2) .

(3) If πp is unramified (i.e. p∈ / S) then for any prime v dividing p the representation ρv := ρπ|Ev is crystalline with jumps in the Hodge–Tate filtration occurring at ci + 2n − i i = 1,..., 2n if v is the place fixed by the embedding Q ,→ Qp −c2n+1−i + 2n − i i = 1,..., 2n otherwise ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS 15

∼ Furthermore, via the ﬁxed isomorphism C = Qp, the characteristic polynomial of the crystalline −1 Frobenius ϕ on Dcris(ρv) is equal to the characteristic polynomial of ϕ 1/2−n (Frob ), where Π0,v ⊗|·|v v

ϕ 1/2−n : WEv → GL2n( ) Π0,v ⊗|·|v C 1/2−n is the local Langlands parameter attached to the representation Π0,v ⊗ | · |v . Proof. See Theorem 3.2.3 in [CH13]. The third part follows from the explicit recipe of (1.5) in op.cit.. Note that the representation ρπ, which a priori is associated with Π, only depends on π. This is because the Galois representation constructed by Chenevier–Harris is semisimple (so is determined by its behaviour at all but finitely many primes, for any finite set of primes), and Π is a weak-base change of π (so its local Langlands parameters are determined by π for all but finitely many primes). In [Mor10, Corollary 8.4.6], Morel shows that, up to multiplicity, this representation appears in the intersection cohomology of the Baily–Borel compactification of ShG. However, to be able to construct an Euler system, we would like this representation to occur in the (compactly-supported) cohomology of the open Shimura variety. In fact to obtain the correct Euler factor, we work with the dual of the representation and assume that it appears as a quotient of the (usual) étale cohomology. We would also like ρπ to be absolutely irreducible (which is expected because we have assumed that Π is cuspidal) – however this is currently not known in general. We summarise this in the following assumption:

Assumption 3.5. Let π0 be a cuspidal automorphic representation of G0(A) such that π0,∞ lies in the discrete series, and let π be a lift to G(A). Suppose that π is cohomological with respect to the representation V , and suppose that any weak base-change of π is cuspidal. Then we assume that:

• ρπ is absolutely irreducible. • There exists a GE × G(Af )-equivariant surjective map 2n−1 ∗ ∨ ∗ pr ∨ :H Sh , V (n) ⊗ π ⊗ W (1 − n). π ´et G,Q Qp f π

(Here, we are identifying Qp with C via the fixed isomorphism in section 1.2). Remark 3.6. If π is tempered, non-endoscopic and the global L-packet containing π is stable, then we expect that the above assumptions hold true for the following reasons. Since π is tempered the cohomology should vanish outside the middle degree and π is not a CAP representation.2 In particular, the latter should imply that πf does not contribute to the cohomology of the boundary strata of the Baily–Borel compactification of ShG. Since the L-packet containing π is stable, this will imply that any weak base-change of π is cuspidal, and finally, since π is non-endoscopic then (at least for p large enough) the representation ρπ should be absolutely irreducible of dimension 2n.

Remark 3.7. If we assume that Π0,v is supercuspidal at some non-archimedean place v of E, then the Galois representation ρπ will be absolutely irreducible for local reasons. In fact, by choosing p large enough, we can + even ensure that ρπ is residually absolutely irreducible (see [LTX 19, Lemma E.8.3]).

4. Horizontal norm relations This section comprises of the local calculations needed to prove the “tame norm relations” for our anticyclotomic Euler system at split primes. In [LSZ17], the key input for proving the norm relations is the existence of a Bessel model and a zeta integral that computes the local L-factor of the automorphic representation. The analogous notion in our case will be that of a Shalika model. This section is entirely local and independent of the rest of the paper – in particular, we can relax the assumption that n is odd. For this section only, we consider

G = GL1 × GL2n H = GL1 × GLn × GLn

G0 = GL2n H0 = GLn × GLn which we view as algebraic groups over Z` and come with diagonal embeddings H,→ G and H0 ,→ G0. We ﬁx an irreducible, smooth, admissible, unramiﬁed representation σ of G0(Q`) and let π = 1 σ be its

2CAP = cuspidal but associated to a parabolic. 16 ANDREW GRAHAM AND SYED WAQAR ALI SHAH extension to G(Q`). For our applications later on, it will be necessary to phrase everything in terms of the extended group Ge := G × GL1. The group H lifts to a subgroup of Ge via the map h1 (α, h1, h2) 7→ α, , ν(h1, h2) h2 × × −1 where ν(h1, h2) = det h2/ det h1. In particular, for any character χ: Q` → C , the product π χ is a −1 representation of Ge(Q`) where the action of the subgroup H(Q`) is now twisted by χ ◦ ν. Using the theory of Shalika models, we will prove the following theorem:

Theorem 4.1. Let π = 1 σ be as above. Then there exists a locally constant compactly-supported function φ: Ge(Q`) → Z satisfying the following: × × −1 (1) For any choice of ﬁnite-order unramiﬁed character χ: Q` → C , any element z ∈ HomH (πχ , C) −1 and any spherical vector ϕ ∈ π χ , we have the relation 2 `n z(φ · ϕ) = L(σ ⊗ χ, 1/2)−1z(ϕ) ` − 1 where L(σ ⊗ χ, s) is the standard local L-factor attached to σ ⊗ χ. Pn (2) The function φ is an (integral) linear combination of indicator functions φ = r=0 br ch ((gr, 1)K) where 1 `−1X g = 1 × r r 1

and Xr = diag(1,..., 1, 0,..., 0) is the diagonal matrix having 1 as its ﬁrst r-entries (we set X0 = 0) and K = Ge(Z`). (3) If we set K1 = G(Z`) × (1 + `Z`) and −1 V1,r := (gr, 1) · K1 · (gr, 1) ∩ H(Q`)

then V1,r is contained in H(Z`) and the coefficients in (2) satisfy −1 (` − 1) · br · [H(Z`): V1,r] ∈ Z. 4.1. Shalika models. Let Q denote the Siegel parabolic subgroup of G given by ∗ ∗ Q = GL × 1 ∗ and let Q0 denote its projection to G0. We denote the unipotent radical of Q (resp. Q0) by N (resp. N0) and define the Shalika subgroup S0 ⊂ G0 to be the subgroup of all matrices of the form h hX h × where h ∈ GLn and X ∈ Mn (the space of n-by-n matrices). Let ψ0 : Q` → C denote the standard additive × character on Q` which is trivial on Z` and define ψ to be the character S0(Q`) → C given by h hX ψ = ψ (tr X). h 0

Deﬁnition 4.2. We say that σ has a Shalika model if there exists a non-zero G0(Q`)-equivariant homomorphism G0(Q`) S• : σ → Ind (ψ). S0(Q`) Shalika models are unique when they exist (see [JR96]) and their existence is closely related to whether σ is H0(Q`)-distinguished or not. Indeed, we have the following:

Proposition 4.3. If HomH0 (σ, C) 6= 0, then σ admits a Shalika model.

Proof. Let B0 denote the standard Borel subgroup of G0. Let wn denote the longest Weyl element in GLn(Q`), i.e. the anti-diagonal matrix with all entries equal to 1, and consider the 2n × 2n block matrix 1 u = . wn 1 ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS 17

A routine calculation shows that the B0(Q`)-orbit Y of [u] ∈ H0(Q`)\G0(Q`) =: X is open, dense and the −1 stabiliser Bu = B0(Q`) ∩ u H0(Q`)u of [u] is equal to the subgroup t × Bu = : t = diag(t1, . . . , tn) and ti ∈ Q` . wntwn Suppose first that σ is a principal series representation, i.e. σ = IndG0(Q`) µδ1/2 where µ is an unramified B0(Q`) B0 character of the standard torus in G0(Q`). Then, by applying Frobenius reciprocity twice we have Hom (σ, ) = Hom (σ, C∞(X)) = Hom (C∞(X), σ∨) = Hom (C∞(X), µ−1δ1/2). H0 C G0 G0 c B0 c B0 For the last equality we have used the fact that σ∨ =∼ IndG0(Q`) µ−1δ1/2. If this space is non-zero then we B0(Q`) B0 must have Hom (C∞(Y ), µ−1δ1/2) 6= 0, which implies that B0 c B0 µ| = µδ1/2| = δ = 1 Bu B0 Bu Bu by [Sak08, §4.4.1] (one checks that δB0 (Bu) = 1). This implies that there exists a Weyl element w such that w −1 µ (t) = 1 for all t = diag(t1, . . . , tn, t1, . . . , tn) and thus σ has a Shalika model by applying Proposition 1.3 in [AG94]. In general, let σ be the unique unramified quotient of σ0 = IndG0(Q`) µδ1/2, where µ is unramified. Let B0(Q`) B0 K = GL2n(Z`) denote the standard maximal compact open subgroup of G0(Q`). Since ψ is trivial on S0(Q`) ∩ K, it is enough to show that 0 0 HomH0 (σ , C) 6= 0 ⇒ HomS0 (σ , ψ) 6= 0. But this follows from the above argument, using the fact that one still has an isomorphism (IndG0(Q`) µδ1/2)∨ =∼ B0(Q`) B0 IndG0(Q`) µ−1δ1/2 even if the representation is not irreducible. B0(Q`) B0 From now on, we assume that σ admits a Shalika model. This will allow us to define certain zeta integrals that compute the local L-factors of σ (and of all its twists by unramified characters). Let dx denote the fixed Haar measure on GLn(Q`) normalised so that the volume of GLn(Z`) is equal to 1. × × Proposition 4.4. Let χ: Q` → C be any continuous character. For Re(s) large enough, the integral Z x Z(ϕ, χ; s) := S χ(det x)| det x|s−1/2dx ϕ 1 GLn(Q`) converges. The lower bound on Re(s) can be chosen independently of ϕ. Furthermore, the function Z(ϕ, χ; s) can be analytically continued to a holomorphic function on all of C, and if ϕ0 is a spherical vector such that

Sϕ0 (1) = 1 then Z(ϕ0, χ; s) = L(σ ⊗ χ, s).

Proof. See [DJR18, §3.2]. The zeta integral above provides us with a linear model for σ.

× × Proposition 4.5. Let χ: Q` → C be a finite-order unramified character. Then the linear map given by ϕ 7→ Z(ϕ, χ; 1/2) constitutes a basis of −1 (4.6) HomH (π χ , C). −1 Proof. Since the underlying spaces for σ, π and π χ are the same, the linear map in the statement of the proposition is well-defined. It is straightforward to verify that Z(h · ϕ, χ; s) = χ(ν(h)) |ν(h)|s−1/2 Z(ϕ, χ; s) for any h = (h1, h2) ∈ H(Q`). Taking s = 1/2 gives the desired H-equivariance, and the map is clearly −1 non-zero. Finally, note that HomH (π χ , C) = HomH0 (σ, χ ◦ ν) and χ ◦ ν is the product of two characters −1 γ0, γ1, one for each factor of H0(Q`), each given by the composition of det with χ , χ respectively. The result now follows from [CS20]; since χ has finite-order, the character γ0 γ1 is always “good” and therefore the Hom space is at most one-dimensional (c.f. the terminology preceding Theorem C and the discussion that follows in op.cit.). 18 ANDREW GRAHAM AND SYED WAQAR ALI SHAH

4.2. Recap on lattices and flag varieties. In this subsection we introduce the objects that will be used throughout the rest of the section. Let M denote the submonoid of all matrices in GLn(Q`) with entries in ≥r Z`, and let M be the subset of all matrices in M whose determinant has `-adic valuation ≥ r. For brevity, we set Kn = GLn(Z`). n Definition 4.7. Let Ln denote the set of all lattices inside Q` , i.e. all Z`-submodules which are free of rank n. The set Ln can be identified with the quotient space Kn\ GLn(Q`) via the map ∼ Kn\ GLn(Q`) −→Ln

Kn · g 7→ L[g]. n where L[g] is the lattice spanned by the rows of g (we view elements of Q` as row vectors). For a lattice L ∈ Ln, let ML = {g ∈ GLn(Q`): L[g] ⊂ L} denote the subset of all matrices in GLn(Q`) whose rows span a sublattice inside L. This contains the same amount of information as the original lattice L, but it will be useful to consider the subset ML later on. In particular, ML[g] is simply equal to the translate Mg. ◦ Let Ln ⊂ Ln denote the ﬁnite subset of all lattices satisfying n n `Z` ⊂ L (Z` n which is closed under intersection. This can be of course identiﬁed with all proper subspaces of F` , a perspective we will frequently take.

Deﬁnition 4.8. A chain C of lattices in Ln of length r is a sequence

Lr ( Lr−1 ( ··· ( L0.

We let Cmin = Lr denote the minimal element of the chain C.

Let Bn and Tn denote the standard Borel and torus in GLn respectively, and let X∗(Tn) denote the + cocharacter group Hom(Gm,Tn). Fixing this choice of Borel and torus determines a subset X∗(Tn) ⊂ X∗(Tn) of dominant cocharacters. Explicitly, every dominant cocharacter is of the form

Gm −→ Tn z 7→ diag(za1 , . . . , zan )

where ai are integers satisfying a1 ≥ · · · ≥ an, therefore we will often refer to such a cocharacter by the tuple (a1, . . . , an). Recall that one has the Cartan decomposition of GLn(Q`) G GLn(Q`) = Knχ(`)Kn χ where the disjoint union is over all dominant cocharacters as above.

Deﬁnition 4.9. For a lattice L = L[g] in Ln, we let inv(L) denote its relative position with respect to the n standard lattice Z` , i.e. inv(L) = χ(`) where χ is the unique dominant cocharacter as above, such that KngKn = Knχ(`)Kn. We introduce the following matrices which will play a key role in the rest of this section. Deﬁnition 4.10. For 1 ≤ m ≤ n, let

tm = diag(`, . . . , `, 1,..., 1) ∈ GLn(Q`) denote the diagonal matrix whose ﬁrst m entries are equal to `.

◦ Remark 4.11. The space Ln can be viewed as the subset of Ln generated by all intersections of lattices of relative position t1, or equivalently, as the subset of all lattices with relative position equal to tm, for some 1 ≤ m ≤ n. We will also need to talk about ﬂag varieties of arbitrary signature, so we recall the deﬁnition for the convenience of the reader. ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS 19

Definition 4.12. Let d0 < d1 < ··· < dk be positive integers satisfying d0 = 0 and dk = m. The flag variety Fl(d1, . . . , dk; F`) of signature (d1, . . . , dk) is the space of all flags m {0} = V0 ⊂ V1 ⊂ · · · ⊂ Vk = F` where Vi is a subspace of dimension di.

We will frequently use the fact that Fl(d1, . . . , dk; F`) is a ﬁnite set of order equal to m [m] ! := ` n1 ··· nk ` [n1]`! ··· [nk]`!

where ni = di − di−1, with notation as in section 1.2.

4.3. Lattice counting. To study the zeta integrals introduced in Proposition 4.4 and their translates by certain Hecke operators, it will be necessary to formulate an inclusion-exclusion principle for the monoid M. Essentially, we will use this principle to count the elements in M ≥1 with respect to a suitable measure arising from the Shalika model for π0. Fix a left-invariant Haar measure on GLn(Q`) (which must also be right-invariant), and let f : GLn(Q`) → be any locally constant function such that R f(x)dx < +∞. Then the assignment C GLn(Q`) Z X 7→ µf (X) := f(x)dx X

deﬁnes a (ﬁnite) measure on the σ-algebra generated by all compact open subsets of GLn(Q`).

Proposition 4.13 (Inclusion-Exclusion). Let f : GLn(Q`) → C be any locally constant function. Then   ≥1 [ X l(C) (4.14) µf M = µf  ML = (−1) µf (MCmin ) ◦ L∈Ln C∈C ◦ where C denotes the set of chains in Ln and l(C) is the length of the chain C (Definition 4.8). ◦ ◦ Proof. We follow [Nar74]. For any non-empty finite subset S of Ln, we define a subset S ⊂ Ln as follows: 0 0 S set S0 = S, and for i ≥ 0, set Si+1 = Si ∪ {L ∩ L |L, L ∈ Si}. Then we take S = i Si. The collection of all ◦ ◦ such subsets S obtained from any finite subset S is denoted by O(Ln). Recall that for any L ∈ Ln, we have set ≥1 ML = {g ∈ GLn(Q`) | L[g] ⊂ L} ⊂ M . ≥1 S ≥1 n n n Then, M = ◦ ML. Indeed, for g ∈ M one has L[g] , which implies that L[g] + ` . L∈Ln (Z` Z` (Z` n Therefore, g ∈ ML with L = L[g] + `Z` . By the inclusion-exclusion principle for finite measures, we have ! ≥1 X |S|−1 \ (4.15) µf M = (−1) µf ML . ◦ ∅6=S⊂Ln L∈S ◦ Fix an element Y ∈ O(Ln) and let 0 0 0 0 Y0 := Y − {L ∩ L | L, L ∈ Y,L 6⊂ L ,L 6⊂ L} i.e. the subset obtained by deleting the intersection of incomparable pairs of lattices. Set s = |Y0|, t = |Y |, ◦ l = t − s. Clearly, for any non-empty S ⊂ Ln, we have S = Y if and only if Y0 ⊂ S ⊂ Y . Therefore l X X l (−1)|S| = (−1)s (−1)k k ◦ k=0 S⊂Ln, S=Y ( (−1)s if Y = Y = 0 0 otherwise 20 ANDREW GRAHAM AND SYED WAQAR ALI SHAH

T Moreover, Y = Y0 if and only if elements of Y form a chain C ∈ C. Using the fact that L∈C ML = MCmin , the right hand side of (4.15) is equal to   X X |S|−1 X l(C)  (−1)  µf (MCmin ) = (−1) µf (MCmin ) C∈C S: S=C C∈C

and the claim follows.

We obtain the following corollary.

Corollary 4.16. Keeping the same notation as in Proposition 4.13, suppose that f is invariant under right-translation of the argument by Kn. For an integer 1 ≤ m ≤ n let

m−1 ◦ ◦ X j chains in Ln of length j λm = #{L ∈ Ln : inv(L) = tm}· (−1) · # . with minimal element L[tm] j=0

Then ≥1 Pn (1) µf (M ) = m=1 λmµf (Mtm). (2) We have n λ ≡ (−1)m+1 m m Pn modulo ` − 1, which implies that m=1 λm ≡ 1 modulo ` − 1.

Proof. Since f is invariant under right-translation by Kn, by Proposition 4.13 and the fact that any element ◦ in Ln has relative position tm for some m (Remark 4.11), we have

n m−1 ≥1 X X X j µf (M ) = (−1) Dj,m · µf (ML)

m=1 inv(L)=tm j=0 n X = λm · µf (Mtm) m=1

where Dj,m is the number of chains of length j with maximal element L[tm]. To calculate Dj,m it is enough m ◦ n to study subspaces of F` (since Ln can be identified with all proper subspaces of F` ). Note that Dj,m is equal to the sum over (d1, . . . , dj+1) of # Fl(d1, . . . , dj+1; F`), where the latter space is m the flag variety of signature (d1, . . . , dj+1) associated to the vector space F` (Definition 4.12). The order of this flag variety is congruent modulo ` − 1 to the multinomial coefficient

n1 ··· nj+1

where ni = di − di−1 (and d0 = 0). This implies that Dj,m is congruent to the sum of these coeﬃcients over m all ni such that ni 6= 0 for all i = 1, . . . , j + 1. The total sum of all coeﬃcients is (j + 1) by the multinomial theorem. We compute Dj,m as follows:   m X m X X X X X Dj,m ≡ (j + 1) − = (j + 1) −  − + · · · ±  .

some ni=0 i ni=0 i,j ni=nj =0 all ni=0

This is equal to

j j + 1 j + 1 X j + 1 D ≡ (j + 1)m − jm − (j − 1)m + · · · ± 0 = (−1)k (j + 1 − k)m. j,m 1 2 k k=0 ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS 21

Consider the following variable change (r, t) = (j − k, k). This implies that

m−1 m−1 m−r−1 X X X r + t + 1 (−1)jD ≡ (−1)r(r + 1)m j,m t j=0 r=0 t=0 m−1 X m + 1 = (−1)r(r + 1)m m − r − 1 r=0 m−1 X m + 1 = − (−1)s P (s) s s=0 where P (X) = (X − m)m and for the last equality we have used the change of variables s = m − r − 1. Now we have P (m) = 0 and P (m + 1) = 1, and since P is a polynomial of degree < m + 1, by the standard identities involving binomial coeﬃcients we have

m−1 X m + 1 − (−1)s P (s) = (−1)m+1P (m + 1) = (−1)m+1. s s=0

◦ To complete the proof we note that #{L ∈ Ln : inv(L) = tm} is equal to the number of points in the n Grassmanian of (n − m)-dimensional subspaces of F` . This quantity is congruent to n n = n − m m

modulo ` − 1.

◦ Example 4.1. Let n = 2 and suppose that f is right Kn-invariant. Then there are ` + 2 elements in L2; ` + 1 of them have relative position t1 and we will denote them by L1,...,L`+1, the other has relative position t2, and we will denote it by S. There are ` + 2 chains of length zero and ` + 1 chains of length 1, namely they are all of the form S ⊂ Li. Since f is Kn-invariant we have

≥1 µf (M ) = (` + 1)µf (Mt1) + µf (Mt2) − (` + 1)µf (Mt2) = (` + 1)µf (Mt1) − `µf (Mt2).

4.4. The Um-operators. Let Mm×n(F`) denote the set of (m × n)-matrices with coefficients in F` which can naturally be identified with the subgroup of Mn×n(F`) consisting of matrices with the last n − m rows equal to zero. We also view Mn×n(F`) ⊂ Mn×n(Z`) via the Teichmüllerlift. We emphasise that when talking n m about the ranks of these matrices, we always mean the rank as a linear map F` → F` (as the Teichmüller lift doesn’t preserve ranks). We introduce some operators. For 1 ≤ m ≤ n we let

X 1 X t U = m ∈ [G ( )] m 1 1 Z 0 Q` X

where the sum is over all matrices in Mm×n(F`) (the blocks in the above matrices are of size n × n). These operators interact with the Shalika model as follows.

Lemma 4.17. Let ϕ0 be a spherical vector. Then  0 if x∈ / M x  S = xt Um·ϕ0 1 `mnS m if x ∈ M  ϕ0 1 Consequently, we have 2 Z x Z(χ(`)m`n(n−m)U · ϕ , χ; s) = `n +m(s−1/2) S χ(det x)| det x|s−1/2dx m 0 ϕ0 1 Mtm for Re(s) large enough so that the integrals converge. 22 ANDREW GRAHAM AND SYED WAQAR ALI SHAH

1 X x Proof. Right-translation by elements of the form implies that S is supported on M. 1 ϕ0 1 Furthermore, a direct calculation shows that x X xt S = ψ (tr(x · X)) ·S m Um·ϕ0 1 0 ϕ0 1 X −1 −1 so this function is supported on all x ∈ Mtm . If x ∈ Mtm , then the function X 7→ ψ0(tr(x · X)) deﬁnes a character Mm×n(F`) → C which is trivial if and only if x ∈ M. Therefore, the result follows from the usual −1 character orthogonality relations. The last part follows from the change of variable x 7→ xtm .

Remark 4.18. The action of Um on the function Sϕ0 (−) differs from the action of the more common U`- operator associated with the parabolic of type (m, n − m, n) by a power of `. However, it is more convenient for our purposes to use the above definition instead. Since we are eventually going to consider congruences modulo ` − 1 this makes no difference to the end result. We now apply the combinatorics of the previous section.

