An Euler system for GU(2, 1)

We construct an Euler system associated to regular algebraic, essentially conjugate self-dual cuspidal automorphic representations of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{GL}\,}}_3$$\end{document}GL3 over imaginary quadratic fields, using the cohomology of Shimura varieties for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {GU}}(2, 1)$$\end{document}GU(2,1).


Overview of the results
Euler systems -families of global cohomology classes satisfying norm-compatibility relations -are among the most powerful tools available for studying the arithmetic of global Galois representations. In particular, most of the known cases of the Bloch-Kato conjecture, and of the Iwasawa main conjecture, use Euler systems as a fundamental ingredient in their proofs. However, Euler systems are correspondingly difficult to construct; in almost all known cases, the construction uses automorphic tools, relying on the motivic cohomology of Shimura varieties. Euler systems come in two flavours: full Euler systems, in which we have classes over almost all of the ray class fields E [m], where E is some fixed number field; or anticyclotomic Euler systems, where E is a CM field, and we restrict to ring class fields (the anticyclotomic parts of ray class fields). Full Euler systems are the most powerful for applications, but correspondingly hardest to construct.
In this paper, we'll construct a new example of a full Euler system, associated to Shimura varieties for the group G = GU(2, 1) (Picard modular surfaces). This construction has some novel features compared with previous constructions, such as the GSp 4 case treated in [16]. Firstly, the field E (which is the reflex field of the Shimura datum for G) is not Q, but an imaginary quadratic field, and so an Euler system in this setting consists of classes over all of the abelian extensions of E (most of which are not abelian over Q). Secondly, we introduce here a new strategy for proving norm-compatibility relations, based on cyclicity results for local Hecke algebras; this allows us to show that our classes are norm-compatible in the strongest possible sense, i.e. as classes in motivic cohomology (whereas in [16] we only proved norm relations for the images of Euler system classes in the étale realisation, after projecting to an appropriate Hecke eigenspace). Such cyclicity results for Hecke algebras are closely bound up with the theory of spherical varieties, and we believe that this connection with spherical varieties should be a fruitful tool for studying Euler systems in many other contexts. We refer the reader to §8 for the definition of the Shimura variety Y G (K G ), and the relative Chow motive D a,b {r , s} over it. In the case (a, b, r , s) = (0, 0, 0, 0), this motive is simply the trivial motive E(0), and our classes coincide with those considered in [23]; in particular, the main result of op.cit. shows that the images of these classes under the Deligne-Beilinson regulator map, paired with suitable real-analytic differential forms on Y G (K G )(C), are related to the values L (π, 0) for cuspidal automorphic representations π of G(A). This shows that our motivic cohomology classes are non-zero in this trivial-coefficient case. (We expect that a complex regulator formula similar to [23] should also hold for more general coefficient systems, but we shall not treat this problem here.) After passing to a Shimura variety with Iwahori level structure at p, we can also obtain families of classes over all the fields E[m p t ] for t ≥ 1, satisfying a normcompatibility in both m and t; see Theorem 10.2.2 for the precise statement. Applying the étale regulator map and projecting to a cuspidal Hecke eigenspace, we obtain Euler systems in the conventional sense -as families of elements in Galois cohomologyassociated to cohomological automorphic representations of G(A). Combining this with known theorems relating automorphic representations of G and of GL 3 /E, we obtain the following: Theorem B Let be a RAECSDC 1 automorphic representation of GL 3 /E which is unramified and ordinary at the primes p | p. Let V P ( ) be its associated Galois representation, and suppose this representation is irreducible. Then there exists a lattice T P ( ) *  where P w ( , X ) = det(1 − X Frob −1 w : V P ( ) (1)).
See Theorem 12.3.1 for a precise statement, and for some additional properties of the classes c m . As well as constructing these Euler systems, we also prove interpolation results showing that their p-adic étale realisations are compatible with twisting by padic families of algebraic Grössencharacters, and with variation in Hida families of automorphic representations.
In future work, we will prove an explicit reciprocity law for this Euler system, relating it to values of an appropriate p-adic L-function, and thus prove the Bloch-Kato conjecture in analytic rank 0 for automorphic Galois representations arising from G. However, in the present paper we shall focus solely on the construction of the Euler system classes.

Outline of the paper
After some preliminary material presented in Sect. 2, Sects. 3-6 of this paper are devoted to proving a certain purely local, representation-theoretic statement which we call an "abstract norm relation" (Theorem 5.2.4). This states that, if Z is any map from a certain space of local test data to a representation of G(Q ), satisfying an appropriate equivariance property, then the values of Z on two particular choices of the test data are related by a certain specific Hecke operator P. We prove this in two stages. Firstly, in §4, we prove that such a Hecke operator P must exist (without identifying the operator), using a cyclicity result for Hecke modules inspired by work of Sakellaridis. Secondly, in §5 and §6 we use local zeta integrals to define a directly computable, purely local example of a morphism z with the correct equivariance property, which allows us to identify the relevant Hecke operator P explicitly. We have developed this theory in some detail, since we expect that the strategy developed here will be applicable to many other Euler system constructions, and it might also serve to clarify some possibly confusing details in earlier works of ours such as [16].
In the second part of the paper, Sects. 7-9, we construct a second, much more sophisticated example of a morphism to which the above theory applies: the "unitary Eisenstein map" UE [a,b,r ,s] of Definition 9.2.3, taking values in the motivic cohomology of the GU(2, 1) Shimura variety. Applying the "abstract norm relation" to this specific choice of morphism, we obtain a family of motivic classes satisfying norm-compatibility relations, whose denominators are uniformly bounded in the étale realisation. This is our Euler system.
In the final sections of the paper, we prove that these classes satisfy normcompatibility relations in a suitable tower of levels at p, and that their étale realisations are compatible with certain p-adic moment maps arising from this tower. This can be interpreted as stating that the étale Euler-system classes vary analytically in Hida families for G; this is an important input for studying explicit reciprocity laws for the Euler system, which will be the subject of a forthcoming paper. Finally, we briefly discuss the Euler system for an individual automorphic Galois representation obtained by projecting our classes to a cuspidal Hecke eigenspace.

Fields
Let E be an imaginary quadratic field, of discriminant −D, and let x →x be the nontrivial automorphism. Let O be the ring of integers of E. We fix an identification of E ⊗ R with C such that δ = √ −D has positive imaginary part.
We writeB G andN G for the lower-triangular Borel and its unipotent radical.

The group G 0
We define G 0 = ker(ν) ⊂ G, so G 0 is the group of unitary isometries (as opposed to unitary similitudes) of J . Since g μ(g) ∈ G 0 for all g ∈ G, we have for all Z-algebras R.

The group H
Let H be the group scheme over Z such that for a Z-algebra R This can be identified with a subgroup of G: In particular we can regard μ as a character of H , by composition with ι, and we have simply μ( (g, z) ) =z.

Note 2.4.1
If is a prime split in E, and we fix a prime w | of E as above, then w gives

Open orbits
The following relationship between G and H is crucial for our arguments: Lemma 2.5.1 Let R be a Z[1/D]-algebra, and let Q 0 H be the subgroup {(g, z) ∈ H : g = 0 1 }. Then there exists an element u ∈ N G (R) such that the map is an open immersion. After a mildly tedious matrix manipulation one sees that this map is given by This clearly identifies Q 0 H with the open subscheme of N G consisting of the n(s, t) with s = −1.

Remark 2.5.2
The openness of the image amounts to the claim thatB G × Q 0 H , or equivalently B G × B H , has an open orbit on the homogenous In other words, X is a spherical variety. This fact will play a crucial role in the norm-compatibility relations for our Euler system, both in the "tame direction" (see Theorem 4.2.1) and the " pdirection" (Theorem 10.2.5).

Base change and L-factors
We now relate representations of G with representations of the group Res E/Q (GL 3 × GL 1 ).

Local case
For each prime split in E/Q, and each prime w | of E, the prime w determines an isomorphism of G(Q ) with GL 3 (Q ) × Q × , as above. Definition 2.6.1 If π is an irreducible smooth representation of G(Q ), we let bc w (π ) denote the representation of GL 3 (Q ) × Q × obtained from π via this isomorphism.
If v is a place which does not split (including the infinite place), and w the place above v in E, then there is also a base-change map bc w taking tempered representations of G(Q v ) to tempered representations of (GL 3 × GL 1 )(E w ); this is a consequence of the local Langlands correspondence for unitary groups due to Mok [20,Theorem 2.5.1]. (See [23,Definition 3.5] for explicit formulae when D and π is spherical.) As in the split case, if bc w (π v ) = τ w ψ w , we use the notation L w (π v , s) for L(τ w ⊗ ψ w , s).
In either case we write L(π v , s) = w|v L w (π v , s), which is the L-factor associated to π v and the natural 6-dimensional representation of the L-group of G.