× × Theorem 4.19. Let χ : Q` → C be a ﬁnite-order unramiﬁed character and let ϕ0 be a spherical vector normalised such that Sϕ0 (1) = 1. Then there exist integers λ1, . . . , λn (independent of χ) such that n ! X n(n−m) m n2 Z ( λm` χ(`) Um) · ϕ0, χ; 1/2 = ` [Z(ϕ0, χ; 1/2) − 1] . m=1 n2 Pn n(n−m) Furthermore, the sum ` − m=1 ` λm is divisible by ` − 1. x Proof. Fix s with real part large enough, and set f(x) = S χ(det x)| det x|s−1/2. Then f is right ϕ0 1 Kn-invariant. By Corollary 4.16, there exist integers λ1, . . . , λn which are independent of χ and satisfy n ≥1 X µf (M ) = λmµf (Mtm). m=1 Therefore, by Lemma 4.17, we have n ! n X n(n−m) m X n2+m(s−1/2) Z ( λm` χ(`) Um) · ϕ0, χ; s = λm` µf (Mtm) m=1 m=1 n ! n2 X m(s−1/2) ≥1 = ` λm(` − 1)µf (Mtm) + µf (M ) . m=1

Furthermore, µf (Mtm) can be analytically continued to a holomorphic function in s, so the limit lims→1/2 µf (Mtm) exists and is ﬁnite. Therefore passing to the limit as s → 1/2 we have n ! X n(n−m) m n2 ≥1 Z ( λm` χ(`) Um) · ϕ0, χ; 1/2 = ` lim µf (M ). s→1/2 m=1 Now to conclude the proof, we note that ≥1 µf (M ) = µf (M) − µf (Kn) = Z(ϕ0, χ; s) − 1. The second part of the theorem just follows from Corollary 4.16. 4.5. Congruences modulo ` − 1. We now seek to ﬁnd the element φ in Theorem 4.1. For this, we must prove a congruence relation for the operators Um. As before, we will express the result in terms of the extended group Ge, so we introduce some additional notation:

Deﬁnition 4.20. For an element g ∈ G0(Q`) let ge denote the element −1 (1, g, det g ) ∈ Ge(Q`).

We also let Uem denote the corresponding operator in Z[Ge(Q`)]. ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS 23

We note that we have a left action of H(Z`) = GL1(Z`) × GLn(Z`) × GLn(Z`) on Mn×n(F`) given by (α, A, B) · X = AXB−1 which will be useful in the proof of the following proposition.

Proposition 4.21. We retain the notation of Theorem 4.1. Let ϕ0 ∈ π be a spherical vector. Then there exists a Z-valued right K-invariant smooth compactly supported function ψm on Ge(Q`) such that Pm (1) ψm = r=0 cr,m ch ((gr, 1)K) and the coeﬃcients cr,m equal the number of rank r matrices in Mm×n(F`); explicitly we have

r−1 Y (`m − `j)(`n − `j) c = r,m `r − `j j=0

for r ≥ 1, and c0,m = 1. −1 (2) For any unramiﬁed χ and z ∈ HomH (π χ , C)

z(Uem · ϕ0) = z(ψm · ϕ0). (3) For r ≥ 0 we have

Stab (X ) = V := (g , 1)K(g , 1)−1 ∩ H( ) H(Z`) r r r r Q`

which implies that [H(Z`): Vr] = cr,n. Furthermore, we have [Vr : V1,r] = ` − 1 when 0 ≤ r ≤ n − 1 and Vn = V1,n. 1 X t Proof. Let g = m be an element appearing the sum U . Since ϕ is spherical, we X 1 1 m 0 have t A−1 1 t−1(t At−1)XB−1 g · ϕ = m m m m · ϕ X 0 B−1 1 0

−1 −1 −1 for A, B ∈ GLn(Z`). We can choose A, B such that tmAtm ∈ GLn(Z`) and (tmAtm )XB ≡ Xr modulo n m 3 `, where r is the rank of X (viewed as a linear map F` → F` ). Hence we have

z(geX · ϕ0) = z((gr, 1) · ϕ0). −1 −1 −m −1 Indeed, this follows from that fact that (1, (tmA ,B ), ` det A det B ) ∈ H(Q`) and χ(det A) = χ(det B) = 1. This proves the ﬁrst and second parts of the proposition – the formula for cr,m is well-known. The last part of the proposition follows from an explicit calculation and the orbit–stabiliser theorem. Furthermore the induced map × ν : Vr → Z` given by ν(α, A, B) = det A−1 det B is surjective when r < n (for example, take A = 1 and B equal to a diagonal matrix with the ﬁrst n − 1 entries equal to 1) so we have [Vr : V1,r] = ` − 1.

4.6. Proof of Theorem 4.1. We begin with the following lemma:

Lemma 4.22. For 1 ≤ r ≤ n set n 0 X n(n−m) br = ` λmcr,m m=r where λm and cr,m are the integers in Corollary 4.16 and Proposition 4.21 respectively. Then (` − 1) · cr,n 0 0 divides br when 1 ≤ r ≤ n − 1, and bn = λncn,n.

3Working modulo `, we can ﬁrst apply column operations to ensure that X has its last (n − r) columns equal to zero. Then apply row operations to ensure that X is only non-zero in its top-left (r × r) block (such an A exists for this step because r ≤ m). This block must be invertible, so we can apply column operations to put X in the form Xr, as required. 24 ANDREW GRAHAM AND SYED WAQAR ALI SHAH

Proof. We ﬁrst show that cr,n divides λmcr,m. Indeed the number of F`-points of the Grassmanian of n m-dimensional subspaces of n-dimensional space is equal to , so by Corollary 4.16 the integer λm is m ` n divisible by . Using the formula in Proposition 4.21 we have m ` n c n m n−1 n − r · r,m = · · = . m ` cr,n m ` r ` r ` n − m ` n Furthermore, by Corollary 4.16 we have λ ≡ (−1)m+1 modulo ` − 1, which implies that m m n − r `n(n−m) · λ · c · c−1 ≡ (−1)m+1 modulo ` − 1 m r,m r,n n − m 0 −1 and thus by the binomial formula, brcr,n ≡ 0 (mod ` − 1) in the case r < n. This completes the proof of the lemma. We now construct the element φ. We set n ! −1 n2 X n(n−m) b0 = (` − 1) ` − λm` m=1 −1 0 which is an integer by Theorem 4.19, and for r ≥ 1 we set br = −(` − 1) br which is an integer by the above lemma and the fact that ` − 1 divides cn,n. Then we set n X φ := br ch ((gr, 1)K) ∈ H Ge(Q`), Z r=0 −1 which naturally acts on the smooth representation π χ of Ge(Q`). Let ϕ0 be a normalised spherical vector and set z(−) = Z(−, χ; 1/2). Then we have n ! n ! n2 X n(n−m) X 0 z((` − 1)φ · ϕ0) = z ` − λm` ϕ0 − br(gr, 1) · ϕ0 m=1 r=1 n ! n n ! n2 X n(n−m) X X n(n−m) = z ` − λm` ϕ0 − ` λmcr,m(gr, 1) · ϕ0 m=1 r=1 m=r n m ! n2 X n(n−m) X = z ` ϕ0 − ` λm cr,m(gr, 1) · ϕ0 m=1 r=0 n ! n2 X n(n−m) (4.23) = z ` ϕ0 − ` λm · Uem · ϕ0 m=1 n ! n2 X n(n−m) m (4.24) = Z ` ϕ0 − ( λm` χ(`) Um) · ϕ0, χ; 1/2 m=1 n2 −1 = ` L(σ ⊗ χ, 1/2) Z(ϕ0, χ; 1/2)(4.25) m where the equalities in (4.23), (4.24) and (4.25) follow from Proposition 4.21, the fact that Uem·ϕ0 = χ(`) Um· ϕ0 and Theorem 4.19 respectively. The latter equality also uses the fact that Z(ϕ0, χ; 1/2) = L(σ ⊗ χ, 1/2). −1 Since Z(−, χ; 1/2) is a basis of the hom-space HomH (π χ , C), this completes the proof of the ﬁrst part of Theorem 4.1. Furthermore, by construction φ is an integral linear combination of indicator functions ch((gr, 1)K), so we have also proven the second part of the theorem. Finally, by Proposition 4.21 we have 0 −1 −1 −1 −br · cr,n · (` − 1) if 1 ≤ r ≤ n − 1 (` − 1) · br · [H(Z`): V1,r] = 0 −1 −bn · cn,n if r = n which in both cases was shown to be an integer. For r = 0 the statement follows from the fact that b0 is an integer. ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS 25

5. Interlude: cohomology functors In [Loe19], a class of functors on subgroups of locally proﬁnite groups is deﬁned in an abstract setting that captures the key properties of the cohomology of locally symmetric spaces. In this section, we study these functors in more detail, and develop some machinery that will be useful later on.

5.1. Generalities. Let G be a locally proﬁnite topological group, Σ ⊂ G an open submonoid and let Υ be a collection of compact open subgroups of G contained in Σ. We will impose the following conditions on (G, Σ, Υ) (T1) For all g ∈ Σ ∪ Σ−1 and K ∈ Υ, gKg−1 ⊂ Σ implies gKg−1 ∈ Υ. (T2) For all K ∈ Υ, Υ contains a topological basis of open normal subgroups of K. (T3) For all K,L ∈ Υ and g ∈ Σ ∪ Σ−1, we have K ∩ gLg−1 ∈ Υ. To such a triplet, we associate a category of compact opens P(G, Σ, Υ) whose objects are elements of Υ and whose morphisms are given by −1 HomP(G,Σ,Υ)(L, K) := g ∈ Σ| g Lg ⊂ K

g hg for L, K ∈ Υ, with compositions given by (L −→ K) ◦ (L0 −→h L) = (L0 −→ K). We will denote a morphism g (L −→ K) by [g]L,K ; if we have L ⊂ K and g = 1, we shall also denote the corresponding morphism by prL,K . If no confusion can arise, we will suppress the subscripts from these morphisms. Omission of Υ from the triplet (G, Σ, Υ) will mean that we take all compact open subgroups of Σ, and omission of Σ means Σ = G. Deﬁnition 5.1. A cohomology functor M on a triplet (G, Σ, Υ) as above and valued in R-Mod for a commutative ring R is a pair of covariant functors

? op −1 M : P(G, Σ, Υ) → R-Mod M? : P(G, Σ , Υ) → R-Mod satisfying the following: ? (C1) M?(K) = M (K) for all K ∈ Υ. We denote this common value by M(K). ? ? (C2) If φ is a morphism then we set φ? = M?(φ) and φ = M (φ). We require that ? −1 [g]L,K = [g ]K,L,? ∈ Hom(M(K),M(L)) for all g ∈ Σ and L, K ∈ Υ satisfying g−1Lg = K. (C3)[ γ]K,K,? = id for all γ ∈ K and K ∈ Υ.

We will denote the cohomology functor as M :(G, Σ, Υ) → R-Mod, and sometimes abusively as M : PG → R-Mod if the categories of compact opens are clear from context. Our cohomology functors will often be enhanced with the following additional axioms. Deﬁnition 5.2. Let M :(G, Σ, Υ) → R −Mod be a cohomology functor. (G) We say M is Galois if for all K,L ∈ Υ with L/K,

? ∼ K/L prL,K : M(K) −→ M(L) ? where the (left) action of γ ∈ K/L on x ∈ M(L) is via x 7→ [γ]L,L(x). (Co) We say that M is a covering functor if for all L, K ∈ Υ, L ⊂ K,

pr pr ? [K:L] (L −→ K)? ◦ (L −→ K) = (K −−−→ K) i.e. the composition is multiplication by index [K : L] on M(K). (M) We say M is Cartesian if for all K, L, L0 ∈ Υ with L, L0 ⊂ K, we have a commutative diagram P L pr? γ M(Lγ ) M(L)

P γ? pr? M(L0) M(K) pr? 26 ANDREW GRAHAM AND SYED WAQAR ALI SHAH

where the direct sum in the top left corner is over a ﬁxed choice of coset representatives γ ∈ L\K/L0 0 −1 and Lγ = γL γ ∩ L ∈ Υ. This condition is independent of the representatives, since if γ are 0 0 0 0 −1 replaced with γ = δγδ for δ ∈ L, δ ∈ L , Lγ is replaced with Lγ0 = δLγ δ , and

0 pr γ ? pr γ ? (Lγ −→ L)? ◦ (Lγ −→ L) = (Lγ0 −→ L)? ◦ (Lγ0 −→ L) If M satisfies both (M) and (Co), we will say that M is CoMack. It is easily seen that if M is CoMack and R is a Q-algebra, then M is Galois. Remark 5.3. If one takes Υ to be all compact opens in Σ, then one obtains the definition of a cohomology functor given in [Loe19]. The addition of Υ is made since we only want to vary over sufficiently small compact open subgroups of G(Af ) for a reductive group G (Definition B.5), and the conditions (T1)–(T3) on Υ are included so that taking products is well-behaved. Axiom (G) is inspired by Galois descent in étale cohomology and axiom (Co) reflects the corresponding property covering maps in étalecohomology [AGV73, Tome 3, Expose IX, §5]. Axiom (M) is based on Mackey’s decomposition formula e.g. see [Dre73]. In [Loe19], this property was named Cartesian after the Cartesian condition for proper/smooth base change in étale cohomology. The terminology ‘CoMack’ is standard e.g. see [TW95]. Example 5.1. Let X be a set with a right action of Σ, and Υ any collection as above. Let M(K) = ∞ Cc (X/K; R) be the set of all functions ζ : X → R which factor through X/K and are non-zero only on g ζ finitely many orbits. For g ∈ Σ, we denote by g ? ζ the composition X −→ X −→ R, which gives a left action on the set of functions from X to R. Then, for elements σ, τ −1 ∈ Σ and morphisms σ : L → K, τ : L0 → K0, we set [σ]? : M(K) → M(L) ζ 7→ σ ? ζ 0 0 X −1 [τ]? : M(L ) → M(K ) ζ 7→ γτ ? ζ γ where the sum runs over representatives in K0 of the coset space K0/(τ −1L0τ). Then M is CoMack. Example 5.2. The prototypical example of a cohomology functor for us will be the cohomology of Shimura varieties as also explained in [Loe19, §3]. Let (G,X) be a Shimura–Deligne datum and set G = G(Af ) and Υ to be the set of all sufficiently small compact open subgroups of G (Definition B.5). If one chooses Zp-sheaves on these varieties and compatible isomorphisms between the pullbacks of these sheaves under Hecke actions (after choosing a suitable monoid and shrinking Υ if necessary), the degree j cohomology for any fixed j are then cohomology functors under pullbacks and trace maps of Hecke actions. Since the maps between Shimura varieties of two levels form covering maps of degree given by the index of the corresponding compact opens, they necessarily satisfy (Co) by [AGV73, Tome 3, Expose IX, §5]. If moreover, the group G satisfies axiom (SV5) of [Mil03], i.e. Z(Q) is discrete in Z(Af ) where Z = ZG is the center of G, then these also satisfy axiom (M). 5.2. Completions. Let M :(G, Σ, Υ) → R-Mod be a cohomology functor. For any compact subgroup C ⊂ Σ of G, we can define two completions M(C) = lim M(K),M (C) = lim M(K) −→ Iw ←− K⊃C K⊃C ? where the limits are taken with respect to the morphisms pr and pr? respectively over all K ∈ Υ containing C. We call these the inductive and Iwasawa completions respectively. When C = {1}, we also denote the inductive completion by Mc. We denote by jK : M(K) → M(C) the natural map. If C is central (i.e. contained in the center of Σ), the space M(C) naturally carries a smooth action of Σ as follows. For an element g ∈ Σ and x ∈ M(C), choose a compact open K ∈ Υ such that there exists xK ∈ M(K) −1 with jK (xK ) = x. By replacing K with a subgroup of K ∩ g Kg contained in Υ, we may assume that −1 gKg ∈ Υ. We then define g · x to be the image of xK under the composition

[g]? M(K) −−→ M(gKg−1) → M(C). It is a routine check that this is well-deﬁned, and the action is smooth by property (C3) in Deﬁnition 5.1. ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS 27

∼ K Lemma 5.4. Suppose M is a Galois functor, C is central. Then jK : M(K) −→ M(C) for any K ⊃ C and K ∈ Υ. In particular, if we choose a left invariant Haar measure on G giving K measure one and Σ = G, then M(K) is a (left) module over the Hecke algebra H(K\G/K) with the action of the Hecke operator ch(KσK) on x ∈ M(K) ,→ M(C) given by X ch(KσK) · x = α · jK (x) α∈KσK/K

K Proof. Let x ∈ M(C) . Then x is in the image of the map jL for some L ∈ Υ such that L/K and L ⊃ C. This implies that x ∈ M(L)K/L ←−∼ M(K) by (G). The explicit action of the Hecke operator can then be read oﬀ via the integral in [BH19, Eqn 4.2.2]. 5.3. Hecke correspondences. Let M :(G, Σ, Υ) → R-Mod be a cohomology functor. For every K,K0 ∈ Υ and σ ∈ Σ, we have a diagram [σ] K ∩ σ−1K0σ σKσ−1 ∩ K0 pr pr

K K0 This induces a map pr? [σ]? pr [K0σK]: M(K) −−→ M(K ∩ σ−1K0σ) −−→ M(σKσ−1 ∩ K0) −−→? M(K0) which we refer to as the Hecke correspondence induced by σ. It is straightforward to verify that [K0σK] only depends on the double coset K0σK. Lemma 5.5. Suppose that M :(G, Σ, Υ) → R-Mod is Cartesian. Let K,K0,K00 ∈ Υ and σ, τ ∈ Σ. (a) If we write K0σK = t αK. then we have 0 X jK0 ◦ [K σK] = α · jK . 00 0 0 (b) The composition jK00 ◦ [K τK ] ◦ [K σK] is given by the convolution product of double cosets, i.e. if we write K0σK = t αK and K00τK0 = t βK0 then 00 0 0 X jK00 ◦ [K τK ] ◦ [K σK] = (βα) · jK . (c) If M is moreover Galois, Σ = G and a left invariant Haar measure on G is chosen giving K measure 1, then the actions on M(K) of the correspondence [KσK] and the operator ch(KσK) deﬁned in the previous subsection agree. Proof. Let L0 := σKσ−1 ∩ K0 which is an element of Υ. We can (and do) choose a compact open subgroup 0 0 0 L ∈ Υ satisfying L/K and L ⊂ L , and let {γ}γ∈I denote any set of representatives in K of the coset 0 0 0 0 0 −1 space K /L = L\K /L ; for any such γ we set Lγ = γL γ ∩ L = L. By the Cartesian property, we obtain the following commutative diagram

P L pr? γ M(L) M(L)

P[γ]? pr? [σ]? pr M(K) M(L0) ? M(K0)

[K0σK]

? 0 P ? which implies that prL,K0 ◦ [K σK] = γ [γσ] . It is then easily veriﬁed that γσ are distinct representatives of K0σK/K, and that the action so deﬁned is independent of the choice of representatives. This completes the proof of part (a). For (b), we note that the composition of the Hecke correspondences in the limit is given by P[α]? P[β]? M(K) M(L) M(J) 28 ANDREW GRAHAM AND SYED WAQAR ALI SHAH

00 −1 where J ∈ Υ is such that J/K and J ⊂ τLτ . Part (c) is immediate. The next lemma is a useful criterion for when two Hecke correspondences commute with each other.

Lemma 5.6. For i = 1, 2 set Pi = P(Gi, Σi, Υi) and P = P(G, Σ, Υ) = P1 × P2. Let M : P → R-Mod be a Cartesian cohomology functor, and take

• σ ∈ Σ2 • J ∈ Υ2 • K = K1 × J and L = L1 × J both elements in Υ • g = (g1, 1): L → K a morphism in P. Then we have [g]? ◦ [KσK] = [LσL] ◦ [g]?. A similar result holds for pushforwards. Proof. Let Kg = gKg−1. Then we have g(σKσ−1 ∩ K)g−1 = σKgσ−1 ∩ Kg. It suﬃces to prove that

pr M(σLσ−1 ∩ L) ? M(L)

pr? pr? pr M(σKgσ−1 ∩ Kg) ? M(Kg)

But this is immediate from the Cartesian property since

g g −1 g g1 g1 −1 L\K /(σK σ ∩ K ) = (L1 × J)\(K1 × J)/K1 × (σJσ ∩ J) = {idK } g −1 g −1 and L ∩ (σK σ ∩ K ) = σLσ ∩ L. 5.4. Pushforwards between cohomology functors. Let ι: H,→ G be a closed subgroup, and suppose that MH ,MG are cohomology functors for the triplets (H, ΣH , ΥH ) and (G, ΣG, ΥG) respectively. We require ι(ΣH ) ⊂ ΣG and that for all U ∈ ΥH , K ∈ ΥG, we have U ∩ K ∈ ΥH . We say that an element (U, K) ∈ ΥH × ΥG forms a compatible pair if U ⊂ K.A morphism of compatible pairs (V,L) → (U, K) is a pair of morphisms [h]: V → U,[h]: L → K for h ∈ ΣH .

Deﬁnition 5.7. A pushforward ι? : MH → MG is a collection of maps MH (U) → MG(K) for all compatible pairs (U, K), which commute with the pushforward maps induced by morphisms of compatible pairs coming −1 from ΣH .A Cartesian pushforward ι? : MH → MG is a pushforward such that for all U ∈ ΥH , L, K ∈ ΥG and U, L ⊂ K, we have a commutative diagram P L [γ]? γ MH (Uγ ) MG(L)

P pr? pr?

ι? MH (U) MG(K) −1 where γ ∈ U\K/L is a ﬁxed set of representatives, Uγ = γLγ ∩ U and [γ]? : MH (Uγ ) → MG(L) is the −1 composition MH (Uγ ) → MG(γLγ ) −→ MG(L).

Remark 5.8. When G = H and ι? = pr?, one recovers the conditions for MG to be Cartesian. We point out that what we refer to as a Cartesian pushforward here is simply called a pushforwad in [Loe19].

Assume for the rest of this section that H is unimodular, P(H, ΣH , ΥH ), P(G, ΣG, ΥG) are categories of compact opens with ΣH = H,ΣG = G and R is a Q-algebra. We ﬁx left invariant Haar measures dh, dg on H, G respectively that take values in Q on the respective compact opens, and denote by Vol(−) the volume of the corresponding compact opens. We consider H(G, R) as an algebra under the usual convolution product ∗ (see [BH19, §1.4.1]) and a left representation of G with g ∈ G acting on ξ ∈ H(G, R) via ρg(ξ): x 7→ ξ(xg). The action ρ of G induces an action of the Hecke algebra on itself [BH19, 1.4.2], which we denote by ξ ∗ρ ζ for ξ ∈ H(G, R) (considered as a ring element) and ζ ∈ H(G, R) (considered as a module element). Finally, for ξ ∈ H(G, R), we let ξt denote its transpose x 7→ ξ(x−1). Deﬁnition 5.9. Given smooth representations τ of H, σ of G and C a central compact subgroup of G which acts trivially on σ, we consider τ ⊗ H(G, R)C and σ smooth representations of H × G by the following extended action. ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS 29

• (h, g) ∈ H × G acts on x ⊗ ξ ∈ τ ⊗ H(G, R)C via x ⊗ ξ 7→ hx ⊗ ξ(h−1(−)g). • H × G acts on σ via projection to G which acts via its natural action. An intertwining map Z : τ ⊗ H(G, R)C → σ is just a morphism of H × G representations. Lemma 5.10. Let τ, σ and Z be as above. C (a) For all x ∈ τ, ξ1, ξ2 ∈ H(G, R) , t Z(x ⊗ ξ1 ∗ ξ2) = ξ2 · Z(x ⊗ ξ1). (b) (Frobenius Reciprocity) Let σ∨ denote the smooth dual of σ and denote by h·, ·i : σ∨ × σ → R the induced pairing. Consider τ ⊗ σ as a smooth H-representation via h · (x ⊗ ϕ) = h · x ⊗ ι(h)ϕ for ∨ ∨ h ∈ H, x ∈ τ, ϕ ∈ σ . There is a unique element z ∈ HomH (τ ⊗ σ ,R) (depending on the choice of dg) such that hϕ, Z(x ⊗ ξ)i = z(x ⊗ ξ · ϕ) for all ϕ ∈ σ∨, x ∈ τ, ξ ∈ H(G, R)C .

t t Proof. For (a), note that the G-equivariance of Z tells us that Z(x ⊗ (ξ2 ∗ρ ξ1)) = ξ2 · Z(x ⊗ ξ1) [BH19, t Proposition 1, p.35]. Since ξ1 ∗ ξ2 = ξ2 ∗ρ ξ1, we have the ﬁrst claim.

∨ −1 (b) Given x ∈ τ, ϕ ∈ σ , choose a compact open K ⊃ C such that the idempotent eK = Vol(K; dg) ch(K) ∈ C H(G, R) fixes ϕ, and set z(x ⊗ ϕ) := hϕ, Z(x ⊗ eK )i. It is easily verified that z is well-defined, satisfies the property claimed above and is uniquely determined by its properties.

We now construct a “completed pushforward” from a given pushforward MH → MG i.e. an intertwining map of H × G representations as above. This is a straightforward generalisation of the “Lemma–Eisenstein” map in [LSZ17, Lemma 8.2.1].

Proposition 5.11 (Completed Pushforward). Suppose MH , MG are cohomology functors with MH satisfy- C ing (Co), MG satisfying (M). Consider McH ⊗ H(G, R) and M G(C) as smooth H × G representations via the extended action. Then for any pushforward ι? : MH → MG, there is a unique (depending on the choice of dh) intertwining map of H × G representations

C ˆι? : McH ⊗ H(G, R) → M G(C)

satisfying the following compatibility condition: for all compatible pairs (U, K) ∈ ΥH × ΥG with K ⊃ C, x ∈ MH (U), y ∈ MG(K) such that ι?(x) = y, we have ˆι?(jU (x) ⊗ ch(K)) = Vol(U) · jK (y).

Proof. We first assume that C = {1}. Elements of McH ⊗ H(G, R) are spanned by pure tensors of the form x ⊗ ch(gK) for x ∈ McH , g ∈ G and K ∈ ΥG. Indeed, for any compact open subgroup K of G, there exists a compact open subgroup such that K0 ⊂ K and K0 ∈ Υ, because we require that Υ contains a basis of the identity. We will define the map on these pure tensors and extend linearly. −1 Choose a compact open subgroup U ∈ ΥH such that U fixes x, U ⊂ gKg ∩ H, and let xU ∈ MH (U) be an element that maps to x under jU . Then, we define ˆι? (x ⊗ ch(gK)) to be the image of Vol(U)xU under the composition

ι? −1 [g]? jK MH (U) −→ MG(gKg ) −−→ MG(K) −−→ McG

To see that this is independent of choice of xU (and U), assume that U is replaced by another open compact 0 subgroup V and xU by xV ∈ MH (V ). We can then choose an even smaller compact open V ∈ ΥH , 0 V ⊂ U ∩ V such that xU , xV 7→ xV 0 under pullbacks, and we may therefore assume that V ⊂ U, and ∗ xU 7→ xV under prV,U . Since MH is a covering functor, xV 7→ [U : V ]xU under prV,U,∗ Hence both Vol(U)xU and Vol(V )xV map to the same element in McG. It remains to show that that the various relations among elements of H(G, R) do not give conﬂicting images on the sums of these simple tensors. To this end, let L, K ∈ ΥG such that L is a normal subgroup of K. We want to verify that X ˆι?(x ⊗ ch(K)) = ˆι?(x ⊗ ch(γL)). γ∈K/L 30 ANDREW GRAHAM AND SYED WAQAR ALI SHAH

The general case for two different representations of ξ ∈ H(G, R) as sums of characteristic functions of left cosets can be reduced to this case by successively choosing normal subgroup for pairs and twisting by the action of G, by establishing G-equivariance of such a (a priori hypothetical) map first. −1 Choose U ∈ ΥH such that U fixes x and U ⊂ L ∩ H. Note that as L/K, γLγ ∩ H = L ∩ H for any γ ∈ K. As before, let xU ∈ MH (U) be an element mapping to x. By definition

ˆι?(x ⊗ ch(K)) = jK ◦ ιU,K,?(Vol(U)xU ), ˆι?(x ⊗ ch(γL)) = jL ◦ [γ]? ◦ ιU,L,?(Vol(U)xU ).

? As jK = jL ◦ prL,K , it suﬃces to prove that

? X X ? prL,K ◦ ιU,K,? = [γ]? ◦ ιU,L,? = [γ] ◦ ιU,L,?. γ γ

as maps MH (U) → MG(L), where the last equality follows from the fact that L is normal in K, so we can replace the set of representatives by their inverses. This equality is then justiﬁed by replacing ιU,K,? = prL,K,? ◦ ιU,L,? and invoking the axiom (M) for MG. Therefore, ˆι? is well-deﬁned. We now check that ˆι? is H × G equivariant with the said actions: let

(h, g) ∈ H × G, v = x ⊗ ch(g1K) ∈ McH ⊗ H(G, R). Then, (h, g) · v = hx ⊗ ch(hg0K0) 0 −1 0 −1 −1 where g = g1g , K = gKg . Choose U ∈ ΥH such that U ﬁxes x, U ⊂ g1Kg1 ∩ H and xU ∈ MH (U) −1 −1 0 0 0 −1 ? −1 that maps to x. Then, hUh ⊂ hg1K(hg1) ∩ H = hg K (hg ) ∩ H, and [h] xU ∈ MH (hUh ) maps ? −1 to hx. Since [h] = [h ]?, we obtain a commutative diagram

ι? −1 MH (U) MG(g1Kg1 )

[h]? [h]?

−1 ι? 0 0 0 −1 MH (hUh ) MG(hg K (hg ) )

Now,

(h, g) · ˆι?(v) = Vol(U) · [g · (jK ◦ [g1]? ◦ ι?(xU ))] −1 0 ? ˆι?((h, g) · v) = Vol(hUh ) · (jK0 ◦ [hg ]? ◦ ι?([h] xU )) .

−1 −1 As H is unimodular, Vol(U) = Vol(hUh ) and it therefore suffices to verify that [g ]? ◦ [g1]? ◦ ι?(xU ) = 0 ? 0 [hg ]? ◦ι?([h] xU ) as elements of MG(K ). But this follows immediately from the commutativity of the above diagram and the composition law [hg]? = [g]?[h]?. This finishes the checking of equivariance. The statement involving C is obtained by taking C invariants on both sides, and noticing that the map factors through M(C) ⊂ McC . It is easily seen that this map is uniquely determined with the prescribed actions of H × G and the compatibility condition with ι?. Ideally one would like an integral version of the above proposition but such a map doesn’t exist in general due to the presence of the volume factors in the definition. It is however possible to relate the pushforward map on finite level to an integral version, as follows. Let O be an integral domain with field of fractions Φ, and let MO and MΦ be cohomology functors for (G, Σ, Υ) valued in O and Φ respectively. We say that MO and MΦ are compatible under base change if there exists a collection of O-linear maps MO(K) → MΦ(K) which are natural in K and commute with pullbacks and pushforwards. Let ι: H,→ G be as above, and let (MH,O,MH,Φ) and (MG,O,MG,Φ) be pairs of compatible cohomology functors. Suppose that we have a pushforward ι? : MH,Φ → MG,Φ that restricts to a pushforward MH,O → MG,O. Then we obtain the following corollary.

Corollary 5.12. Let ΣG = G and ΣH = H. Keeping with the same notation as above, let U ∈ P(H, ΣH , ΥH ) and K ∈ P(G, ΣG, ΥG) be compact open subgroups, and suppose that we have an element g ∈ G such that ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS 31

U ⊂ gKg−1 ∩ H. Then we have a commutative diagram:

x7→ˆι∗(x⊗ch(gK)) McH,Φ McG,Φ

jU Vol(U)·jK

ι∗ −1 [g]∗ MH,O(U) MG,O(gKg ) MG,O(K)

6. Vertical norm relations at split primes In this section we construct classes in the “p-direction” following the general method outlined in [Loe19]. As a result we will obtain cohomologically trivial cycles which are compatible as one goes up in the anticyclotomic tower. We will also need results of this section in the construction of the Euler system map. We do not require n to be odd in this section.

6.1. Group theoretic set-up. Let p be an odd prime which splits in E/Q. Then we can consider the local versions of the groups in section 2.1 at the prime p – we will denote these groups by unscripted letters. We have identifications ∼ G = GQp = GL1 × GL2n ∼ H = HQp = GL1 × GLn × GLn ∼ T = TQp = GL1 Since all of these groups are unramified, they all admit a reductive model over Zp which we fix and, by abuse of notation, denote by the same letters. ∗ ∗ Definition 6.1. Let Q denote the Siegel parabolic subgroup of G given by Q = GL × and let 1 ∗ J ⊂ G(Zp) denote the parahoric subgroup associated with Q (i.e. all elements in G(Zp) which lie in Q modulo p). We identify the Levi of Q with the subgroup ι(H). The automorphic representations that we will consider will be “ordinary” with respect to a certain Hecke operator associated with the parabolic subgroup Q × T ⊂ Ge, which we will now define. Let τ and τe be the elements p p τ = 1, ∈ G( ) τ = 1, × 1 ∈ G( ). 1 Qp e 1 e Qp Then for any compact open subgroup Ke ⊂ J × T , we let T denote the Hecke operator associated with the −1 double coset [Keτe Ke]. More concretely, T is the operator associated with the following correspondence: [τ −1] Sh (K ∩ τKτ −1) e Sh (K ∩ τ −1Kτ) Ge e e e e Ge e e e e

Sh (K) Sh (K) Ge e Ge e

Explicitly, if p1 and p2 denote the left-hand and right-hand projection maps respectively, then T = p2,∗ ◦ −1 ∗ 0 [τe ]∗ ◦p1 (see also section 5.3). In all choices Ke ⊂ Ke that we will consider, this operator will be compatible with the pushforward morphism (pr ) , where pr : Sh (K0) → Sh (K) is the natural map; so we Ke 0,Ke ∗ Ke 0,Ke Ge e Ge e will often omit the level when referring to it (see [Loe19, §4.5]). Note that if Ke = K × C is decomposible, then T only acts on the cohomology of the first factor of Sh (K) ∼ Sh (K) × Sh (C). We will denote the Ge e = G T Hecke correspondence induced on ShG(K) also by T . 6.2. The flag variety. We now consider the spherical variety that underlies the whole construction in this section. Let Q denote the opposite of Q (with respect to the standard torus) with unipotent radical N. We will also abusively write N for the unipotent radical of Q × T . Then we consider the flag variety F = G/e (Q × T ) = G/Q over Spec Zp, which has a natural left action of Ge (which factors through the projection to G). Let ue ∈ Ge(Zp) denote the element 1 1 u = 1, × 1 e 1 32 ANDREW GRAHAM AND SYED WAQAR ALI SHAH

and consider the following reductive subgroup L◦ / ι(H) × T given by

X L◦ = GL × × 1. 1 Y

Then we verify the following properties.

Lemma 6.2. Keeping the same notation as above, the following are satisﬁed: −1 ◦ −1 (1) The stabiliser StabH ([ue]) = (ι, ν)(H) ∩ ue(Q × T )ue is contained in ue N · L ue . (2) The H-orbit of ue in F (over Spec Zp) is Zariski open. Proof. The ﬁrst part is a routine check that is left to the reader. For the second part we note that we have the following equality

X (ι, ν)(H) ∩ u(Q × T )u−1 = GL × × 1 : X ∈ GL e e 1 X n

which implies that the orbit of [ue] can be identiﬁed with a copy of GLn ⊂ F. One can check that this is open.

Remark 6.3. In the notation of [Loe19, §4.4, §4.6], we have ◦ • QH = H • QG = Q × T with unipotent radical N G = N and Levi LG = ι(H) × T . e ◦ e e • L◦ = L◦ and Q = N · L◦. Ge Ge Proceeding as in [Loe19, §4.4], we deﬁne the following level subgroups r ◦ −r r • Ur = {g ∈ Ge(Zp): g (mod p ) ∈ N · L and τe gτe ∈ Ge(Zp)} −r r • Vr = τe Urτe r for r ≥ 0. We let Dpr ⊂ T (Zp) denote the group of all elements x ≡ 1 modulo p . Then Vr is a compact open subgroup of J × Dpr for r ≥ 1.

0 −1 Lemma 6.4. Let r ≥ 1 and set Vr = Vr ∩ τVe rτe . Then 0 0 0 (a) Vr \Vr/Vr+1 is a singleton and Vr ∩ Vr+1 = Vr+1. 0 −1 (b) Set (J × Dpr ) = τe(J × Dpr )τe ∩ (J × Dpr ). The double coset space 0 (J × Dpr ) \J × Dpr /Vr

0 0 is a singleton and Vr ∩ (J × Dpr ) = Vr .

−1 −1 Proof. In terms of Ur, the ﬁrst claim in part (a) amounts to showing that (Ur ∩ τUe rτe )\Ur/τe Ur+1τe is a singleton. Notice that we have the Iwahori decomposition

Ur = N 0 · Lr · Nr

where N, N are the unipotent radicals of Q × T , Q × T respectively, L = ι(H) × T ⊂ G × T is the Levi, and r −r −r r ◦ Nr := τe N(Zp)τe , N r := τe N(Zp)τe and Lr = L (Zp)(1 × Dpr ) ⊂ G(Zp) × T (Zp). One then easily sees that −1 −1 Ur ∩ τUe rτe = N 0 · Lr · Nr+1, τe Ur+1τe = N 1 · Lr+1 · Nr −1 −1 which implies that Ur = Ur ∩ τUe rτe · τe Ur+1τe as required. As

0 Vr = N r · Lr · N1,Vr+1 = N r+1 · Lr+1 · N0

0 their intersection equals N r+1 · Lr+1 · N1 = Vr+1. The proof of part (b) is similar. ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS 33

6.3. Integral sheaves. In this section we introduce the integral versions of the sheaves constructed in sections 2.5–2.6 and discuss the corresponding branching law. Unfortunately, just choosing a Zp-lattice inside the representation is not suﬃcient in general, since there does not exist a lattice inside the self-dual −1 representation V which is stable under both G(Zp) and τ (unless V is the trivial representation). Therefore we must consider a modiﬁed action on the lattice – essentially, this amounts to renormalising the T operator so that the action on the cohomology of V is integral. The construction is the same as that in [DJR18, §1.2], however we are using opposite conventions to stay consistent with the construction in [Loe19]. We continue to use the notation introduced in sections 6.1–6.2. Let V be an algebraic representation of

G = GQp that is self-dual of weight c = (0; c1, . . . , c2n) as in Deﬁnition 2.12. We ﬁx once and for all a highest weight vector vc ∈ V , so that the action of the standard diagonal torus inside G is given by

diag(a0; a1, . . . , a2n) · vc = µ(a0; a1, . . . , a2n)vc

c1 c2n where µ is the character diag(a0; a1, . . . , a2n) 7→ a1 ··· a2n . This vector can also be viewed as a highest weight vector in Ve := V 1 – in this case, we denote it by vec.

Deﬁnition 6.5. Let T denote the G(Zp)-stable lattice

T = U(uZp ) · vc inside V , where U(uZp ) is the universal enveloping algebra of the subgroup lower triangular U = {1} × . matrices

This extends trivially to a Ge(Zp)-stable lattice Te ⊂ Ve

Let τ0 denote the following (2n × 2n)-block matrix p τ := ∈ G ( ). 0 1 0 Qp

Let J0 ⊂ G0(Zp) denote the parahoric subgroup of elements which land in Q0 modulo p, and let J := × × Zp × J0 ⊂ G(Zp) and Je := J × Zp ⊂ Ge(Zp). We introduce the following monoids and actions on the lattices introduced in Deﬁnition 6.5. Lemma 6.6. Consider the following sets: −r • Σ0 := J0 ·{τ0 : r ∈ Z≥0}· J0 × • Σ := Qp × Σ0 as a subset of G(Qp) • Σe := Σ × T (Qp) as a subset of Ge(Qp).

Then Σ and Σe are open submonoids of G(Qp) and Ge(Qp) respectively, and the actions of J and Je extend to actions • of Σ and Σe on T and Te which factor through Σ0 and satisfy: −1 −1 −1 −1 τ • v = µ(τ)(τ · v) τe • ve = µ(τ)(τe · v) for all v ∈ T and ve ∈ Te. Proof. Recall from the proof of Lemma 6.4 that we have the Iwahori decompostion

J = N 1 · H(Zp) · N(Zp) = N(Zp) · H(Zp) · N 1 −r r where N ⊂ Q is the unipotent radical and N r = τ N(Zp)τ . The result now follows from the fact that τ −1Qτ ⊂ Q and τNτ −1 ⊂ N, which allows us to extend the action of J by letting τ −1 act trivially on the highest weight vector (c.f. also [DJR18, §1.2]). The statement for Ge is similar.

Using the above action, we can now define equivariant sheaves for the integral lattices T and Te which are ? compatible with the rational versions by relaxing the condition that σ TK → TL are isomorphisms. More precisely, we let T correspond to the collection of the sheaves TK with morphisms ? σ• σ TK −→ TL 34 ANDREW GRAHAM AND SYED WAQAR ALI SHAH induced from the •-action, for all σ ∈ Σ satisfying σ−1Lσ ⊂ K. In particular, we can still define pullbacks and pushforwards on the cohomology of these sheaves as follows. Let σ ∈ Σ and K,L ⊂ G(Qp) compact open subgroups satisfying σ−1Lσ ⊂ K. Then we define the pullback morphism as the composition ? i i ? i [σ] :Hét (ShG(K), TK (j)) → Hét (ShG(L), σ TK (j)) → Hét (ShG(L), TL(j)) . −1 −1 Similarly if we take σ ∈ Σ and K,L ⊂ G(Qp) compact open subgroups satisfying σ Lσ ⊂ K, then we can define the corresponding pushforward morphism as the composition

−1 ? i [σ ] i −1 [σ]? :Hét (ShG(L), TL(j)) −−−−→ Hét ShG(σ Lσ), Tσ−1Lσ(j) ∼ i ! −→ Hét ShG(K), pr! pr TK (j) i → Hét (ShG(K), TK (j)) −1 where the middle isomorphism is induced from the pushforward along pr: ShG(σ Lσ) → ShG(K) and the ! ∗ ∼ fact that we have pr TK = pr TK = Tσ−1Lσ, and the last map is the trace map. We have similar maps for the sheaves corresponding to the group Ge . Lemma 6.7. The natural maps Hi (Sh , (j)) → Hi (Sh , (j)) Hi Sh , (j) → Hi Sh , (j) ét G T ét G V ét Ge Tf ét Ge Ve p p are equivariant for the actions of G(Af ) × J and Ge (Af ) × Je respectively. Furthermore, the action of T on 0 the left-hand side is intertwined with the action of US := µ(τ)T on the right-hand side. Proof. The first part follows from the fact that T ⊂ V and Te ⊂ Ve are equivariant embeddings for the actions −1 −1 of J and Je respectively. The last assertion follows from the fact that τe • ve = µ(τ)(τe · ve) for all ve ∈ Te −1 (recall that T is the Hecke operator [Keτe Ke]). Since we have slightly modified the action of Σ we have to modify the branching law discussed in section Pn 2.6 when working integrally. Let λ = i=1 ci and consider the following character λ n × × | det |p :Σ ∩ H(Qp) = Σe ∩ H(Qp) → Q>0 ⊂ Qp λ n (α; h1, h2) 7→ | det h1 · det h2|p and let Zp[λ/n] denote the trivial equivariant sheaf on ShH whose action of Σ ∩ H(Qp) is twisted by the λ/n character | det |p . Since the representation Ve is self-dual, we can fix a H(Qp)-equivariant embedding Qp ,→ Ve which restricts to a H(Zp)-equivariant embedding Zp ,→ Te. This gives rise to an embedding of relative Chow motives ? br: E,→ (ι, ν) AncE(Ve) ? with étalerealisation Qp ,→ (ι, ν) Ve, which we also denote by br. The following is an immediate application of the description of the •-action in Lemma 6.6. Pn ? Corollary 6.8. Let λ = i=1 ci. The morphism br: Qp → (ι, ν) Ve restricts to a morphism ? Zp[λ/n] ,→ (ι, ν) Tf p of H(Af ) × Σe ∩ H(Qp) -equivariant sheaves. 6.4. Functors from étalecohomology. For any prime `, we denote ` ` G` := G(Q`),G := G(Af ),Gf := G(Af ) and similarly for the groups H and T. Let ΣGp ⊂ Gp denote the open submonoid Σ in Lemma 6.6, and set Σ = Σ ∩ H. We let Υ be the collection of all compact open subgroups in Σ and take Υtot to be Hp Gp Gp Gp Gp the collection of all compact open subgroups in Gp. p Lemma 6.9. There exists a (non-empty) maximal collection ΥGp of compact open subgroups of G , closed p tot under inclusions and conjugation by G , such that the product Υ p × Υ consists of sufficiently small G Gp subgroups of Gf . Moreover, for any ` 6= p and compact open subgroup K` ⊂ G`, there exists an element in ΥGp whose `-adic component is equal to K`. ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS 35

0 Proof. Let Kp denote any representative of the unique conjugacy class of maximal compact open subgroups × p p 0 in Gp = Qp × GL2n(Qp). Then there exists a compact open subgroup K such that K Kp is sufficiently p small and therefore, K Kp is sufficiently small for any Kp ⊂ Gp. We take ΥGp to be the collection of all such groups Kp. This collection is closed under conjugation by Gp and contains all compact open subgroups p of any element K ∈ ΥGp . By deepening the level at a finite set of places S not containing ` and p, we can p S∪{`,p} arrange that K = K`KSK ∈ ΥGp for any arbitrary K`. We now define the monoids Σ and collections Υ that will be used throughout the paper. Firstly, we deal with the “tame level”: set p p p p • Σ p , Σ p , Σ p and Σ to be G ,H ,T and G respectively. G H T Gep e • ΥGp is the collection appearing in Lemma 6.9. • ΥHp is defined similarly to ΥGp . tot p • ΥT p = ΥT p denotes the collection of all compact open subgroups of T . p p p • Υ = Υ p ×Υ p (we will only be interested in compact open subgroups of G of the form K ×C ). Gep G T e At the prime p we fix the following data:

• ΣGp and ΣHp are as in the start of this section. We set ΣTp = Tp and

Σ = ΣG × ΣT Gep p p so Σ equals the monoid denoted Σ in Lemma 6.6. Gep e • For = G ,H ,T or G , we set Υ to be the collection of all compact open subgroups contained p p p p ep p in Σ (so in particular, elements of Υ are not necessarily of the form Kp × Cp). p Gep

Finally, for = G, H, T or G, we set Σ = Σ p × Σ and Υ = Υ p × Υ . For brevity, we set e p p P := P ( , Σ , Υ ) f

and similarly for P p and P . p

Remark 6.10. Although not explicitly included in the notation, the collections Υ p depend on the group f . The subscript simply refers to the ambient group of which the elements are compact open subgroups. The following lemma will allow us to apply the results of section 5.

Lemma 6.11. For a group ∈ {G, H, T, G} and a pair (Σ, Υ) ∈ {(Σ , Υ ), (Σ , Υ ), (Σ p , Υ p )} the e p p axioms (T1)–(T3) at the start of section 5.1 are satisﬁed. Moreover, for any elements U ∈ ΥH and Ke ∈ ΥG, the intersection satisﬁes U ∩ Ke ∈ ΥH .

Proof. By construction, if properties (T1)–(T3) are satisﬁed for both (Σ , Υ ) and (Σ p , Υ p ) then they p p are also satisﬁed for the pair (Σ , Υ ). For the “tame level”, the properties follow from the fact that the collection in Lemma 6.9 is closed under inclusions and conjugation. For the level at p, the properties follow from the fact that Υ is taken to be the collection of all compact open subgroups of Σ. The last part of the lemma follows from the analogous statement at levels away and at p; the former is true because the collection in Lemma 6.9 is closed under inclusion, the latter is true because ΥHp is the collection of all compact open subgroups of ΣHp , which is also closed under inclusion.

Let V be a self-dual algebraic representation of GE with highest weight c = (0; c1, . . . , c2n) as in Definition 2.12. With notation as in section 6.3, we define the following cohomology functors on P where is to be replaced by the symbol of the group appearing in the subscript of the cohomology functor: : 0 : 0 MH,Zp (−) = Hét (ShH(−), Zp)[λ/n] MH,Qp (−) = Hét (ShH(−), Qp) M (−) := H2n Sh (−), Tf(n) M (−) := H2n Sh (−), Ve(n) G,e Zp ét Ge G,e Qp ét Ge : 2n−1 : 2n−1 MG,ét,Zp (−) = Hét (ShG(−)E¯ , T (n)) MG,ét,Qp (−) = Hét (ShG(−)E¯ , V (n)) 2n−1 2n−1 M (−) := H Sh (−) ¯ , Tf(n) M (−) := H Sh (−) ¯ , Ve(n) G,e ét,Zp ét Ge E G,e ét,Qp ét Ge E

λ/n where [λ/n] denotes the twist by the character | det |p as in section 6.3. If we do not wish to specify whether we are working integrally or rationally, we will simply leave the ring out of the subscript. 36 ANDREW GRAHAM AND SYED WAQAR ALI SHAH

Remark 6.12. Note that the construction of the pullbacks and pushforwards in section 6.3 are valid in this

setting because we have taken ΥGp to be the collection of all compact open subgroups in ΣGp ; so we do indeed obtain cohomology functors. Lemma 6.13. The functors M ,M ,M and M are CoMack functors on their respective categories. H Ge G,ét G,e ét Moreover, we have a Cartesian pushforward ι : M → M . ? H Ge Proof. As explained in Example 5.2, all of these are cohomology functors satisfying axiom (Co). Note that Z = GU(1) × U(1), Z = GU(1), Z = GU(1) × U(1) and since GU( ) and U(1)( ) are finite, the - H G Ge Z Z Z points of all these central tori are finite too. As a result GU(1)(Q) (resp. U(1)(Q)) is discrete in GU(1)(Af ) (resp. U(1)(Af )), so all the Shimura data satisfy axiom SV5 of [Mil03] (or (SD5) of Appendix B). This implies the action of any sufficiently small compact open subgroup (in the sense of Definition B.5) on the respective infinite-level Shimura variety is free, and the proper base change theorem applied to pullbacks of these Shimura varieties gives axiom (M). For any compatible pair (U, K) ∈ Υ × Υ , we have a morphism of E-varieties Sh (U) → Sh (K), and H Ge H Ge therefore we have a pushforward (ι , br): (Sh (U), [λ/n]) → (Sh (K), ) ? H Zp Ge Tf in the sense of Definition A.4.4 By [Del71, Proposition 1.15], this forms an étalesmooth E-pair of codimension 5 n, whence we also have a pushforward ι? : MH, → M of cohomology functors by Proposition A.5, which Zp G,e Zp is also Cartesian (in the sense of Definition 5.7) by the second point of Proposition A.5. Indeed, for any K,L ∈ Υ , U ∈ Υ with U ⊂ K, and any choice of representatives γ ∈ U\K/L, the diagram given by Ge H

F γ Sh (U ) Sh (L) H γ Ge

(ι,ν) Sh (U) Sh (K) H Ge

is Cartesian, where U = U ∩ γLγ−1 and γ : Sh (U ) → Sh (L) is the morphism given by the composition γ H γ Ge γ Sh (U ) → Sh (γLγ−1) −→ Sh (L). H γ Ge Ge

Remark 6.14. Note that (MH, ,MH, ) and (M ,M ) do not form pairs of compatible cohomology Zp Qp G,e Zp G,e Qp p functors as defined in the discussion preceding Corollary 5.12, because the natural maps are only H(Af ) × p H(Zp)-equivariant and Ge (Af ) × Je-equivariant respectively. 6.5. Local finiteness of the Hecke action. In the next subsection, we will need certain finiteness hypothesis on the classes in the image of ι?. We recall the following notion from [Pil18].

Definition 6.15. Let M be a Zp-module and A an endomorphism of M. We say that the action of A on M is locally finite if for all n ∈ and all v ∈ M, the elements Akv generate a finite /pn -submodule N k∈N Z Z of M/pnM.

Lemma 6.16. Let M be a Zp-module and A an endomorphism of M. There exists a unique maximal sub-module M lf ⊂ M such that the action of A on M lf is locally ﬁnite. Furthermore, there exists an lf endomorphism eord of M such that A commutes with eord and gives a decomposition lf lf lf M = eord · M ⊕ (id − eord)M with A acting invertibly on the ﬁrst factor. Moreover, this construction is functorial in the pair (M,A), i.e. if (N,B) is another such pair and ϕ: M → N is a homomorphism of Zp-modules that intertwines the actions of A and B, then there is an induced homomorphism ϕlf : M lf → N lf which respects the decomposition above.

4 To ease notation we will simply denote this by ι? rather than the more cumbersome notation (ι, ν)?. 5More precisely, [Del71, Proposition 1.15] implies that there exists a compact open subgroup L containing U such that Sh (U) → Sh (L) is a closed immersion. This map factors through the morphism Sh (U) → Sh (L ∩ K), which by the H Ge H Ge cancellation property, is also a closed immersion. The desired factorisation is then given by Sh (U) → Sh (L∩K) → Sh (K). H Ge Ge ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS 37

lf k Proof. We deﬁne M to be subset of all x ∈ M such that for all n, the elements A x¯n k generate a ﬁnite n n sub-module of M/p , wherex ¯n is the reduction of x modulo p . Clearly this is a A-stable sub-module of M. lf k k lf If x ∈ M and y = ϕ(x), then B y¯n k = ϕ( A x¯n k) which implies that y ∈ N . The existence of eord and the corresponding decomposition is [Pil18, Lemma 2.1.3], and its funtoriality is straightforward.

The following lemma shows that the image of the pushforward ι? lands in the locally-ﬁnite subspace of M . Ge Lemma 6.17. Let (U, K × C) ∈ Υ × Υ be a compatible pair. Then the image of H Ge

ι? : MH, (U) → M (K × C) Zp G,e Zp is contained in M (K × C)lf , the locally ﬁnite sub-module with respect to the endomorphism T . G,e Zp Proof. To ease notation, we denote X := Sh (U),Y := Sh (K × C) = Sh (K) × Sh (C) H Ge G Spec E T

and f : X → Y the finite unramified morphism induced from the map on Shimura datum. Let Kp denote the projection of K to G(Qp) and fix a nested collection of normal compact open subgroups

Kp,i ⊂ Kp (i ≥ 1)

that has trivial intersection. Set Kp,0 = Kp and let Ki denote set of γ ∈ K such that γp ∈ Kp,i. If we set Y = Sh (K ×C), then there exists a ﬁnite set of primes S of E and a pro-system of smooth quasi-projective i Ge i ﬁnite-type schemes (Yi)i≥0 over OE,S such that: • For each i, there exist isomorphisms ∼ pi : Yi,E −→ Yi which are compatible under varying i. −1 ·g • For each Kp,j ⊂ gKp,ig with g ∈ Gep, the morphism Yj −→ Yi extends to a morphism ·g Yj −→Yi with respect to the isomorphisms above. • For j ≥ i, the map

Yj → Yi

is ´etalewith Galois group Kp,i/Kp,j.

Indeed, since (G,XG) is a PEL-type Shimura variety, the existence of integral models for ShG(Ki) is guaranteed by [Lan13, Theorem 1.4.1.12] once we fix a choice of PEL datum, and we can replace ShT(C) with the union of the spectra of the ring of integers of a collection of finite extensions of E localised at all the places above S (in fact, it will turn out that we only need one, as ShT will be a torsor over the anticylotomic extension). Similarly, there exists an integral model for X and a pro-system of schemes (Xi)i≥0 for (H,XH) satisfying the analogous conditions as above (see Example B.2). Set Y := Y0 and X := X0. The quadruplet (Y, {Yi} ,Kp, Te) determines a lisse Zp-sheaf Te on YK (e.g. see [HT01, §III.2]) and it is easily verified that the pullback of Te under p: Y → Y is isomorphic to Tf. A similar construction holds for the quadruplets (X , (Xi),Up, Te) and (X , (Xi),Up, Zp[λ/n]) with respect to the map q : X → X , where Zp[λ/n] is the equivariant sheaf defined in section 6.3. We note that since the branching law holds on X, it also holds on X . By Theorem 3.2.1 in [Poo17], we may choose (after possibly enlarging S) an extension of f to a morphism g : X → Y which makes (X , Y) into an étalesmooth OE,S-pair of codimension n, as in Definition A.4. We ? may also assume that the sheaf on X is identified with f Te and that π0(X ) is identified with π0(X). This data therefore determines a pushforward

(g, br)? :(X , Zp[λ/n]) → (Y, Te). Since the diagram on the left below is Cartesian, the diagram on the right is commutative by Proposition A.5. 38 ANDREW GRAHAM AND SYED WAQAR ALI SHAH

f ι? X Y MH, (U) M (K × C) Zp G,e Zp

p q q?

g 0 (g,br)? 2n X Y H´et(X , Zp)[λ/n] H´et (Y, Te(n))

? Our choices above imply that q is a Zp[T ]-module homomorphism. 2n Finally, to conclude the proof of the lemma, we will show that Hét (Y, Te(n)) is a finitely-generated Zp- module. Enlarge S to include all primes above p, if necessary. By [Mil06, Proposition II.2.9], the étalecoho- mology of Spec OE,S for constructible étale sheaves is identified with the group cohomology of Gal(ES/E) (where ES is the maximal unramified outside S extension of E). Therefore, for any integer r ≥ 1 and r i any constructible étalesheaf F of Z/p Z-modules, the cohomology group Hét (Spec OE,S, F ) is a finite r Z/p Z-module. Indeed, this is because Gal(ES/E) satisfies condition (F) (see Lemma A.11). r t By [Kat93, §3.1.3], the higher direct images of Te(n)/p under the morphism Y −→ Spec OE,S are constructible. Combining this with the Grothendieck spectral sequence associated with the composition

RΓ´et(Y, −) = RΓ´et(Spec OE,S, −) ◦ Rt∗

(where RΓét(Z, −) denotes the right-derived global sections of constructible sheaves on Z), we see that 2n r r Hét Y, Te(n)/p is a finite Z/p Z-module, for r ≥ 1. This system of finite-type modules therefore satisfies the Mittag-Leffler property ([Jan88, Remark 1.14]), and we have H2n Y, T(n) = lim H2n Y, T(n)/pr . ét e ←− ét e r

The right-hand side is an inverse limit of ﬁnite-type modules, so is ﬁnite-type over Zp (see the proof of Proposition 4.2.3 in [Nek06], for example).

6.6. Ordinary and ﬁnite-slope parts. To obtain classes in the cohomology of ShG that are cohomologically trivial, it turns out it is enough to construct universal norms out of them in the “p-direction”. Once we have accomplished this, we can apply the construction in Proposition A.10 to map to the cohomology of the Galois representation attached to π. We obtain these universal norms by projecting these classes to the ordinary part of the cohomology of Sh under the action of a suitable idempotent. As a consequence, we Ge also obtain the vertical norm relations for the Euler system.

We fix the level in Gp component to be J ∈ ΥGp in this section (although results apply for any fixed level p Kp). We note that each M (K J × C) is endowed with an action of T , and these actions commute with G,e Zp pullbacks and pushforwards. More precisely p Lemma 6.18. Let M : PGp × PTp → Zp-Mod denote the functor G,e Zp e p p p p (K × C ,Cp) 7→ M (K J × C Cp). G,e Zp p p Then T is an endo-functor of M . In other words, the functor M is valued in Zp[T ]-Mod. G,e Zp G,e Zp Proof. This is a consequence of Lemma 5.6. The ordinary idempotent is now constructed on the sub-functor on which T acts locally finitely. lf Lemma 6.19. The mapping M : PGp × PTp → Zp-Mod given by G,e Zp e p p p p lf (K × C ,Cp) 7→ M (K J × C Cp) G,e Zp p is a Zp[T ]-valued CoMack sub-functor of M . Moreover, there exists an idempotent endofunctor eord of G,e Zp M lf giving a decomposition G,e Zp lf lf lf M = eord · M ⊕ (id − eord) · M G,e Zp G,e Zp G,e Zp which commutes with T , and the action of T on the first component is invertible. Proof. Axioms (C1), (C2), (C3) and (M) hold by the first part of Lemma 6.16 and Lemma 5.6. Axiom (Co) holds by the additivity of taking locally finite parts. The rest follows from the second part Lemma 6.16. ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS 39

Definition 6.20. We consider the following cohomology functors ord −1 fs −1 M : P p × PTp → Zp[T , T ]-Mod M : P p × PTp → Zp[T , T ]-Mod G,e Zp Ge G,e Zp Ge p p lf p p p p lf p p (K × C ,Cp) 7→ eord · MG, (K J × C Cp)(K × C ,Cp) 7→ MG, (K J × C Cp) Zp Zp T which we refer to as ordinary part and the finite slope part of M respectively. Here the subscript denotes G,e Zp fs ord localisation. We denote the natural transformation M → M by eord as well. G,e Zp G,e Zp ord −1 Lemma 6.21. The map M : PGp → Zp[T , T ]-Mod given by G,e Iw,Zp e p p ord p p K × C 7→ lim M (K J × C Cp) ←− G,e Zp Cp defines a CoMack functor. ord lf Proof. All the axioms hold for M because they hold for M and eord is additive. The pushforwards G,e Zp G,e Zp N along P p commute with pushforwards/pullbacks along P (because the operators T also do) by Lemma T Gep fs 5.6, and therefore they also hold for M by functoriality of the inverse limit. G,e Iw,Zp p p p Let M : PHp → p-Mod denote the functor U 7→ MH, (U H( p)). H,Zp Z Zp Z Proposition 6.22. There is a (Cartesian) pushforward of cohomology functors ord p ord ι? : MH, → M . Zp G,e Iw,Zp

Proof. We apply the results of [Loe19] to the pushforward ι? : MH, → M . Let Vr ∈ P and Dpr ∈ PT Zp G,e Zp Gep p be as in section 6.2. By Lemma 6.2 and Theorem 4.5.3 in op.cit., we have a (Cartesian) pushforward p (6.23) M → lim M (− · Vr)T H,Zp ←− G,e Zp r where the target is a functor on P . Explicitly, let η = uτ r and U p = Hp ∩ Kp. For all r ≥ 1, we have a Gep r ee map θr,Kp given by the composition: ? p pr p −1 MH,Zp (U · H(Zp)) −−→ MH,Zp (U · (Hp ∩ ηrVrηr )) ι? p −1 −→ M (K · (ηrVrη )) G,e Zp r [ηr ]? p −−−→ M (K · Vr) G,e Zp −r T p −−−→ M (K · Vr)T G,e Zp p p p For any r, the maps θr,Kp , θr+1,K commute with the pushforward map from level K · Vr+1 to K · Vr – this amounts to showing the commutativity of the bottom left square of the diagram in §4.5 of op. cit, which is where the Cartesian property of ι? is invoked. Moreover, this pushforward is also compatible with the T action – this follows from Lemma 6.4 (a) and the Cartesian property of M . The upshot is that we G,e Zp −1 p have an inverse system of maps (θr,Kp )r≥1 of Zp[T , T ]-modules. As the dependence on K is functorial by Lemma 5.6, this gives the pushforward claimed in (6.23). Now Vr ⊂ J × Dpr , so there also exists projection maps

pr : M (− · V ) −→ M (− · (J × D r )). ? Ge r Ge p Again, by Lemma 5.6, this is a pushforward of cohomology functors, and moreover commutes with the action of T by Lemma 6.4 (b), whence also with the localisation with respect to T . We therefore obtain a pushforward of cohomology functors fs p ι : M → lim M (− · (J × Dpr )) ? H,Zp ←− G,e Zp T r by composing (6.23) with pr? (after localizing at T ). Finally, we claim that this functor factors through fs lim M (− · (J × Dpr )). Indeed, it suﬃces to show that the image of the map ←− G,e Zp r p p p p M (U · H( p)) → M (K J × C Dpr ) H,Zp Z G,e Zp 40 ANDREW GRAHAM AND SYED WAQAR ALI SHAH

lf p p lands in M (K J × C Dpr ) which was proven in Lemma 6.17. We then set G,e Zp ord p ord ι? : MH, → M Zp G,e Iw,Zp fs to be eord ◦ ι? , which completes the proof.

6.7. Pushforward to Iwasawa cohomology. For a compact open subgroup C ∈ ΥT , we consider the finite group ∆(C) := T(Q)\T(Af )/C ab which carries an action of Gal(E /E) via translation by the character κC = κ, where we set κC (σ) = × s¯f /sf for any idele s ∈ AE satisfying ArtE(s) = σ. We let E(C) denote the finite abelian extension of E corresponding to the kernel of this character. Since M and MG,ét, are finitely-generated over p (see G,e ét,Zp Zp Z [Del77, Corollaire 1.10, Chapitre 7]), the action of T on both of these modules is locally finite, and we can therefore define ordinary and finite-slope parts on the whole functor. We note the following application of Shapiro’s lemma. Lemma 6.24. We have an isomorphism 1 ∼ 1 H E,M (K × C) = H E(C),MG,ét, (K) G,e ét,Zp Zp −1 which intertwines the action of Gal(E(C)/E) and T(Af ) via the character κC (explicitly, the action of an −1 element σ ∈ Gal(E(C)/E) on the right-hand side corresponds to the action of κC (σ) on the left-hand side). The same statement holds for ordinary and finite-slope parts.

ab Proof. One sees from the deﬁnition of the Shimura datum for T that Gal(E /E) acts on ShT(C)E¯ = ∆(C) via translation by the character κ = κC . One has an isomorphism 0 (6.25) H´et ShT(C) ×Spec E Spec E, Zp = Maps (ShT(C)(C), Zp) = Zp[Gal(E(C)/E)] X φ 7→ φ(κ(σ−1)) · σ σ∈Gal(E(C)/E)

which intertwines the action of t ∈ T(Af ) on the left-hand side (given by right-translation of the argument (t · φ)(−) = φ(− · t)) with the action of σ ∈ Gal(E(C)/E) on the right-hand side (the action is by left- multiplication), where σ is the unique element such that the image of t in ∆(C) equals κ(σ). We have the following isomorphisms 1 ∼ 1 0 (6.26) H E,M (K × C) = H E,MG,ét, (K) ⊗ H ShT(C) ×Spec E Spec E, p G,e ét,Zp Zp Zp ét Z ∼ 1 (6.27) = H E,MG,ét,Zp (K) ⊗Zp Zp[Gal(E(C)/E)] ∼ 1 (6.28) = H E(C),MG,ét,Zp (K) where (6.26), (6.27) and (6.28) follow from the Künnethformula, (6.25) and Shapiro’s lemma respectively. The action of an element σ ∈ Gal(E(C)/E) on the right-hand side of (6.28) is intertwined with left multipli- −1 cation by σ on Zp[Gal(E(C)/E)] appearing on the right-hand side of (6.27) (see [Nek06, §8.2]). By (6.25), −1 this is precisely intertwined with the action of an element t ∈ T(Af ) on the left-hand side of (6.26), where the image of t in ∆(C) coincides with κ(σ). The same result holds for the finite-slope and ordinary parts because these operations commute with the action of T. Definition 6.29. We let N : P → [T .T −1]-Mod denote the map Iw,Zp Gep Zp p p 1 ord p p K × C 7→ lim H E,M (K J × C Dpr ) . ←− G,e ét,Zp r

Lemma 6.30. NIw,Zp is a CoMack functor. p p fs Proof. The functor NIw, is obtained by composing the functor M (K J ×C Dpr ) with eord, applying Zp G,e ´et,Zp H1(E, −) and then taking the inverse limit over r. All of these operations are additive endomorphisms of −1 the category of Zp[T , T ]-modules. ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS 41

p We now obtain the vertical norm relations for our Euler system. Set MH, := M ⊗ p and Qp H,Zp Zp Q : NIw,Qp = NIw,Zp ⊗Zp Qp. p p p S p Theorem 6.31 (Vertical norm relations). Set Er(C ) = E(C Dpr ) and E∞(C ) = Er(C ). For R = Zp or Qp, there is a (Cartesian) pushforward of cohomology functors p ι?,Iw : MH,R → NIw,R which is compatible with respect to base change of coeﬃcients (as in the paragraph preceding Corollary 5.12).

Proof. We ﬁrst note that by Lemma 6.24 that the functor NIw,Zp is isomorphic to the functor

p p 1 p ord p K × C 7→ lim H Er(C ),M (K J) ←− G,ét,Zp r p where the inverse limit is with respect to corestriction. Moreover, the fields Er(C ) for r ≥ 0 form Γ-towers × p for Γ = Zp in the sense of Appendix A.2. By Proposition A.10 applied to the Shimura varieties of level K and Γ-towers of level Cp, we get an Abel–Jacobi map: p p 2n p p AJIw : lim M (K J × C Dpr ) = M (E∞(C ), ShG(K J)) ←− G,e Zp Iw r 2n−1 → H1 E (Cp), M (Sh (KpJ)) = lim H1 E (Cp),M (KpJ) Iw ∞ G ←− r G,ét,Zp r p because the p-adic Lie extension E∞(C ) has positive dimension. By the functoriality properties in loc.cit., this map commutes with taking ordinary parts and assembles into a map of cohomology functors:

ord 1 ord AJIw : M → lim H Er(−),MG,´et, (− · J) = NIw, p . G,e Iw,Zp ←− Zp Z r ord The map ι?,Iw is then deﬁned as the composition ι?,Iw := AJIw ◦ ι? . The result for Qp follows by inverting p.

7. The anticyclotomic Euler system

We now have all the ingredients needed to deﬁne the Euler system. Let π0 be a cuspidal automorphic representation of G0 such that π0,∞ lies in the discrete series, and let π be a lift to G. From now on, we make the following assumption: Assumption 7.1. We assume that:

(1) π0,f admits a H0(Af )-linear model. In particular, we can choose a lift π with trivial central character (see section 2.4). (2) n is odd and π satisfies Assumption 3.5. In particular, let K be a compact open subgroup of G(Af ) K such that πf 6= 0. (3) Let S be the finite set of primes containing all primes that ramify in E and all primes where K is not hyperspecial. We assume that p∈ / S and splits in E/Q. In this case we set S0 = S ∪ {p}. (4) We have S0 K = Kp · KS · K S0 S0 with KS satisfying the following property: for any compact open subgroups Lp ⊂ Gp and L ⊂ G , S0 the subgroup LpKSL is sufficiently small (see section 2.2). (5) Let c = (0; c1, . . . , c2n) denote the weight corresponding to the discrete series L-packet containing Pn π∞, and set λ = i=1 ci. We assume that π is “Siegel ordinary”, i.e. there exists an eigenvector ! p O O K` KS J K J ϕ = ϕ` ⊗ ϕS ⊗ ϕp ∈ π` ⊗ πS ⊗ πp = πf `/∈S0 `/∈S0 λ of the Hecke operator US := p [JτJ] with eigenvalue α a p-adic unit. Here, J is the Siegel parahoric subgroup and τ is the element defined in section 6.1. In particular, this implies that α occurs as an 0 λ −1 ∨ KpJ eigenvalue for the transpose operator US := p [Jτ J] on (πf ) . 42 ANDREW GRAHAM AND SYED WAQAR ALI SHAH

∼ Remark 7.2 (Justification of Assumption 7.1, (5)). Suppose that π0,p is generic, in the sense that π0,p = I(χ), ∼ where I(−) denotes the (normalised) induction from the (standard) Borel to G0(Qp) = GL2n(Qp), and 2n × χ = χ1 ··· χ2n : Qp → C is an unramified character, naturally viewed as a character of the (standard) torus. Let βi = χi(p)(i = 1,..., 2n) denote the Satake parameters of π0,p. For any rational number s, we let ps denote the unique element of Q which maps to es log p under the fixed embedding Q ,→ C. Via the s fixed embedding Q ,→ Qp, we can also view p as an element of Qp.

Let v|p be the prime of E ﬁxed by the embedding Q ,→ Qp, and set ρv := ρπ|Ev . Then by Theorem 3.4(3), the Galois representation ρv is crystalline and one has an equality of polynomials 2n Y n−1/2 det (1 − ϕ · X|Dcris(ρv)) = (1 − p βi · X). i=1

Let vp denote the p-adic valuation on Qp, normalised so that vp(p) = 1, and order the Satake parameters so n−1/2 Q that γi := p βi satisfy vp(γ1) ≤ · · · ≤ vp(γ2n). For any subset I ⊂ {1,..., 2n} of size n, set γI = i∈I γi. Recall that attached to D = Dcris(ρv) one has the Hodge polygon Hodge(D), which is deﬁned to be the convex hull of the points

 i    X  {(0, 0)} ∪ i, c2n+1−j + (j − 1) : i = 1,..., 2n  j=1  using the explicit recipe for the jumps in Hodge–Tate ﬁltration in Theorem 3.4(3). One also has the Newton polygon Newt(D), which is deﬁned to be the convex hull of the points

 i    X  {(0, 0)} ∪ i, vp(γj) : i = 1,..., 2n .  j=1  Since D is weakly admissible, the Newton polygon lies on or above the Hodge polygon (and has equal endpoints). In particular, this implies that n n n X X X (7.3) vp(γI ) ≥ vp(γj) ≥ c2n+1−j + (j − 1) = c2n+1−j + n(n − 1)/2. j=1 j=1 j=1 J On the other hand, we can express the eigenvalues of US acting on πp in terms of γI , using the results of J J0 ∼ J0 [OST19, §4.1.1]. Indeed, note that we have πp = π0,p = I(χ) and, in the notation of loc.cit., the action of the operator [JτJ] corresponds to the action of the operator Iχ(1γˇ), withγ ˇ(p) = diag(p, . . . , p, 1,..., 1) (there are n lots of p). Therefore, one has an equality of polynomials

2 2 −n /2 J −n /2 1 J0 Y det 1 − p [JτJ] · X|πp = det 1 − p Iχ( γˇ) · X|I(χ) = (1 − βI · X) I Q where the product runs over all subsets I ⊂ {1,..., 2n} of size n, and βI = i∈I βi. This implies that the J eigenvalues of US acting on πp are of the form λ n2/2 λ+n2/2−n(n−1/2) λ−n(n−1)/2 (7.4) p p βI = p γI = p γI .

It turns out that, under Assumption 7.1(1), the algebraic representation with highest weight (c1, . . . , c2n) is Pn self-dual (see Lemma 7.5 below), which implies that λ = − j=1 c2n+1−j. Therefore, combining (7.3) and (7.4), it is reasonable (at least generically) to assume that US may have an eigenvalue which is a p-adic unit. In particular, if πp is Borel ordinary (or equivalently, Newt(D) = Hodge(D)) then πp is Siegel ordinary. ∗ Let V be the algebraic representation of GC (which arises from an algebraic representation of GE) parameterising the discrete series L-packet of π∞. By [BW13, §II.5], this implies that π is cohomological with respect to the representation V . Lemma 7.5. The representation V is self-dual. 0 Proof. Let `∈ / S be a split prime. Since π0,f admits a H0(Af )-linear model, the representation π0,` is H0(Q`)-distinguished, therefore by [JR96] ∨ ∼ π0,` = π0,`. ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS 43

This implies that π0 is self-dual by the mild Chebotarev density theorem proven in [Ram15, Corollary B]. Since duality is preserved under base-change and we have assumed that any weak base-change Π of π is cuspidal, by strong multiplicity one (see [JS81, Theorem 4.4] for example) we must have Π =∼ Π∨.

In particular, this implies that V is self-dual since the algebraic representation of ResC/R GL2n corresponding 0 ∗ 0 to Π0 has weight (c, −c ), where c = (c1, . . . , c2n) is the weight of V and c = (c2n, . . . , c1) (see Proposition 3.1). Let R be the set of all square-free products of rational primes that lie outside the set S0 and split in the × × extension E/Q (we allow 1 ∈ R). For any positive integer m, let Obm ⊂ AE,f be the compact open subgroup consisting of the units of the profinite completion of Om = Z + mOE, and let D[m] ⊂ T(Af ) denote the × image of Obm under the map × N : AE,f → T(Af ) z 7→ z/z¯ × By Lemma 2.3, D[m] is a compact open subgroup. If we set D` = N ((Om ⊗ Z`) ) for `|m, then D[m] Q Q × decomposes as D` × N ((OE ⊗ `) ). Let E[∞] be the compositum of all ring class extensions `|m `-m Z ab E[m] for m ≥ 1 and let ResE[∞] : Gal(E /E) → Gal(E[∞]/E) denote the restriction map. Since the infinite place of E is complex, the restriction of the Artin map ArtE to the finite ideles is surjective onto the abelianisation of GE. Lemma 7.6. There exists a continuous surjective homomorphism

Art0 : T(Af ) → Gal(E[∞]/E) with kernel T(Q), which satisﬁes the following properties: • One has the following equality of compositions

ResE[∞] ◦ Art = Art0 ◦ N ∼ • For each m ≥ 1, the map Art0 induces an isomorphism T(Af )/T(Q)D[m] = Gal(E[m]/E). ∼ × • If ` - m is split in E, then under the identification T(Q`) = Q` , the map Art0 sends ` to the geometric Frobenius at λ in Gal(E[m]/E), where λ is the prime above ` distinguished by the fixed embedding Q ,→ Q` (the identification is also with respect to this fixed embedding). Proof. It is well known (e.g. see [Kü17,Lemma 2]) that E[∞] is the fixed field of the transfer map Ver: Gab → Q ab GE , so have a commuative diagram

× × N 1 Af AE,f T(Af ) 1

Art Q ArtE Art0 1 Gab Ver Gab Res Gal(E[∞]/E)) 1 Q E

with exact rows, by Lemma 2.3. The surjectivity of Art0 follows from that of ArtE. The isomorphism in the ∼ × × × second assertion is induced from restricting the isomorphism Gal(E[m]/E) = AE,f /E Obm via N and Res. The third assertion follows by tracking the uniformiser under Res ◦ ArtE.

Let V be the (self-dual) representation of GE with highest weight c = (0; c1, . . . , c2n) associated with π as above, and let Ve := V 1 denotes its trivial extension to Ge E. We ﬁx choices of highest weight vectors and lattices as in section 6.3. By abuse of notation, we let V (resp. Ve) denote both the relative Chow motive

AncE(V ) (resp. AncE(Ve)) and the lisse Qp-sheaf µ(VQp ) (resp. µ(VeQp )). We also let T (resp. Tf) denote the equivariant sheaf associated with the representation T (resp. Te) as constructed in section 6.3. Since V is self-dual, the ﬁxed parahoric vector in Assumption 7.1 gives rise to a “modular parametrisation”, i.e. an equivariant linear map 2n−1 ∗ vϕ ◦ pr ∨ :H (Sh , V (n)) ⊗ → W (1 − n) π ´et G,Q Qp π where vϕ is evaluation at ϕ. Since we have assumed that π is Siegel ordinary, this map factors through the 0 ordinary part with respect to the operator US. 44 ANDREW GRAHAM AND SYED WAQAR ALI SHAH

Definition 7.7. We let ∗ p ∗ : ∨ Tπ (1 − n) = vϕ ◦ prπ MG,ét,Zp (K J) ⊗ Zp ⊂ Wπ (1 − n) ∗ which is by construction a Galois stable Zp-submodule inside Wπ (1 − n). If Φ is a finite extension of Qp, ∗ ∗ with ring of integers O, such that Wπ (1 − n) and vϕ ◦ prπ∨ are defined over Φ, then Tπ (1 − n) is a Galois p stable lattice with a model over O given by vϕ ◦ prπ∨ MG,ét,Zp (K J) ⊗ O . Such an extension Φ always 1 ∞ ∗ exists. We let Hlf,Iw(E[mp ],Tπ (1 − n)) denote the locally finite Iwasawa cohomology (Definition A.13), and Ψ :H1 (E[mp∞],T ∗(1 − n)) → lim H1(E[mpr],W ∗(1 − n)) m lf,Iw π ←− π r denote the natural maps. These maps are injective by Corollary A.19 and Assumption 3.5.

For every (sufficiently small) compact open subgroup U ⊂ H(Af ), fix elements 0 1U ∈ Hmot (ShH(U), Z) 0 that are compatible under pull-backs, such that their common image 1 ∈ Hmot (ShH, Z) is fixed by the action 0 of H(Af ). By abuse of notation, we also let 1U denote the étalerealisation in Hét (ShH(U), Zp).

p p Definition 7.8. Let m ∈ R and g ∈ Ge , and set Ug = Ug · H(Zp) where p p p −1 p Ug := g (K × D[m] ) g ∩ H . We define the following class 1 ∞ ∗ z := v ◦ pr ∨ ◦ [g] ◦ ι 1 ∈ lim H (E[mp ],W (1 − n)) g,m,Iw ϕ π ? ?,Iw Ug ←− π r p This definition extends by linearity to elements zφ,m,Iw for functions φ ∈ H(Ge ) fixed by the action of Kp × D[m]p.

1 ∞ ∗ Lemma 7.9. Let m ∈ R and φ as above. Then there exists a unique class in Hlf,Iw(E[mp ],Tπ (1 − n)) that maps to zφ,m,Iw under Ψm.

Proof. By construction, the class zφ,m,Iw is in the image of H1 (E[mp∞],T ∗(1 − n)) → lim H1(E[mpr],W ∗(1 − n)) Iw π ←− π r ∗ as [g]∗ and ι?,Iw are pushforwards of Zp-valued cohomology functors and Tπ (1 − n) was deﬁned to be the (Galois-stable) image of values of these functors under the Qp-linear maps vϕ and prπ∨ . The fact that these classes are in the image of Ψm follows from our arguments in the proof of Lemma 6.17: the classes lift to classes in the cohomology of appropriate integral models of our Shimura–Deligne varieties, and therefore by Lemma A.14, are in the image of Ψm. As Ψm is injective by Corollary A.19, the result follows. 1 ∞ ∗ Abusing notation, we will denote both the class in HIw,lf (E[mp ],Tπ (1−n)) and its image (under inﬂation) 1 ∞ ∗ in HIw(E[mp ],Tπ (1 − n)) by zφ,m,Iw. The following proposition will allow us to prove norm compatibility relations for our Euler system classes.

p p Proposition 7.10. Keeping with the same notation as in Deﬁnition 7.8, there exists a H(Af ) × Ge (Af )- equivariant linear map

p L : H(Gp)1×D[m] −→ π∨ ⊗ lim H1 (E[mpr],W ∗(1 − n)) m e f ←− π r such that: (m) P p p p (a) For any φ = i ai ch (gi(K × D[m] )) in the Hecke algebra H(Ge , Zp), the image of zφ(m),m,Iw in the inverse limit lim H1 (E[mpr],W ∗(1 − n)) satisﬁes ←−r π (m) zφ(m),m,Iw = vϕ ◦ Lm(ψ )

(m) P ai p p where ψ = ch (gi(K × D[m] )). Vol(Ugi ) ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS 45

(b) For a prime ` such that `m ∈ R, one has

E[`mp∞] (`m) (`m) coresE[mp∞] L`m(ψ ) = Lm(pr∗(ψ )).

p 1×D[`m]p p 1×D[m]p where pr∗ : H(Ge ) → H(Ge ) is the natural pushforward.

Proof. The map Lm is obtained by composing the completed pushforward of ι?,Iw (see Proposition 5.11) p p with prπ∨ as follows. Since 1×D[m] lies in the centre of Ge , the completed pushforward gives rise to a map ! p p 1×D[m]p p 1 r p ord (7.11) ˆι?,Iw : Mc ⊗ H(Ge ) −→ N Iw, p (1 × D[m] ) → lim H E[mp ], lim MG,´et, p (JL ) H,Qp Q ←− −→ Q r Lp where the last map is the natural one. We deﬁne Lm to be the map

Lm(φ) := prπ∨ ◦ bι?,Iw(1 ⊗ φ).

The equivariance property for the map Lm follows from the equivariance of ι?,Iw and the fact that the class p b 1 is fixed by the action of H(Af ). Since the pushforward ι?,Iw is compatible with base-change of coefficients, one obtains the required property from the commutative diagram in Corollary 5.12. This gives us (a). For p p claim (b), notice that there are induced maps N Iw,Qp (1 × D[`m] ) → N Iw,Qp (1 × D[m] ) coming from the p pushforward of NIw,Qp in the T component. By Lemma 6.24, the norm map corresponds the corestriction under the second morphism in (7.11). 7.1. The Euler system. We continue with the same notation as in the previous sections. We consider the p p following test data for our Euler system: for m ∈ R, define an element φ(m) ∈ H(Gep, Z)K ×D[m] as the product     (m) O  O φ :=  φ`,0 ⊗  φ` ⊗ φS   `/∈S `|m `-m where × KS ×N ((OE ⊗ S ) ) • φS ∈ H(GeS, Z) Z is fixed. × • φ`,0 = ch (K` × T(Z`)) ∈ H(Ge`, Z) (note that N ((OE ⊗ Z`) ) = T(Z`) for primes ` that do not ramify in E/Q, by the proof of Lemma 2.3). Pn • Using the notation in Theorem 4.1, we define φ` = i=0 ai ch ((gi, 1)(K` × D`)), where the element gi is 1 `−1X g = 1 × i ∈ GL ( ) × GL ( ) = G( ). i 1 1 Q` 2n Q` Q`

The integers ai are deﬁned to be −1 ai := (` − 1) · bi · [H(Z`): V1,i] −1 where V1,i = (gi, 1)(K` × D`)(gi, 1) ∩ H` (this quantity ai is an integer by part (3) of loc.cit.).

Deﬁnition 7.12. Let π0 be a cuspidal automorphic representation of G0 satisfying Assumption 7.1. For m ∈ R, we deﬁne the Iwasawa Euler system class to be

−n2 1 ∞ ∗ Zm,Iw := m · zφ(m),m,Iw ∈ HIw (E[mp ],Tπ (1 − n)) which are well-deﬁned elements by Lemma 7.9. We now prove our main result of this article. Theorem 7.13. Let `, m ∈ R such that ` is prime and their product satisﬁes `m ∈ R. The Iwasawa Euler system classes satisfy

E[`mp∞] −1 (7.14) coresE[mp∞] Z`m,Iw = P (Frλ ) ·Zm,Iw −1 where P (X) = det 1 − Frobλ X|ρπ(n) and λ is the unique prime of E lying above ` ﬁxed by the embedding Q ,→ Q`. 46 ANDREW GRAHAM AND SYED WAQAR ALI SHAH

Proof. We will perform a series of reduction steps to reduce the proof to that of Theorem 4.1.

1 ∞ ∗ Step 1. Since the classes Zm,Iw are elements of the submodule Hlf,Iw(E[mp ],Tπ (1 − n)) and the maps Ψ are injective by Corollary A.19, it suﬃces to verify the claimed relations in lim H1(E[mpr],W ∗(1 − n)) m ←−r π considered as a Qp-vector space. By Proposition 7.10 (b), there exist a commutative diagram

p H(Gp)1×D[`m] L`m π∨ ⊗ lim H1 (E[`mpr],W ∗(1 − n)) e f ←−r π ∞ pr∗ E[`mp ] coresE[mp∞]

p H(Gp)1×D[m] Lm π∨ ⊗ lim H1(E[mpr],W ∗(1 − n)) e f ←−r π and functions ψ(`m) and ψ(m) such that: −n2 −n2 (`m) •Z `m,Iw = ` m (vϕ ◦ L`m)(ψ ) −n2 (m) •Z m,Iw = m (vϕ ◦ Lm)(ψ ). 2 By the diagram above, and cancelling the m−n factor, this reduces the theorem to proving the statement (`m) n2 −1 (m) (7.15) (vϕ ◦ Lm) (pr∗(ψ )) = ` P (Frλ ) · (vϕ ◦ Lm)(ψ ). (`m) (m) (`m) ` (m) ` We note that both functions pr∗(ψ ) and ψ decompose as pr∗(ψ` ) ⊗ ψ and ψ` ⊗ ψ for some ﬁxed element ψ` ∈ H(Ge`p), where (`m) X ai ψ = ch ((gi, 1)(K` × T(Z`))) ∈ H(Ge`) ` Vol(V ) i 1,i (m) 1 ψ` = ch(K` × T(Z`)) ∈ H(Ge`) Vol ((K` × T(Z`)) ∩ H`) using the explicit formulae in Proposition 7.10(a) and at the start of section 7.1.

Step 2. We now simplify the expression in (7.15) further by cutting out by a finite-order character. Since it is enough to prove the relation (7.15) at each finite layer of the inverse limit lim H1 (E[mpr],W ∗(1 − n)), it ←−r π is enough to show the relation in (7.15) after post-composing both sides by any Galois equivariant morphism (7.16) lim H1 (E[mpr],W ∗(1 − n)) → (χ) ←− π Qp r where χ is a finite-order character of Gal(E[mp∞]/E). More precisely, let χ be a finite-order character of Gal(E[mp∞]/E). Since the Galois action and the action p p −1 p of T(Af ) = T are intertwined by the character κ (see Lemma 6.24), we can view χ as a character of T by setting χ(t) = χ(σ), where σ ∈ Gal(E[mp∞]/E) is an element such that κ(σ−1) equals the image of t in p ` ` lim ∆(D[m] D r ). This character factorises as χ · χ , where χ is a character of T and χ is a character ←−r p ` ` ` `p `p of T = T(Af ). −1 −1 −1 −1 Furthermore, the Frobenius element Frλ satisfies κ(Frλ ) = ` , so the action of P (Frλ ) is intertwined with multiplication by −n −1 P (χ`(`)) = Pλ(χ`(`)` ) = L(π` ⊗ χ`, 1/2)

under the map in (7.16), where Pλ(X) is as in Theorem 3.4. χ p p Let Lm denote the composition of Lm with the map 1⊗(7.16). This is a H(Af ) × Ge (Af )-equivariant p −1 ∨ linear map H(Ge ) → (πf χ ) , and to prove the theorem, we are required to prove the statement χ (`m) n2 −1 χ (m) (7.17) (vϕ ◦ Lm)(ψ ) = ` L(π` ⊗ χ`, 1/2) (vϕ ◦ Lm)(ψ ) for all ﬁnite-order characters χ.

Step 3. We now reduce the statement in (7.17) to one in local representation theory. The map 1×D[m]p −1 ∨ Z: H(Ge`) −→ (π` χ` ) χ ` υ 7→ vϕ` ◦ Lm (υ ⊗ ψ ) ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS 47

` ` −1 ∨ −1 ∨ −1 ∨ ` where vϕ` :(πf (χ ) ) ⊗ (π` χ` ) → (π` χ` ) is the evaluation map at the vector ϕ = N q∈ /S0∪{`} ϕq ⊗ ϕS ⊗ ϕp, is H` × Ge`-equivariant by construction. Note that we have

−1 ai (` − 1) · Vol ((K` × T(Z`)) ∩ H`) · = bi Vol(V1,i) where bi are the integers appearing at the start of section 7.1. So by multiplying the relation in (7.17) by −1 (` − 1) · Vol ((K` × T(Z`)) ∩ H`) and using the explicit formulae in Step 1, we are reduced to proving the relation: ! 2 X `n (7.18) (v ◦ Z) b ch ((g , 1)(K × T( ))) = L(π ⊗ χ , 1/2)−1 · (v ◦ Z) (ch(K × T( ))) . ϕ` i i ` Z` ` − 1 ` ` ϕ` ` Z` i

P Step 4. We now apply Frobenius reciprocity. Let υ denote the element i bi ch ((gi, 1)(K` × T(Z`))). −1 By Lemma 5.10 (b), the map Z corresponds to an element z ∈ HomH (π` χ` , C) (implicitly using the identiﬁcation of Qp and C), and the relation in (7.18) takes the form 2 `n z(υ · ϕ ) = L(π ⊗ χ , 1/2)−1 · z(ϕ ) ` ` − 1 ` ` `

noting that ϕ` is spherical. This relation now follows from Theorem 4.1 (using the fact that χ` is unramfied) and this completes the proof of Theorem 7.13. r Definition 7.19. We define the anticyclotomic Euler system class at level mp , denoted cmpr , to be the image of Zm,Iw under the natural map 1 ∞ ∗ 1 r ∗ HIw (E[mp ],Tπ (1 − n)) −→ H (E[mp ],Tπ (1 − n)) . As a corollary of Theorem 7.13, we immediately obtain Theorem A in the introduction with the classes cmpr .

7.2. Motivic interpretation. It will come as no surprise to the reader that the Euler system classes cmpr arise from elements in the motivic cohomology (see section 1.2) of Sh . More precisely, we consider the Ge following deﬁnition.

−1 Definition 7.20. Let g ∈ Ge (Af ) and L ⊂ Ge (Af ) a compact open subgroup. Set U := gLg ∩ H(Af ). Then we define Z := ([g] ◦ (ι, ν) ◦ br)(1 ) ∈ H2n Sh (L), (n) g,mot ? ? U mot Ge Ve where = Anc (V ) as before. This definition can be extended by linearity to elements Z where Ve Ge ,E e φ,mot L φ ∈ H(Ge (Af )) .

Unwinding the definition of the classes Zm,Iw we have the following result. (m) Proposition 7.21. For r ≥ 1, set φ = φ ⊗ ch (ηr(J × Dpr )) where ηr is the element defined in the proof 1 r ∗ of Proposition 6.22. Then the image of (c ∞ ) in lim H (E[mp ],W (1 − n)) is equal to the image of Z mp ←− π φ,mot r under the following composition: lf 2n p p rét 2n p p lim H Sh (K J × D[m] D r ), (n) −−→ lim H Sh (K J × D[m] D r ), (n) ←− mot Ge p Ve ←− ét Ge p Ve r r −n2 0 −r fs (m (US ) )· 2n p p −−−−−−−−−−→ lim H Sh (K J × D[m] D r ), (n) ←− ét Ge p Ve r −−−−−−→eord◦AJIw lim H1 E[mpr], H2n−1 (Sh (KpJ), (n))ord ←− ét G V r pr −−−→π∨ π∨ ⊗ lim H1 (E[mpr],W ∗(1 − n)) f ←− π r v −→ϕ lim H1 (E[mpr],W ∗(1 − n)) ←− π r 48 ANDREW GRAHAM AND SYED WAQAR ALI SHAH

0 where AJIw denotes the Abel–Jacobi map in Proposition A.10, US is the Hecke operator introduced in As- 0 0 sumption 7.1, and the “ordinary/finite-slope parts” are with respect to US (recall that US is intertwined with the action of T on the integral cohomology). The fact that these classes come from motivic cohomology will allow us to investigate the image of our Euler system classes in other cohomology theories, e.g. in Deligne cohomology or syntomic cohomology as defined by Nekováˇrand Niziol([NN16]). We will investigate these questions in a future paper. We end with the following result showing that, under suitable conditions, the anticyclotomic Euler system classes actually live in the Bloch–Kato Selmer group (after inverting p).

Proposition 7.22. Let π0 be a cuspidal automorphic representation satisfying Assumption 7.1 and let Sns ⊂ S denote the subset of all primes which are either inert or ramiﬁed in E/Q. Then the image of the 1 r ∗ r anticyclotomic Euler system classes cmpr in H (E[mp ],Wπ (1−n)) are unramiﬁed at all primes λ of E[mp ] which do not lie above a prime in Sns ∪ {p}, i.e. the image of cmpr under the map

1 r ∗ 1 r ∗ 1 ∗ H (E[mp ],Wπ (1 − n)) → H (E[mp ]λ,Wπ (1 − n)) → H (Iλ,Wπ (1 − n)) is trivial, where Iλ ⊂ GE[mpr ] denotes the inertia subgroup at λ. r Furthermore, for any prime P of E[mp ] dividing p, the restriction of the class cmpr at P lies in the 1 r ∗ Bloch–Kato local group Hf (E[mp ]P,Wπ (1 − n)). Proof. It is enough to prove the statement for r ≥ 1. In this case, if λ is a prime not dividing a rational prime in Se := S ∪ { primes dividing mpr} then the class cmpr is constructed from sub-Shimura varieties which have hyperspecial level outside Se. By −1 [Lan13], these Shimura varieties have smooth integral models over OE[S ], which implies that the class cmpr is unramified at λ. If λ is a prime lying above `∈ / Sns ∪ {p} and ` splits in E/Q, then the decomposition group of λ in E[mp∞]/E[mpr] is infinite. Indeed, it is enough to check that the prime of E lying below λ doesn’t split completely in the anticyclotomic Zp-extension of E, and this is standard. In this case the result follows from [Rub00, Corollary B.3.5], since the classes cmpr are universal norms in this extension. The comparison of syntomic cohomology and étalecohomology in [NN16] and the fact that the classes 1 come from motivic classes imply that, at primes dividing p, the classes cmpr lie in the local group Hg .A 1 1 −1 criterion for when Hg and Hf coincide is that p is not an eigenvalue of the crystalline Frobenius ϕ on ∗ n−1 −1 Dcris(Wπ (1 − n)). By part (3) of Theorem 3.4 the eigenvalues of ϕ are given by p αi where 2n −1 Y −s L(Πv, s + (1 − 2n)/2) = (1 − αip ). i=1

Now αi are Weil numbers of weight 2n − 1, because the representation Π satisfies the Ramanujan–Petersson conjecture (see [Mor10, Corollary 8.4.9]). This implies that the eigenvalues of ϕ are Weil numbers of weight n−1 −1 −1 −1, hence p αi cannot equal p . Corollary 7.23. Suppose that the weak base-change Π is unramified outside a finite set of primes that split in E/Q. Then 1 r ∗ cmpr ∈ Hf (E[mp ],Wπ (1 − n)) i.e. the classes live in the Bloch–Kato Selmer group after inverting p.

Proof. Let L be a prime of E[mpr] lying above a prime ` that ramifies in E/Q, and let λ denote the prime of E lying below L. By assumption, the representation Πλ is unramified, so [CH13, Theorem 3.2.3] implies −1 ∗ that the representation ρπ is unramified at λ and the eigenvalues β1, . . . , β2n of Frobλ acting on ρπ(1 − n) satisfy 2n 2n −1 Y −1 s+n−1 Y −1 s+n−1 L(Πλ, s + (1 − 2n)/2) = 1 − βi (Nm λ) = 1 − βi ` . i=1 i=1 ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS 49

n−1 −1 As in the proof of Proposition 7.22, the quantities ` βi are Weil numbers of weight 2n − 1, and since ∗ FrobL is a power of Frobλ, we see that the eigenvalues of FrobL acting on ρπ(1 − n) cannot possibly equal 1. This implies that 0 r ∗ (7.24) H (E[mp ]L,Wπ (1 − n)) = 0. ∼ ∨ ∼ c ∼ ∗ Since Π = Π = Π we have Wπ(n) = Wπ (1−n) (see the proof of Lemma 7.5 and Theorem 3.4). Combining this with (7.24) implies 1 r ∗ H (E[mp ]L,Wπ (1 − n)) = 0 so in particular, any local condition at L is vacuous. The result then follows from Proposition 7.22. Remark 7.25. The conditions in Corollary 7.23 are completely analogous to the case of Heegner points. Indeed, in this setting one considers an elliptic curve whose conductor is divisible only by primes that split in the imaginary quadratic extension.

Appendix A. Continuous etale´ cohomology In this appendix, we explicate in a general setting two constructions in Jannsen’s continuous étaleco- homology that we have made use of in this article. The first of these are the “pushforward functors” – compositions of certain Gysin and trace morphisms. In [KLZ17, §2.1], these were defined in cases when continuous étalecohomology coincides with the usual definition of (p-adic) étalecohomology. The second is an Iwasawa theoretic version of the (p-adic) étaleAbel–Jacobi map that allows us to “bypass” cohomological triviality when the p-adic Lie extension along which the limit is taken has positive dimension. In §A.3, we record some results on torsion in the Iwasawa cohomology of such p-adic Lie extensions. A.1. Pushforwards maps. For any abelian category A, we let D+(A) denote the derived category of left- bounded complexes in A, and if C is any category, we denote by CN the category of inverse systems of objects in C. If C is Grothendieck abelian, so is CN, in which case D+(CN) = D+(C)N. Moreover, if F : C → D is a left exact functor, then so is F N : CN → DN, and if C has enough injectives, so does CN, in which case the right derived functors of F and F N are related by Ri(F N) = (RiF )N [Jan88, Proposition 1.2]. If X is a scheme, we denote by Sh(Xét) the category of étalesheaves of abelian groups on the small étalesite Xét. 0 We let ΓX : Sh(Xét)N → Ab denote the functor F = (Fn) 7→ lim H (X, Fn). Then, following [Jan88, 3], ←−n § the continuous étalecohomology of F is defined to be j j Hét(X, F ) := R ΓX (F ). Lemma A.1. Let X, Y be quasi-compact, quasi-separated (qcqs) schemes. For any quasi-finite separated morphism f : X → Y , there exists a pair of functors ! f! : Sh(Xét)N → Sh(Yét)N f : Sh(Yét)N → Sh(Xét)N satisfying the following properties: ! N (1) f! is exact and f! a f i.e. for any F ∈ Sh(Xét) and G ∈ Sh(Yét)N, there are functorial isomorphisms ∼ ! Hom(f!F , G ) −→ Hom(F , f G ). ∼ (2) If f = gh with g, h both quasi-finite and separated, then there are canonical isomorphisms f! = g! ◦h! and f ! =∼ h! ◦ g!. Moreover, these isomorphisms are compatible with any third such composition. (3) f! is a sub-functor of f∗ and if f is proper, f! = f∗. (4) f ! is a sub-functor of f ∗ and if f is etale, f ! = f ∗. Moreover, these functors induce the usual derived functors + + ! + + f! : D (Sh(Xét)N) → D (Sh(Yét)N) Rf : D (Sh(Yét)N) → D (Sh(Xét)N). Proof. See Proposition 3.1.4 and 3.1.8 in [AGV73, Exp. XVIII] and Proposition 6.1.4 and Theorem 5.1.8 in [AGV73, Exp. XVII]. In a more general setting, these are also found in [Sta20, Tag 0F4U]. We will be interested in applying these functors to the following sub-class of sheaves.

N Definition A.2. Let X be a qcqs scheme. An étale Zp-sheaf, or a p-adic étalesheaf F ∈ Sh(X)ét is an inverse system {Fn}n≥1 where n • Fn is a constructible Z/p Z-module on Xét, 50 ANDREW GRAHAM AND SYED WAQAR ALI SHAH

n n+1 ∼ • the transition maps Fn+1 → Fn induce isomorphisms Fn+1 ⊗(Z/` Z) (Z/` Z) = Fn. We say that F is lisse or that F is a local system if each Fn is locally constant. A morphism of such sheaves is just a morphism of these objects in Sh(X´et)N.

Definition A.3. Let X, Y be schemes, F ∈ Sh(Xét)N and G ∈ Sh(Yét)N.A pushforward (f, φ)? :(X, F ) → (Y, G ) is a morphism f : X → Y of schemes and a morphism φ : F → f ∗G of sheaves on X. Definition A.4. Let S be a scheme. An étalesmooth S-pair of codimension c is a morphism f : X → Y of smooth qcqs S-schemes satisfying the following condition: there exists a smooth S-scheme Y¯ and a g factorization f : X −→h Y¯ −→ Y such that h is a closed immersion with fibers over each point of S of pure codimension c in Y¯ and g is étale.If f 0 : X0 → Y 0 is another such pair, then a morphism from f to f 0 is a pair of étalemaps p: X0 → X and q : Y 0 → Y that commute with f, f 0.

Proposition A.5. Let f : X → Y be an étalesmooth S-pair of codimension c and let F , G be lisse étale Zp-sheaves on X, Y respectively. Assume that p is invertible on S. Then for any pushforward (f, φ)∗ : (X, F ) → (Y, G ) and any j ∈ Z≥0, there is an induced “pushforward” on cohomology j j+2c (f, φ)∗ :Hét(X, F ) → Hét (Y, G (c)) that is functorial and Cartesian: • (Functoriality) if f 0 : X0 → Y 0 is another such pair and (p, q): f → f 0 is a morphism, then we have a commutative diagram

0 j 0 ∗ (f ,ψ)∗ j+2c 0 ∗ H´et(X , p F ) H´et (Y , q G (c))

Trp Trq

j (f,φ)∗ j+2c Hét(X, F ) Hét (Y, G (c)) where ψ = p∗φ. 0 0 0 0 0 0 0 • (Cartesian) if q : Y → Y is any finite étalemorphism, X = X ×Y Y , f : X → Y , p : X → X are the natural morphisms, then we have a commutative diagram

0 j 0 ∗ (f ,ψ)∗ j+2c 0 ∗ H´et(X , p F ) H´et (Y , q G (c))

p∗ q∗

j (f,φ)∗ j+2c H´et(X, F ) H´et (Y, G (c)) where ψ = p∗φ.

When f is a closed immersion, (f, φ)? is deﬁned in [Jan88, Theorem 3.17]. The proof below is along similar lines except that we account for the independence of the factorization of f.

g Proof. Let f : X −→h Y¯ −→ Y be a factorization. As g is ´etale, g! = g∗ by Lemma A.1 (4). By [Jan88, Eqn. ! ! ∗ ! ∗ 3.20], h Zp(c) = Zp[−2c]. Thus, by equation (3.19) of op.cit, h G (c) = h G ⊗h Zp(c) = h G [−2c]. Therefore, Rf !G (c) = Rg! ◦ Rh! G (c) = g∗h∗G [−2c] = f ∗G [−2c]

! ! ! Deriving Trf :ΓX f → ΓY , we obtain a natural transformation R(Trf ): R(ΓX f ) = RΓX ◦ Rf → RΓY ∗ j ∗ ∼ between the two derived functors. Evaluating R(Trf ) at G (c), we obtain an induced map Hét(X, f G ) = j+2c ! Trf j+2c R (ΓX f )(G (c)) −−→ Hét (Y, G (c)) via passage to cohomology. The pushforward map is now defined to be j φ j ∗ Trf j+2c (f, φ)∗ = Hét(X, F ) −→ Hét(X, f G ) −−→ Hét (Y, G (c)). Lemma A.1 (ii) then gives the independence of this map on the choice of Y¯ . The functoriality of these pushforwards follows from that of the trace, i.e. RTrf ◦ RTrp = RTrq ◦ RTrf 0 . The Cartesian property 0 follows by applying proper base change ([Sta20, Tag 095S]) to q : Y → Y . ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS 51

A.2. Abel–Jacobi map in Iwasawa cohomology. Let E be a field of characteristic 0, and let FtSchét be the category of schemes of finite-type over E with morphisms given by étalemaps. Let X be such a ¯ scheme and F a lisse étale Zp-sheaf on X. We denote by X the base change X ×Spec E Spec E. Set i i i i M (X) := Hét(X, F ), M (X) := Hét(X, F ), i where Hét is continuous étalecohomology as defined in the previous subsection. Theorem A.6 (Jannsen, Deligne). There is a first quadrant cohomological spectral sequence

i,j i j i+j E2 :H (E, M (X)) ⇒ M (X) which we refer to as the Hochschild–Serre spectral sequence. We say an element in M i(X) is cohomologically trivial if it is in the kernel of the natural base-change i 0 i i map M (X) → H (E, M (X)). We write M (X)0 for the subspace of all cohomologically trivial elements in M i(X).

Deﬁnition A.7 (Abel–Jacobi map). Let AJ´et denote the edge map

i 1 i−1 AJ´et : M (X)0 −→ H E, M (X) arising from the Hochschild–Serre spectral sequence.

For any ﬁnite extension F/E, set XF = X ×Spec E Spec F and consider it as an E-scheme. Then XF = F τ XE where τ runs over the elements of HomE(F, E), therefore i i i (A.8) M (XF ) = M (XF ) = Zp[∆F ] ⊗Zp M (X)

where ∆F = HomE(F, E) is considered as a Gal(E/E)-set (with the left action given by post-composition). 0 If F /F is a ﬁnite extension, then the natural map pr: XF 0 → XF induces two maps on cohomology, namely i i ? i i pr? : M (XF 0 ) → M (XF ), pr : M (XF ) → M (XF 0 ) i i ? i i pr? : M (XF 0 ) → M (XF ), pr : M (XF ) → M (XF 0 ). In the geometric case, these maps are the ones induced by ? pr? : ∆F 0 ∆F , pr : ∆F ,→ ∆F 0 X τ 7→ τ|F S 7→ η

η|F =S Fix a pro-system (Γ ) of ﬁnite groups and let Γ = Γ = lim Γ . Let Γ = Γ be the category deﬁned as r r≥1 ∞ ←−r r E follows:

• the objects of Γ are Γ-towers E ⊂ F0 ⊂ F1 ⊂ F2 ⊂ ... contained in E, i.e. Fr/E are finite Galois for all r ≥ 0, and Γr = Gal(Fr/F0) for r ≥ 1. 0 • the morphisms of Γ are reverse inclusions of towers, i.e. a morphism (Fr)r → (Fr)r is a system of 0 inclusions Fr ,→ Fr for r ≥ 0 compatible with the inclusion morphisms of the two systems. ∼ We let F∞ denote the compositum of Fr. Then F∞/E is Galois, and Gal(F∞/F0) = Γ. If we let Γ0 r r be the trivial group and let Γ := ker (Γ → Γr) for r ≥ 0, then (Γ )r≥0 forms decreasing (exhaustive and separated) filtration of open normal subgroups of Γ and we may equivalently describe objects of Γ as r towers (Fr)r≥0 Galois over E such that Gal(F∞/Fr) = Γ for r ≥ 0. We have a distinguished embedding Γ = Gal(F∞/F0) ,→ Gal(F∞/E), and thus a third description of objects of Γ is as Galois extensions F∞/E contained in E that come with an embedding of the filtered group Γ as an open normal subgroup of their 0 0 Galois group, and the morphisms F∞ → F∞ are now inclusions F∞ ,→ F∞ such that the induced morphism 0 Gal(F∞/E) → Gal(F∞/E) induces the identity map on Γ. We will follow this perspective in the sequel, and abusively say that F∞ ∈ Γ is an object. × r × Example A.1. Let p be an odd prime and Γr = Zp /(1 + p Zp) for r ≥ 1. Then Γ = Zp . Let E be an imaginary quadratic number field in which p splits, and let m be a positive integer satisfying (m, p) = 1. r r 0 For r ≥ 1, we let Fr = E[mp ] denote the ring class field of conductor mp . Then (Fr)r ∈ Γ. If m|m and 0 0 ∞ 0 F∞ = E[m p ], we obtain a morphism F∞ → F∞. 52 ANDREW GRAHAM AND SYED WAQAR ALI SHAH

Definition A.9. For X ∈ FtSchét, F∞ ∈ Γ, denote X ×Spec E Spec Fr by Xr. We set j • Hi (F , M (X)) := lim Hi(F ,M j(X)), Iw ∞ ←−r r • M i (F ,X) := lim M i(X ), Iw ∞ ←−r r i • M (F ,X) := lim M i(X ). Iw ∞ ←−r r j Here, the first limit, known as the Iwasawa cohomology of M (X), is with respect to corestriction, and other i two are with respect to pr?. By (A.8), the last limit is also equal to Zp Gal(F∞/E) ⊗Zp M (X). These are then covariant (resp. contravariant) bifunctors (i.e. functors in each variable)J on ΓK× FtSchét with respect to pushforwards (resp. pullbacks).

∞ Proposition A.10. Suppose that p divides the proﬁnite order of Γ. Then for any F∞ ∈ Γ, X ∈ FtSch´et, there is an induced “Abel-Jacobi map”

i 1 i−1 AJIw : MIw(F∞,X) → HIw(F∞, M (X)).

that is bifunctorial with respect to both pushforwards and pullbacks in each variable, i.e. AJIw is a natural transformation of the (four) bi-functors on FtSch´et × Γ.

i j i j i j Proof. By Shapiro’s lemma, we know that H (E, M (Xr)) = H E, Zp[∆Fr ] ⊗ M (X) = H (Fr, M (X)) i for each r, with pushforwards corresponding to corestriction. The exact sequences 0 → M (Xr)0 → i 0 i M (Xr) → H (Fr, M (X)) therefore form an inverse system, and so we get

i i 0 i 0 → MIw(F∞,X)0 → MIw(F∞,X) → HIw(F∞, M (X)) i i i where MIw(F∞,X)0 is defined as the inverse limit of M (Xr)0. As M (X) is of finite type over Zp ([Del77, 0 i Corollaire 1.10, Chapitre 7]), we have HIw(F∞, M (X)) = 0 by [Rub00, Lemma 3.2, Appendix B], so we see i i that MIw(F∞,X)0 = MIw(F∞,X). As we also have an inverse system of Abel–Jacobi maps i 1 i−1 AJét : M (Xr)0 −→ H Fr, M (X)

for each level, we obtain the result by passing to the limit. A.3. Torsion in Iwasawa cohomology. Let p be an odd prime and G a profinite group. We say that G satisfies condition (F) if: • G has finite p-cohomological dimension. i • For any finite discrete Fp[G]-module T , and any integer i ≥ 0, H (G, T ) is finite. If G satisfies condition (F), then the continuous Galois cohomology groups Hi(G, −) are finitely-generated (see [Nek06, Proposition 4.2.3]). We record the following well-known lemma which will be used in the latter half of this section. Lemma A.11. Let F be a number field and Σ a finite set of places of F containing all primes above p and all archimedean places. Let FΣ denote the maximal extension of F which is unramified outside Σ. Then Gal(FΣ/F ) satisfies condition (F).

Proof. See [NSW08, Proposition 8.3.18, Theorem 8.3.20].

Let E be a number field and Φ a finite extension of Qp with ring of integers O and uniformiser π. Let T be a free O-module of finite rank with a continuous action of Gal(E/E) such that T is ramified at only a finite set of places of E. We set W := T ⊗O Φ and A := W/T the divisible Galois representation. Let Γ be fin the category as above and let Γ be the sub-category of extensions F∞ such that only finitely many primes Γ × fin of F0 = (F∞) are ramified in F∞. When Γ = Zp or Zp , we have Γ = Γ [NSW08, Proposition 11.1.1]. fin Let F∞ ∈ Γ and set F = F0. For any set of places Σ of F , we let FΣ denote the maximal extension of F unramified away from Σ. Let S(F ) denote the finite set of places of F containing all primes above p, all archimedean places, all places where T is ramified and all places ramified in F∞. Then for any finite set of places Σ ⊃ S(F ), FΣ contains F∞ and T is a Gal(FΣ/Fr)-module for all r ≥ 0, r = ∞. Definition A.12. Let Σ ⊃ S(F ) be as above. We define the following cohomology groups for i = 0, 1, 2. ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS 53

• The unramified away from Σ-cohomology of T is defined to be i i HΣ(F,T ) := H (Gal(FΣ/F ),T ). • The unramified away from Σ-Iwasawa cohomology (or just Σ-Iwasawa cohomology) of T is defined to be Hi (F ,T ) := lim Hi(Gal(F /F ),T ) Σ,Iw ∞ ←− Σ r r with limit taken with respect to corestriction. If Σ ⊂ Σ are two such sets, then there are induced inflation morphisms Inf :Hi (F,T ) → Hi (F,T ). 1 2 Σ1,Σ2 Σ1 Σ2 As inflation commutes with corestriction, there are also induced maps Inf :Hi (F ,T ) → Hi (F ,T ) Σ1,Σ2 Σ1,Iw ∞ Σ2,Iw ∞ Definition A.13. The locally finite Iwasawa cohomology of T is defined to be the inductive limit Hi (F ,T ) = lim Hi (F ,T ). lf,Iw ∞ −→ Σ,Iw ∞ Σ over all Σ with respect to the induced inflations maps. When i = 1, it is equal to the union of the images of 1 1 all InfΣ :HΣ,Iw(F∞,T ) ,→ HIw(F∞,T ). The above groups are relevant in light of the following result.

Lemma A.14. Let X be smooth E-scheme and F ∈ Sh(Xét)N a lisse étale Zp-sheaf. Suppose that for some finite set of places Σ of F , there exist

• a projective system of smooth OF,Σ-schemes (Xr)r with Galois transition morphisms and Galois group isomorphic to (Γr)r, • maps p : X ×Spec F Spec Fr → Xr of Spec OE-schemes such that Xr → Spec F is the pullback of Xr → OF,Σ under p. • lisse ´etale Zp-sheaves Fr on Xr compatible under pullbacks on the projective system such that F on Xr is the pullback of Fr along p. Suppose moreover that p∞ divides the proﬁnite order of Γ. Then the image of

p? i−1 lim Hi (X , F ) −→ M i (F ,X) −−−→AJIw H1 (F , M (X)) ←− ´et r r Iw ∞ Iw ∞ r

1 i−1 is contained in HΣ,Iw(F∞, M (X)). Proof. This follows from [Jan88, Theorem 3.3], and functoriality of the Hochschild-Serre spectral sequence under morphisms of schemes. For the rest of this section, our general reference is [NSW08, Chapter V] and [Jan89].

Definition A.15. The Iwasawa algebra of Γ, denoted Λ = Λ(Γ) = ΛO(Γ) = O Γ is defined to be the completed group ring lim O[Γ ]. We equip Λ(Γ) with the projective limit topologyJ inducedK by the π-adic ←−r r topology of O[Γr] which makes Λ(Γ) a profinite ring. A left topological module over Λ(Γ) is a Hausdorff abelian topological group M with a continuous action Λ(Γ) × M → M. A right module is defined similarly. Via the involution Γ → Γ, γ 7→ γ−1, one can view a right module as a left module. If M is also locally ∨ compact, we let M = Homcts(M, Qp/Zp) denote its Pontrjagin dual, which carries a natural Λ(Γ)-module structure. If Γ is a compact p-adic Lie group, then Λ(Γ) is (left and right) Noetherian [Ven02, §1.2]. Moreover, if Γ is pro-p, then Λ(Γ) is a local ring [NSW08, Proposition 5.2.16 (iii)] and if Γ is torsion free, then Λ(Γ) is an integral domain, i.e. it does not have zero divisors [Ven02, §1.2]. Definition A.16. Let M be a topological Λ(Γ)-module. The (generalized) Iwasawa adjoints of M are defined to be i i E (M) := ExtΛ(M, Λ) which are (right) Λ-modules via functoriality and the right-module structure of Λ. We denote the Λ-dual 0 + ++ E (M) = HomΛ(M, Λ) by M . Then, there is a canonial map ϕM : M → M and the torsion submodule of M is defined to be the kernel of ϕM . 54 ANDREW GRAHAM AND SYED WAQAR ALI SHAH

If M is finitely generated and Γ is a p-adic Lie group, then the torsion can equivalently be described as the set of torsion elements (in the ring theoretic sense) in M over the (necessarily integral) Iwasawa algebra Λ(Γ0) of an open pro-p, p-torsion free subgroup Γ0 ⊂ Γ. [Ven02, §2]. In particular, Λ and M + are Λ-torsion free, as is any sub-module of M +. Proposition A.17 (Jannsen). Suppose Γ is a compact p-adic Lie group. For any Σ ⊃ S(F ), there is a first quadrant cohomological spectral sequence of finitely generated Λ-modules p,q p q ∨ p+q E2 = E (H (Gal(FΣ/F∞),A) ) ⇒ HΣ,Iw(F∞,T ) Proof. See [Jan14, Theorem 1] (or [TV17, Theorem 8.3] and the reference within). The finiteness claim uses Lemma A.11. 1,0 Corollary A.18. Suppose E2 = 0. Then we have a canonical injection 1 1 ∨+ HΣ,Iw(F∞,T ) ,→ H (Gal(FΣ/F∞), V/T ) .

× Gal(E/F∞) 1 In particular, if Γ = Zp and T = 0, then HΣ,Iw(F∞,T ) is Λ-torsion free. Proof. The spectral sequence gives rise to the following 5-term exact sequence 1,0 1 0,1 2,0 2 0 → E2 → HΣ,Iw(F∞,T ) → E2 → E2 → HΣ,Iw(F∞,T )

× Gal(E/F∞) Gal(FΣ/F∞) 0 ∨ from which the first claim follows. If Γ = Zp , and T = T = 0, then H (Gal(FΣ/F∞),A)) 1,0 is finite. By [Jan89, Proposition 2.6(ii)], E2 = 0 and hence the final claim.

× r r Corollary A.19. Let Γ be the filtered group Zp (with filtration Γ = 1 + p Zp for r ≥ 1) and suppose that W is absolutely irreducible as a Gal(E/E)-representation with dimΦ W ≥ 2. Then for any F∞ ∈ Γ with F∞ an abelian extension of E, the canonical map Ψ:H1 (F ,T ) → lim H1(F ,W ) lf,Iw ∞ ←− r r is injective. Proof. First notice that H0(Eab,T ) = 0. Suppose for the sake of contradiction that v ∈ T ⊂ W is a non-zero vector fixed by Gal(E/Eab). As W is irreducible, the Gal(E/E)-span of any non-zero vector (in particular, v) must be all of W . But this implies that the action of Gal(E/E) on W factors through Gal(Eab/E), and hence W ⊗Φ Qp contains a one-dimensional subrepresentation – a contradiction. By Corollary A.18, it therefore suffices to show that the kernel of Ψ is contained in the union over all Σ 1 1 of the Λ-torsion submodules (in the sense of Definition A.16) of Hlf,Σ(F∞,T ) ⊂ Hlf,Iw(F∞,T ). Equivalently, 1 ∼ it suffices to show that ker Ψ is contained in the union of the Λ(Γ ) = Λ(Zp)-torsion sub-modules (in the 1 1 1 ring theoretic sense) of Hlf,Σ(F∞,T ). Clearly ker Ψ contains the O-torsion of Hlf,Iw(F∞,T ), which is Λ(Γ )- torsion, so it suffices to verify that for all Σ ⊃ S(F ), the kernel of Ψ :H1 (F ,T ) ⊗ → lim H1(F ,W ) Σ Σ,Iw ∞ Zp Qp ←− r r 1 1 is contained in the R := Λ(Γ )[1/p]-torsion. To this end, set M := H (F ,T )⊗ . Then M = lim M r Σ,Iw ∞ Zp Qp ←−r Γ where MU denotes the Φ[Γ/U]-module of U-coinvariants for a compact open subgroup U ⊂ Γ, and moreover 1 ΨΣ is given by the inverse limit of the induced maps Ψr : MΓr → H (Fr,W ). The remainder of this proof follows [LLZ14, Proposition A.2.1, Proposition A.2.6]. By Shapiro’s lemma (see [Nek06, §8.2]), we have 1 1 1 M = HΣ(F, Λ(Γ) ⊗O T )[1/p]H (Fr,W ) = H (F, O[Γr] ⊗O T )[1/p]

with appropriate actions of Gal(FΣ/F ) and Gal(F /F ) on Λ(Γ) ⊗O T and O[Γr] ⊗O T respectively. Let r (γr)r≥1 denote a compatible choice of topological generators for (Γ )r≥1. One has a short exact sequence of Λ(Γ)[Gal(FΣ/F )]-modules

[γr ]−1 0 → Λ(Γ) ⊗O T −−−−→ Λ(Γ) ⊗O T → O[Γr] ⊗O T → 0. ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS 55

After passing to the long exact sequence on cohomology and inverting p we see that, for r ≥ 1, the ho- 1 momorphism Ψr is the composition of an injective homomorphism and inﬂation (which is injective on H ). Therefore the map Ψr is injective for r ≥ 1. This implies that the kernel of ΨΣ is contained in \ ([γr] − 1)M ⊂ Mtors r≥1

where Mtors is the R-torsion submodule (in the ring-theoretic sense), and the last inclusion follows from the fact that M is ﬁnitely generated over R =∼ O X [1/p] (by Lemma A.11 and [Nek06, Proposition 4.2.3]), J K which is a PID (so we can write M as the direct sum of a free module and a torsion module).

Appendix B. Shimura–Deligne varieties In this article, we have worked with varieties that are strictly speaking not Shimura varieties in the sense of [Del79]. Speciﬁcally, axiom 2.1.1.3 of op.cit., which we will refer to as (SD3), fails for the morphism h: S → H considered in §2.2 (since Had has a compact Q-simple factor SU(n)). More generally, for a Shimura datum arising from PEL moduli problems, (SD3) usually does not hold, e.g. see [Mor10, §1.1], or [Lan12, Remark 2.5.8]. The primary reason for imposing it is that it allows one to apply strong approximation, which is used in describing the reciprocity law on the geometric connected components [Del71, §3.4]. Another application is in reducing the problem of the existence of canonical models to the case of connected Shimura varieties [Del79, §2.7]. Assuming (SD3) however excludes the case of so-called Shimura sets, for example the Gross curve in [Zha14, 3.1], which are instrumental in applications to Euler systems. The purpose of this appendix is to record results that continue to hold for datum that do not necessarily satisfy (SD3). We make no claim of originality here as most proofs carry over verbatim. The proofs that we have chosen to include are primarily for purposes of exposition of the originals. We shall freely use results from [Del71] when they do not invoke the hypothesis in §2.1 where (SD3) assumed.

Notation. We fix an algebraic closure C of R, and take Q to be the algebraic closure of Q in C. We denote × × the Deligne torus by S = Res / Gm and let w : Gm, → S defined via the inverse of the inclusion R ,→ C . ∼C R R We fix the identification SC = Gm × Gm such that the inclusion S(R) → S(C) is given by z 7→ (z, z), and we take µ: Gm,C → SC to be the cocharacter z 7→ (z, 1). For an algebraic group G, we denote by ZG its centre, der ad + G its derived group, and G its quotient by ZG. A superscript “+” (e.g. G(R) ) denotes the connected component of the identity in the analytic topology. B.1. Preliminaries. Definition B.1. A Shimura–Deligne datum is a pair (G,X) consisting of a connected reductive algebraic

group G over Q and a G(R)-conjugacy class X of homomorphisms h: S → GR satisfying

(SD1) For all h ∈ X, the Hodge bigrading of the complex vector space Lie(G)C under the adjoint action of SC is of type {(−1, 1), (0, 0), (1, −1)}. In particular, the cocharacter h ◦ w : Gm → GR is central and independent of h. √ (SD2) For any h ∈ X, ad h( −1) : GR → GR (which is an involution by (SD1)) is a Cartan involution of Gder, i.e. the real Lie group R √ √ der −1 g ∈ G (C) | h( −1)gh( −1) = g is compact.

A morphism (G1,X1) → (G2,X2) of Shimura–Deligne datum is a homomorphism u: G1 → G2 such that u(X1) ⊂ X2. We say that such a morphism is injective if u is. Remark B.2. These are the axioms (1.5.1), (1.5.2), (1.5.3) of [Del71], or (2.1.1.1), (2.1.1.2) of [Del79]. We have borrowed the above terminology from [GN09]. Remark B.3. It suffices to verify the axiom for one h ∈ X, and we may write (G, h) for the Shimura–Deligne der ad datum.√ Notice also that since G → G is a (central) isogeny, (SD2) is equivalent to requiring that ad h( −1) induces a Cartan involution of Gad. R Fix h0 ∈ X and let K∞ be the centraliser of h0 in G(R), so that X√= G(R)/K∞. Then K∞ contains the (0,0) centre of G(R) and Lie(K∞)C coincides with Lie(G) . Since h0( −1) acts as −1 on Lie(G)/Lie(K∞), 56 ANDREW GRAHAM AND SYED WAQAR ALI SHAH √ we see that h0 is central if and only if h0( −1) is. Since Cartan involutions are unique up to conjugacy, this is equivalent to Gder(R) being compact. Such a pair will be referred to as a trivial Shimura–Deligne datum. 0 0 Let X be the connected component of X containing h0. By [Del79, Corollary 1.1.17], X is either a singleton (which happens if and only if (G, h0) is trivial) or a Hermitian symmetric domain. More precisely, ad 0 if G = G1 × ... Gk is the decomposition into R-simple factors, then X = X1 × · · · × Xk where each Xi R + is a quotient of Gi(R) by a maximal compact subgroup. 0 der + 0 Lemma B.4. The Lie group K∞ = K∞ ∩ G (R) is connected, and K∞ = Z(R) · K∞. 0 ∗ Proof. As K∞ is a maximal compact subgroup√ in a connected Lie group, it is connected. Let G denote the compact real form of Gad defined by h( −1). Let h0 : → Gad be the induced map, and let C be R 0 S the centraliser of h0 in Gad. Then C is a Cartan subgroup of Gad, and is therefore (Zariski) connected. 0 R Moreover, C(R) is a closed subgroup of G∗(R), so C is R-anisotropic and hence C(R) is connected by [BT65, ad + der + ad + 14.3]. As K∞ lands in C(R) ⊂ G (R) and G (R) surjects onto G (R) , the second claim follows. Definition B.5. Let (G,X) be a Shimura–Deligne datum. We say that a compact open subgroup K ⊂ −1 G(Af ) is sufficiently small if, for every g ∈ G(Af ), the image of the discrete subgroup G(Q) ∩ gKg in Gad(Q) has no non-trivial stabilisers for its action on any connected component of X. For instance, this happens when K is neat, by [Pin88, Proposition 3.3(b)].

Deﬁnition B.6. Given a Shimura–Deligne datum (G,X), and a compact open subgroup K ⊂ G(Af ), we deﬁne the Shimura–Deligne variety to be the double quotient

G(Q)\[X × (G(Af )/K)] which we denote by ShG(X,K)(C) or just ShG(K)(C).

Arguing as in [Lan12, §2.5] (or [GN09, Lemma 4.6.1]), ShG(K)(C) is a disjoint union of quotients of Hermitian symmetric domains/ﬁnite discrete sets by arithmetic subgroups, and therefore are the complex

points of a quasi-projective C-variety ShG(K)C by the theorem of Baily–Borel. By [Pin88, §0.1], these varieties are also smooth for K sufficiently small. As K gets smaller, these varieties form a projective system whose limit is a quasi-compact separated scheme carrying a continuous action of G(Af ) [Del71, §1.8]. We shall denote this scheme by ShG(X)C or just ShG,C. Let u:(G1,X1) → (G2,X2) be a morphism of Shimura–Deligne datum. For compact open subgroups Ki ⊂ Gi(Af ) such that u(K1) ⊂ K2, we have an induced C-morphism of corresponding C-varieties by Borel’s Theorem (see [Del71, §1.14]). If one starts with a closed immersion of reductive groups, this morphism is finite unramified for sufficienlty small compatible compact opens, and factors as a composition of a closed immersion followed by a finite étalemorphism [Del71, Proposition 1.15].

B.2. Canonical models. Let G be a reductive Q-group. For any Q-algebra R, the group G(R) acts on the left on the space HomR(Gm,R, GR) of algebraic group homomorphisms over R, by conjugation on the target. Let Y = YG be the (fppf sheaﬁﬁcation of) the functor

Q-algebras → Sets R 7→ G(R)\ HomR(Gm,R, GR). If F/Q is a finite Galois extension over which G splits, then restricted to F -algebras, Y is a constant functor, F hence representable as Spec F . By Galois descent, one sees that it is representable by an étale Q-scheme. Suppose moreover that (G,X) is Shimura–Deligne datum. The cocharacters hC ◦ µ: Gm,C → GC for h ∈ X lie in the same G(C)-conjugacy class (as h lies in a single G(R)-conjugacy class). Therefore, one obtains a geometric point µX ∈ Y ( ) = Y ( ) ⊂ Y . C Q Q Definition B.7. The reflex field E(G,X) of a Shimura–Deligne datum (G,X) is the field of moduli of the point µX ∈ YG(Q).

The reflex field is also the field of definition of the conjugacy class µX , which is how it is usually defined and computed. We give an example below. Example B.1. Let G = GU(p, q) be the unitary group defined in §2.1 using the imaginary quadratic field E and let h: S → GR be the map z 7→ (z, . . . , z, z,¯ . . . , z¯) where there are p copies of z, q copies ofz ¯. Then (G, {h}) is a Shimura–Deligne datum. The cocharacter µh associated to h over C is given by µh : Gm → Gm × GLn,C z 7→ (z, diag(z, , . . . , z, 1,... 1)) which is defined over E. If σ ∈ Gal(E/Q) is the complex ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS 57

conjugation, then σ(µh): z 7→ (z, diag(1,..., 1, z, . . . , z)). These are in two different conjugacy classes if and only if p 6= q, i.e. the reflex field is E if p 6= q and Q otherwise.

For the notion of models of ShG,C over subfields E ⊂ C, we refer the reader to [Del71, §3]. If a model exists, we denote it by ShG,E. As in [Pin88, §11.4], one can define a model of ShG,C over its reflex field when G is a torus (and X = {h} is a singleton). Moreover, if (G1,X1) → (G2,X2) is a morphism, then

E(G1,X1) ⊃ E(G2,X2) because one obtains a morphism YG1 → YG2 with µX1 mapping to µX2 . This motivates the following deﬁnition.

Definition B.8. A canonical model of ShG,C is a model ShG,E defined over the reflex field E = E(G,X) such that for every Shimura–Deligne datum (T,X0) with reflex field E0, where T is a torus, and any injective morphism (T,X0) → (G,X), the induced map

ShT,C → ShG,C 0 is the pullback of a morphism ShT,E0 → ShG,E ×Spec E Spec E . Lemma B.9. Let (G,X) be a Shimura–Deligne datum and E = E(G,X) its reflex field. Let S be the collection of all subfields F ⊂ C that arise as follows: there exists an injective morphism of Shimura–Deligne 0 0 T datum (T,X ) → (G,X) such that F = E(T,X ). Then S is non-empty and F ∈S F = E. This result is [Del71, Theorem 5.1] and is the key input of the next theorem. While (SD3) is a running hypothesis in §5 of op.cit., the proof of this result does not need it. As this might not be immediately obvious, we provide an exposition of the key steps. ∼ Proof. Let YG be the Q-scheme as above and let Y0 = Spec E be the finite étale Q-subscheme whose Q-points consist of the Galois translates of µX ∈ YG(Q). Let V be the Q-scheme of regular elements in Lie(G), i.e.

points in V (R) are elements vR ∈ Lie(G)R whose centraliser is a maximal torus TvR in GR. Let W be (fppf sheafification of) the functor that associates to each Q-algebra R the set of all triplets (TR, vR, λR) where • TR is a (fibrewise) maximal torus in GR. • vR ∈ Lie(T)R is a regular element of Lie(G)R, i.e. the centraliser of v in GR is TR. • λR : Gm,R → TR is a cocharacter such that the class of the induced morphism Gm,R → GR is in Y0,R. Then W → V has constant finite fibres. By Galois descent, W is representable by a finite-type -scheme Q Q Q admitting a finite étalesurjective map f : W → V and a map p: W → Y0. This makes W a E-scheme. f W V (B.10) p

Y0 Spec Q

For all v ∈ V (R), w ∈ W (C) with f(w) = v and λw : Gm,C → Tv,C the cocharacter part of w, there is a unique hw : S → TR such that hw,C ◦ µ = λw; on R-points, it is given by × hw(z) = λw(z) · λw(z), z ∈ S(R) = C . We note that if v ∈ V (Q) is such that −1 • the Q-scheme Zv := f (v) is the spectrum of a ﬁeld Ev, • for some w ∈ Zv(C), hw ∈ X. then, Ev ∈ S. Indeed, the second point means that there is an injective morphism (Tv, {hw}) ,→ (G,X)

of Shimura–Deligne datum, and the first guarantees that the cocharacter λw = µ ◦ hw,C associated to the datum (Tv, {h}) has field of definition Ev. The goal therefore is to show that there are many such v.

Step 1. W is geometrically irreducible as an E-scheme.

0 Let W be the geometric fibre of p over µX and C the scheme over of cocharacters of G whose conjugacy Q Q 0 class is µX . Let q : W → C be the natural map. Then C, being a quotient of G , is irreducible and the fibers Q 00 0 of q are isomorphic. Let W ⊂ W be the fibre of q over a given λ: m → G ∈ C( ) and D the scheme G Q Q of all maximal tori of G that contain the image of λ. Let r : W 00 → D be the natural map. Any maximal Q 58 ANDREW GRAHAM AND SYED WAQAR ALI SHAH torus coming from D centralizes λ( m) and therefore must be contained in the centralizer ZG (λ( m)). G Q G Now, ZG (λ( m)) is connected by [Mil17, Theorem 17.38] and D is a quotient of ZG (λ( m)) by [Mil17, Q G Q G Theorem 17.10]. Therefore, D irreducible. As W 00 → D is open in the vector bundle over D associated to the Lie algebra of the universal torus over D, it has geometrically irreducible fibers over D, and is therefore irreducible too. Collecting these statements together, we see that the fibers of q are irreducible, whence W 0 is irreducible. As the geometric fibres of p are isomorphic and Y0 is irreducible, the claim follows.

Step 2. In the analytic topology of V (R), there is a non-empty open subset U such that for all v ∈ U, Tv contains the image of some h ∈ X.

We let U ⊂ V (R) denote the subset of all v ∈ V (R) such that (Tv/ZR)(R) is compact. Then U is open since the association v 7→ Tv/ZR is continuous and the set of anisotropic (equivalently compact) tori is an open subspace6. of the moduli space of all maximal tori of Gad. Moreover, U is non-empty as follows: for any R h ∈ X, the map had : → Gad factors via had :U → Gad. Hence any maximal torus of Gad( ) containing S R 1 R R ad ad the image of h is contained in the centralizer StabG(h ), and is therefore compact. Any regular element in the Lie algebra of the inverse image in G of T is then the desired element. It now remains to show that for all v ∈ U, Tv contains the image of some h ∈ X. To this end, ﬁx h0 ∈ X and let K∞ be the stabiliser of 0 der + der + h0 in G(R). Then K∞ = K∞ ∩ G (R) is a maximal compact Lie subgroup of G (R) , and K∞/Z(R) the maximal compact subgroup of Gad(R)+ by Lemma B.4. As any two maximal compact subgroups in ad + der + −1 G (R) are conjugate and Tv/ZR(R) is compact, there exists g ∈ G (R) such that gTv(R)g ⊆ K∞. −1 Since gTv(R)g is a maximal torus in K∞, it contains the centre of K∞, and in particular the image of h0. −1 Therefore, Tv contains the image of g h0g.

Step 3. For any ﬁnite extension E0 of E, there exists v ∈ V (Q) ∩ U such that f −1(v) is the spectrum of a ﬁeld linearly disjoint from E0.

This is (a slighly general form) of Hilbert’s irreducibilty theorem applied to the diagram in (B.10), and is proved in [Del71, Lemma 5.1.3]. Remark B.11. In fact, the proof shows that there are two extensions in S whose intersection is E.

Theorem B.12 (Functoriality). Let (G1,X1) → (G2,X2) be a morphism of Shimura–Deligne datum and, for i = 1, 2, let Ei be the reﬂex ﬁeld of (Gi,Xi). Suppose ShGi,Ei is a canonical model over Ei of ShGi,C. Then the morphism

ShG1,C → ShG2,C is the pullback of a morphism ShG1,E1 → ShG2,E2 ×Spec E2 Spec E1. Proof. This is [Del71, Corollary 5.4] whose proof relies on Proposition 5.2, 5.3, and Theorem 5.1 in op.cit. Proposition 5.2 of op.cit. holds in our case as we still have real approximation, and Lemma 5.3 is a general fact about descent data of schemes (see the explanation in [Pin88, Lemma 11.8] and the reference therein). The key input of Theorem 5.1 is that the intersection of the reflex fields of tori admitting a map to (G1,X1) is E1 (without assuming (SD3)), and this was shown in Lemma B.9 above. Corollary B.13 (Uniqueness). Canonical models are unique up to a unique isomorphism. 0 Proof. If ShG,E and ShG,E are two canonical models over the reflex field E of a datum (G,X), then the 0 identity map on ShG,C is the pullback of a morphism ShG,E → ShG,E. Corollary B.14 (Existence criterion). Let (G,X) → (G0,X0) be an injective morphism of Shimura–Deligne datum. If (G0,X0) admits a canonical model, then so does (G,X).

Proof. This is [Del71, Corollary 5.7] and the proof carries over verbatim. Deﬁnition B.15. Let (G,X) be a Shimura–Deligne datum. In addition to the axioms introduced in Deﬁnition B.1, some additional axioms are often imposed which we list here:

6 ad The space of conjugacy classes of maximal tori in G (R) form ﬁnitely many connected components, each of which is open, ad + and any two in the same component conjugate by G (R) . See the footnote in [Mil03, p. 117]. ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS 59

(SD3) Gad(R) has no Q-simple factors that are R-anisotropic. (SD4) The weight morphism wX := w ◦ h: Gm,R → GR, a priori defined over R, is defined over Q. (SD5) ZG(Q) is discrete in ZG(Af ). Remark B.16. When (G,X) satisfies (SD3), we arrive at the common definition of Shimura datum in the literature. We note that (SD5) is equivalent to the statement that the maximal anisotropic Q-subtorus of ZG remains anisotropic (equivalently, compact) over R, i.e. ZG is an almost direct product of a split and a compact type torus over Q [Mil03, Theorem 5.36]. (SD5) implies (SD4) (see [Pin92, 5.4]) and is therefore a practical means to check the latter. Axiom (SD5) also implies that the (right) action of any sufficiently small compact open K ⊂ G(Af ) on the quotient G(Q)\[X × G(Af )] is free. B.3. PEL-type Shimura–Deligne varieties. For the notion of (semisimple) PEL-datum, we refer the reader to [Tor19, Definition 7.2]. Here, we simply recall that a PEL datum is a tuple (B, ∗,V, h·, ·i, h) where • B is a finite-dimensional semisimple Q-algebra •∗ is a positive anti-involution of B • V is a finite-dimensional B-module • h·, ·i: V × V → Q a non-degenerate alternating (i.e. skew Hermitian) pairing • h: → EndB V is an -algebra homomorphism C R R R satisfying the conditions in loc.cit.. Given such a PEL datum as above, we can associate to it an algebraic group G (over Q) whose R-points equal × G(R) = AutBR (VR) | ∃µ(g) ∈ R such that hgu, gvi = µ(g)hu, vi .

The group G is reductive and µ: G → Gm is a homomorphism, which we call the similitude factor. We let G1 denote the kernel of µ. Then G1,R splits into a product of unitary (A), symplectic (C) and orthogonal factors (D) [Tor19, Lemma 7.5 (i)]. In particular, if B is simple then only one type can occur. The R-algebra homomorphism h: → EndB (V ) gives rise to a morphism → G, which we still denote by h. C R R S

Lemma B.17. If G1 has no factors of type (D), then the pair (G, h) is a Shimura–Deligne datum. In addition, it satisfies (SD4). Proof. This is [Tor19, Lemma 7.5 (ii)], with the only modification being that we include non-central factors of compact type. The second statement follows by Lemma 7.5 (iii) of op.cit. Definition B.18. A PEL-type Shimura–Deligne datum is a pair (G, h) that arises from a choice of a PEL datum with no orthogonal factors as above.

To any PEL-datum as above, and for any compact open subgroup K ⊂ G(Af ), we have an associated moduli problem MK over the reflex field which classifies abelian varieties with corresponding PEL structure as in [Lan12, §1.2]. If K is neat, then MK is representable by a smooth quasi-projective scheme. Remark B.19. Here, we are assuming that such a rational PEL-datum arises from a choice of integral PEL- datum as in [Lan12, §1.1]. As we are only interested in moduli problems over reflex fields in the sequel, the choice of an integral PEL datum does not matter by Corollary 1.4.3.7 of [Lan13]. Lemma B.20. Let (G,X) be a PEL-type Shimura–Deligne datum arising from (B, ∗,V, h·, ·i, h). Set   1 1 M 1 ker (Q, G) := ker H (Q, G) → H (Qv, G) . v≤∞

1 Then ker (G, Q) is finite and classifies isomorphism classes of skew-Hermitian B-modules (V 0, h·, ·i0) such that the B -module (V 0 , h·, ·i0) is isomorphic to (V , h·, ·i). A A A 1 Proof. That ker (Q, G) is finite follows by [BS64, §7]. By [Mil17, Proposition 3.50]), H1(Q, G) (resp. 1 1 H (Qv, Gv)) classifies isomorphism classes of G-torsors over Q (resp. Gv torsors over Qv), and so ker (Q, G) classifies isomorphism classes of G torsors over Q that are locally trivial. Now G-torsors are just symplectic 0 B-modules V that are isomorphic to V over Q. 60 ANDREW GRAHAM AND SYED WAQAR ALI SHAH

Proposition B.21. Let (G,X) be a PEL-type Shimura–Deligne datum associated with the tuple (B, ∗,V, h·, ·i, h), and ﬁx an integral PEL-datum satisfying [Lan12, Condition 1.2.5]. For any neat compact open subgroup K ⊂ G(Af ), one has an open and closed embedding of C-varieties

(B.22) ShG,C(K) ,→ MK,C. 1 If the group G satisfies the Hasse principle, i.e. ker (Q, G) = 0, then the embedding in (B.22) is an isomorphism. Proof. The embedding in (B.22) follows from [Lan12, Lemma 2.5.6] and the paragraph following its proof. More precisely, it is shown that ShG(K)(C) is isomorphic to the locus in MK (C) parameterising all PEL abelian varieties whose first (rational) homology is isomorphic to V (as polarised symplectic spaces with B-structure). Note that for any point in MK (C), the first homology (over A) of the associated abelian variety is always isomorphic to V ⊗Q A (as polarised symplectic spaces with B-structure), by definition; so to show (B.22) is an isomorphism, it is enough to show that there is a unique polarised symplectic B-module ∼ W such that W ⊗Q A = V ⊗Q A. This follows by Lemma B.20. Remark B.23. The discussion above can also be found in [Kot92, §8] under the assumption that B is simple. 1 1 Remark B.24. The computation of ker (Q, G) can be reduced to ker (Q, G/Gder) by [Mil17, Proposition 25.71], as Gder is always simply connected for PEL-type Shimura Deligne datum ([Mil03, Proposition 8.7]). Corollary B.25. Keeping the same notation as in Proposition B.21, suppose that G satisfies the Hasse principle. Then ShG,C(K) has a canonical model isomorphic to MK,E, where E = E(G,X) is the reflex field. Proof. Any PEL-type Shimura–Deligne datum has an embedding into the standard Siegel Shimura datum, so ShG,C(K) has a unique canonical model by Corollaries B.14 and B.13. The fact that MK,E is a canonical model follows from [Mil03, Proposition 14.12] with the identity map (the proof of this proposition is purely a statement about CM abelian varieties).

Example B.2. Let (G , X) ∈ {(H,XH), (G,XG)} be the Shimura–Deligne datum introduced in section 2.2. Then (G , X) is of PEL-type and satisfies (SD5) and [Lan12, Condition 1.2.5]. Indeed • If G = G, then the semisimple algebra is just B = E and V is the 2n-dimensional Hermitian space with Hermitian form corresponding to the matrix J1,2n−1 (see section 2.1). In particular, this arises from an integral PEL-datum with semisimple algebra OE. • If G = H, then we take B = E × E and consider the product of Hermitian spaces V = W1 ⊕ W2 where W1 (resp. W2) has Hermitian form given by the matrix J1,n−1 (resp. J0,n). As above, this arises from an integral PEL-datum with semisimple algebra OE × OE. In addition to this, the group G satisfies the Hasse principle, for the following reason. By Remark B.24, it is enough to deduce it for the the maximal abelian quotient of G , denoted TG . In either case, TG is a product of factors U(1), GU(1) or Gm. The Hasse principle is known to hold for these three groups; for U(1), see ([Kot92, §7]), while for GU(1), Gm, it follows by Hilbert’s Theorem 90. Appendix C. Ancona’s construction for Shimura–Deligne varieties In this appendix, we document certain results, particularly the functoriality of motivic lifts, of [Anc15] and [Tor19] that hold in the absence of (SD3). The techniques used by these authors involve mixed Shimura varieties, which are a generalisation of the usual (pure) Shimura varieties. The definition of a mixed Shimura datum includes a counterpart of axiom (SD3); axiom (vii) of [Pin88, Definition 2.1]), and as in the case of pure Shimura data, this axiom is usually invoked when strong approximation is needed (for example, when describing the connected components of mixed Shimura varieties in [Pin88, Proposition 3.9]). It also plays a key role in reducing many statements to ones involving pure Shimura data. By replacing this condition with an alternative assumption (Assumption C.3) however, many proofs in [Pin88] carry over verbatim (building on the results in Appendix B). We have not attempted to be thorough here – instead, we content ourselves with a summary of the general results we need from [Pin88], and only use them in the restricted setting of §C.3 that suffices for the purposes of this article. We continue with the notation of Appendix B until §C.5, after which we specialise to the particular groups defined in §2.1. ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS 61

C.1. Mixed Shimura–Deligne data. Consider the following collection of data • P a connected linear algebraic group over Q • W the unipotent radical of P • U a subgroup of W over Q that is normal in P. •X a left homogenous space under the group P(R) · U(C) ⊂ P(C), • h: X → Hom(SC, PC) a P(R) · U(C)-equivariant map. Set V = W/U, G = P/W and π : P → G, π0 : P → P/U the natural projections. Then such a collection is said to be mixed Shimura–Deligne datum if it satisfies axioms (i)–(viii) of [Pin88, Definition 2.1], except possibly (vii). If these axioms are satisfied, the data is determined by the triple (P, X , h) since U is charac- terised by its action on Lie(P). We suppress the dependency on h when a choice has been made, and denote such a datum by (P, X ) only. A morphism of mixed Shimura–Deligne datum (P1, X1, h1) → (P2, X2, h2) is a morphism φ: P1 → P2 and a P1(R) · U1(C)-equivariant map Ψ: X1 → X2 satisfying the commutativity property of [Pin88, Definition 2.3]. We call a morphism injective if both φ and Ψ are. Given a mixed Shimura–Deligne datum (P, X , h) and a compact open subgroup K ⊂ P(Af ), we define the corresponding mixed Shimura–Deligne variety to be the double coset

ShP(X ,K)(C) := P(Q)\ [X × P(Af )/K] . As in [Pin88, Proposition 2.19], every connected component of X is a holomorphic complex vector bundle on a Hermitian symmetric domain, or simply a complex vector space. Therefore, if K is neat, then ShP(X ,K)(C) inherits the structure of a complex analytic variety. A morphism of mixed Shimura–Deligne data induces a morphism on the corresponding mixed Shimura–Deligne varieties when the compact open subgroups are chosen as in [Pin88, Proposition 3.8].

C.2. Hodge structures. Let (G,X) be a Shimura–Deligne datum as in Definition B.1, and let F be a number field. Let RepF (G) denote the category of all finite-dimensional algebraic representations (ρ, V) of GF and set V = V(F ), the underlying F -vector space. Such a representation can equivalently be viewed as an algebraic representation of G defined over Q of dimension dimF (V ) · [F : Q], together with a Q-algebra homomorphism F,→ EndG(Q)(V ). For any (ρ, V) ∈ RepF (G) and h ∈ X, we obtain a Hodge structure on V ⊗Q C via the map ρ ◦ h. In × particular, V ⊗Q C breaks up into one-dimensional subspaces for which C ⊂ SC acts via a character of the −pi −qi form z 7→ z z¯ – the corresponding Hodge bigrading of this subspace is given by the pair (pi, qi), and its weight is defined to be pi + qi. The Hodge type of (ρ, V) is the collection of all such pairs that appear in this decomposition. If pi + qi = n for all i, then we say that V is pure of weight n. In this case, for any n h ∈ X, the weight homomorphism ρ ◦ h ◦ w : Gm,R → GL(V ⊗Q R) is equal to the character λ → λ (recall our conventions on w at the start of Appendix B).

AV Definition C.1. We let RepF (G) denote the full sub-category of RepF (G) whose objects have Hodge type contained in {(−1, 0), (0, −1)}. C.3. Unipotent extensions. We shall primarily be concerned with a specific sub-class of mixed Shimura– Deligne datum, namely those which are unipotent extensions and of a prescribed Hodge type. Definition C.2. Let (G,X) be a Shimura–Deligne datum satisfying (SD5). For a representation V in AV RepF (G) , consider the pair (P, X ), where

• P = ResF/Q(V) o G is the unipotent extension of G, with natural map π : P → G •X is the subset of all t ∈ Hom(SC, PC) which are defined over R and satisfy π◦t ∈ XC = {hC : h ∈ X}. Then the pair (P, X ) satisfies axioms (i)–(viii) in [Pin88, 2.1], excluding axiom (vii) (which is axiom (SD3)) and therefore determines a mixed Shimura–Deligne datum. We refer to such a datum as a unipotent extension of the pure datum (G,X). Under an additional assumption below, we will explain how one can associate mixed Shimura–Deligne varieties to the unipotent extensions in Definition C.2 which satisfy good functoriality properties in both (G,X) and V. Essentially, all the results in [Pin88] that we will need hold by replacing (SD3) with the following assumption. Let S1 ⊂ S denote the circle group. 62 ANDREW GRAHAM AND SYED WAQAR ALI SHAH

Assumption C.3 (c.f. [Pin88, 1.12]). Let (G,X) be a Shimura–Deligne datum satisfying (SD5) and let (P, X ) be a unipotent extension as in Deﬁnition C.2. Let G1 be a normal subgroup of G which contains the image of h(S1), for any h ∈ X. Let (ρ, M) be an algebraic representation of P (over Q) that is pure of n weight n, i.e. for any h ∈ X , the action of ρ ◦ h ◦ w(z) on MC is given by multiplication by z . We assume that there exists

• A one-dimensional algebraic representation N of P, deﬁned over Q, which factors through G/G1 and is pure of Hodge type (n, n).

• A P-equivariant non-degenerate pairing Ψ: M ⊗Q M → N, and • For every h ∈ X , a morphism of rational Hodge structures λh : N → Q(−n), such that λh ◦ Ψ is a polarisation for the Hodge structure on M defined by h. Remark C.4. Assumption C.3 is an assumption on (P, X , ρ, M). By abuse of language, we will say that Assumption C.3 holds for (P, X ) if it holds for (P, X , ρ, M), for every pure algebraic representation (ρ, M) of P. Remark C.5. If (G,X) is a Shimura–Deligne datum satisfying (SD3) and (SD5) then Assumption C.3 holds for irreducible M. If, in addition to this, dim (G/G1) ≤ 1, then Assumption C.3 holds for all M (see [Pin88, 1.13]). In the absence of (SD3), Assumption C.3 provides an alternative way to deduce [Tor19, Lemma 5.5]. In particular, it holds for Shimura–Deligne data of PEL-type satisfying (SD5). Lemma C.6. Let (G,X) be a PEL-type Shimura–Deligne datum as in Definition B.18 that satisfies (SD5). Then Assumption C.3 holds for any unipotent extension of (G,X) as in Definition C.2. In fact, one can take N to be a power of the similitude character. Proof. Let (P, X ) be any unipotent extension, as in Definition C.2, and take any t ∈ X and h ∈ X such

that π ◦ t = hC. Take G1 to be the kernel of the similitude character. Then this is a normal subgroup of G 1 ∼ which contains h(S ), so satisfies the conditions in Assumption C.3. Furthermore, we have G/G1 = Gm. The homomorphism t ◦ w : → P defines a weight filtration W on the category Rep (P) as in the Gm,C C • C end of §1 in [Mil90] (note that we are using ascending, rather than descending, filtrations – see Remark 1.8 in loc.cit.). By the assumptions on (P, X ), the algebraic group PC stabilises the filtration W• (see [Pin88, Proposition 1.4]). By [Mil90, Proposition 1.7], this implies that W0PC = PC and W−1PC = ResF/Q(V)C. As a consequence, we see that any pure algebraic representation of P must factor through G. Combining this with the fact that the pairing Ψ is required to be P-equivariant, it is enough to verify that Assumption C.3 holds in the case (P, X ) = (G,X), which we place ourselves in for the remainder of this proof. It is also enough to show that for any irreducible representation M that is pure of weight n, the assumption is satisfied with N equal to the n-th power of the similitude character. Indeed, the category Rep (G) is semisimple, so we can easily extend the result to arbitrary pure representations after showing Q this. Let M be an irreducible algebraic representation of G. Any such representation is still irreducible after restricting to G , and since the category Rep (G ) is semisimple, we must have 1 Q 1

dimQ ((M ⊗Q M)G1 ) ≤ 1 ∼ where (−)G1 denotes coinvariants by G1. Furthermore this dimension is equal to 1 precisely when M|G1 = ∗ M |G1 , by Schur’s lemma. Another way of saying this is that there exists an integer r such that M∗ =∼ M ⊗ µr where µ is the similitude character of G. But the character µ gives rise to a Hodge structure that is pure of weight −2r, so the above isomorphism implies that r = n. The rest of the lemma now follows from the

proof of [Pin88, 1.12] (using the fact that conjugation by h(i) is a Cartan involution for G1,R). C.4. Summary of results on mixed Shimura–Deligne varieties. Let (G,X) be a Shimura–Deligne datum that satisﬁes (SD5) and Assumption C.3 for any unipotent extension (P, X ) as in Deﬁnition C.2 (see Remark C.4). This assumption will allow us to apply the majority of the results in [Pin88, §1–§3, §11] and [Tor19], which we summarise below: ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS 63

AV (1) Let V be an algebraic representation of GF in RepF (G) which we view as an algebraic representation of G with an F -structure, and let (P = ResF/Q(V) o G, X ) denote the unipotent extension 0 as in Definition C.2. For a neat compact open subgroup K ⊂ P(Af ), we set 0 0 ShP(K )(C) := P(Q)\ [X × P(Af )/K ] which we view with the quotient topology induced from X . By [Pin88, 3.2–3.3] and Baily–Borel this set carries the structure of a complex algebraic variety, which we call the associated mixed Shimura– Deligne variety. Such an example of a neat compact open subgroup is as follows. Let K ⊂ G(Af ) be a neat compact open subgroup, and let L ⊂ V(Af ) denote a (full rank) Zb-lattice that is stable : under K. By [Tor19, Lemma 5.4], L o K is a neat compact open subgroup of P = ResF/Q(V) o G. 0 0 −1 0 0 (2) Let K ,L ⊂ P(Af ) be neat compact open subgroups satisfying σ L σ ⊂ K for some σ ∈ P(Af ). Then we have a finite étalemap

0 [σ] 0 ShP(L )(C) −−→ ShP(K )(C) constructed in exactly the same way as for (pure) Shimura–Deligne varieties (see [Pin88, 3.4]). We can also define morphisms of mixed Shimura–Deligne datum in the usual way, and this defines morphisms between the associated Shimura–Deligne varieties (see [Pin88, 3.8]). AV (3) For V in RepF (G) the mixed Shimura–Deligne variety ShP(L o K)(C) has the structure of an abelian scheme over ShG(K)(C) (c.f. [Pin88, 3.22] – we use Assumption C.3 here). In particular, these abelian schemes satisfy functoriality properties with respect to the datum (G,X) and the representation V (see [Tor19, Lemma 5.7]). (4) There is a natural way to define reflex fields and canonical models associated to mixed Shimura– Deligne data (see [Pin88, §11]). In our setting, the reflex field E of any unipotent extension (P, X ) as in Definition C.2 is equal to the reflex field of (G,X) (11.2 in op.cit.). Suppose that ShG(K) admits a canonical model over E (which is unique by Corollary B.13), then the results of §11 and 0 the reduction lemma (Lemma 2.26) in op.cit. imply that ShP(K ) admits a unique canonical model over E. Furthermore, the morphisms discussed in (2) descend to morphisms between the canonical models, as well as the functoriality properties in (3) (as long as the pure Shimura–Deligne data admit canonical models). Note that we are permitted to apply the reduction lemma in op.cit. by Assumption C.3. Corollary C.7. Take G ∈ {G, H, T, Ge } to be any of the groups defined in section 2.1, and X the associated symmetric space. Then (G ,X) is a Shimura–Deligne datum which satisfies (SD5) and Assumption C.3 for any unipotent extension (P, X ) of (G ,X) as in Definition C.2 (including, of course, the datum (G ,X) itself). In particular, (1)–(4) above hold for any such (P, X ). Proof. The only potentially problematic group is G = H, but we have shown that Assumption C.3 is satisfied in this case (see Lemma C.6). The existence of a canonical model for (H,XH) follows from Corollary B.14 (the other groups gives rise to Shimura data in the usual sense). C.5. Lifts of the canonical construction. We now put ourselves in the situation of Corollary C.7, so ( ,X) ∈ {(G,X ), (H,X ), (T,X ), (G,X )}. Let p be a prime that splits in E/ and recall that we G G H T e Ge Q ∼ have fixed an embedding E,→ Qp, which distinguishes a prime P of E lying above p satisfying EP = Qp. By the general procedure described in section 2.5, for any neat compact open subgroup K ⊂ G (Af ) we have a Qp-linear tensor functor ´ µ ,K : Rep (G ) → Et (Sh (K)) G Qp G Qp from the category of finite-dimensional algebraic representations of GQp to lisse Qp-sheaves on ShG (K). Theorem C.8. Let G ∈ {G, H, T, Ge } as above. Then the results of [Tor19, §10], excluding Lemma 10.6, hold in this setting. If G ∈ {G, H}, then Lemma 10.6 does hold, in addition to the results of §8–§9 of op.cit. where Betti cohomology is replaced with p-adic cohomology and the functoriality statements are with respect to the embedding H ,→ G (as in section 2.1). Proof. We will justify that all the proofs hold in our setting, even though we have not assumed (SD3). Note that (SD5) and Assumption C.3 hold for (any unipotent extension of) the Shimura–Deligne data that we are 64 ANDREW GRAHAM AND SYED WAQAR ALI SHAH

considering, and that the morphism H ,→ G is admissible in the sense of [Tor19, Definition 9.1]. The key references for the proofs are [Pin88], [Pin92] and [Wil97, §I–II]. We justify the use of the results in [Wil97] (the remaining results from [Pin88] and [Pin92] are justified in section C.4). As we are working in the special case of mixed Shimura–Deligne datum in Definition C.2, as remarked at the end of page 10 in [Wil97, §II], one can check that we are in the situation of [Wil97, §I.3] (except for geometric connectedness). Also [Pin88, 3.13] is valid in our setting, so Lemma 1.6 in [Wil97, §II] holds (the fibres of the map ShP (L o K)E → ShG (K)E are “unipotent K(π, 1)s”). These observations mean that [Wil97, §II.4] is valid in our setting. Indeed, we first check that all the cited results in this section hold: • The reference [Pin92, Proposition 3.3.3]. It is an easy check that this proof does not depend on the axiom (SD3). • Theorem 4.3 in [Wil97, §II] (which is a consequence of Propositions 5.5.4, 5.8.2 and 5.6.1 in [Pin92]). For part (a) there is nothing to check, because any torus satisfies (SD3). For parts (b) and (c), the proofs in [Pin92, §5.6] still hold verbatim (we do not need to consider models and compactifications). The arguments involving special points are also valid in our setting, because the results in [Pin88, §11] are valid by replacing any referenced result in [Del71] with the appropriate analogue in Appendix B. • The results in [Wil97, §I] are valid by the above discussion. Then one can check that none of the remaining arguments in [Wil97, §II.4] rely on the axiom (SD3). We note that the opposite convention of left/right actions for the functor µG is used in loc.cit., however this does not affect the validity of the results. One can now check that the arguments in [Tor19] carry over into our situation. Furthermore, Ancona’s construction is valid in our setting, as the results in [Anc15] do not depend on (SD3) (in fact the majority of the paper doesn’t even involve Shimura varieties). Remark C.9. Some of the arguments in [Tor19] do involve passing to connected components of the mixed Shimura varieties. However, this is only ever used in an abstract way; an explicit description of the connected components is not needed, which would require (SD3).

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(Graham) Department of Mathematics, South Kensington Campus, Imperial College London, London SW7 2AZ, UK E-mail address: [email protected] (Shah) Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA 02138, U.S.A. E-mail address: [email protected]