Global case
(The definitions in this section will not be used until §12.) We recall the following definition (see e.g. [2, §1]): Definition 2.6.2 A "RAECSDC" (regular algebraic, essentially conjugate self-dual, cuspidal) automorphic representation of GL 3 /E is a pair ( , ω), where is a cuspidal automorphic representation of GL 3 /E and ω is a character of A × /Q × , such that: Let L be any field of characteristic 0, and write S(G S , L) for the space 2 of compactly-supported, locally-constant L-valued functions on G S . We write S(Q 2 S , L) for the space of Schwartz functions on Q 2 S . Definition 3.1.1 Let V be a smooth L-linear (left) representation of G S . We shall say an L-linear map • G S acts on the left-hand side by g · (φ ⊗ ξ) = φ ⊗ ξ((−)g), and on the right-hand side by its given action on V; , and trivially on the right-hand side. Equivalently, these are the G S -equivariant maps . We can make similar definitions with S replaced with the space S 0 (Q 2 S , L) of Schwartz functions vanishing at (0, 0); we write I 0 (G S , L) for the H S -coinvariants of S 0 Q 2 S , L ⊗ L S (G S , L). In order to avoid unnecessary repetition, we adopt the following notational shortcut: S , L to denote a statement which is valid for either S or S 0 , and correspondingly I (0) . As in [16, §3.9], once a Haar measure on G S is chosen, one can identify I (0) (G S , L) with the compact induction cInd G S H S (S (0) (Q 2 S , L)). It then follows from Frobenius reciprocity that G S -equivariant maps

Integrality
Let us fix a Haar measure vol H ,S on H S , which we suppose to be Q-valued.

Definition 3.2.1
We shall say an element of I (0) (G S /U , Q) is primitive integral at level U if it can be written in the form φ ⊗ ch(gU ) for some φ ∈ S (0) and g ∈ G S , and the function φ takes values in the fractional ideal CZ, where we define An element of I (0) (G S /U , Q) is said to be integral at level U if it is a sum of primitive integral elements at level U ; and we write the set of such elements as I (0) (G S /U , Z).
Clearly, any element of I (0) (G S /U , Q) can be scaled into I (0) (G S /U , Z). More generally, we can replace Q with a number field L, and Z with O L [1/ ] for any set of primes of L.

Remark 3.2.2
This definition may seem bizarre at first sight; its motivation is the following. Later in this paper, we shall construct G S × H S -equivariant maps into the motivic and étale cohomology of Shimura varieties for G, analogous to the "Lemma-Eisenstein map" considered in [16] for the GSp 4 case. However, the definition of these maps involves various volume factors, so it is far from obvious a priori which input data give rise to classes in the integral étale cohomology. The above notion of "integral elements" is designed for exactly this purpose.
Note that the definition of integrality depends on the level U , but we have the following compatibilities.  Proof Evidently, it suffices to check either statement on primitive integral elements. For the trace map this is selfevident, as the trace sends a coset ch(gU ) to ch(gU ), and the corresponding normalising factors C and C satisfy C | C, so primitive integral elements map to primitive integral elements. The reverse-direction map is a little more intricate, and follows by considering the orbits of the group V = gU g −1 ∩ stab H S (φ) on the U -cosets contained in a given U -coset.

Remark 3.2.4
One can interpret the system of abelian groups I (0) (G S /U , Z), for varying U , as a "Cartesian cohomology functor" in the sense of [14]. We would like to prove the following statement (an "abstract norm relation"): if δ 0 = ch(Z 2 ) ⊗ ch(G 0 ) is the natural spherical vector of I(G /G 0 , Z), then there exists an element

Spherical Hecke algebras and cyclicity
where P w (to be defined below) is a certain polynomial over the spherical Hecke algebra, related to local Euler factors. What we shall actually prove, as Theorem 5.2.4 below, is something a little weaker than this, but still sufficient for applications: δ w is only integral up to powers of , and if is inert, the equality norm (1) · δ 0 only holds up to inverting a certain element in the centre of the Hecke algebra.
We shall prove this statement in two stages. Firstly, we shall show that for any open U ⊆ G 0 and any δ ∈ I (G /U , Z), there exists an element P δ lying in (a localisation of) the spherical Hecke algebra of G such that norm U G 0 (δ) = P δ · δ 0 . This relies crucially on a cyclicity result for Hecke algebras due to Sakellaridis (Theorem 4.2.1).
Secondly, we shall write down a candidate for δ w and verify that it is integral at level up to powers of . The aforementioned results then show that norm the image of δ 0 under some Hecke operator P δ w . Via a lengthy but routine computation with local zeta integrals, we show that this Hecke operator must be equal to P w (1). This completes the proof.

Preliminaries
As in the previous section, let D be a prime. From here until the end of Section 4, all Schwartz spaces and Hecke algebras are over C and we omit this from the notation.

Hecke algebras
Let H G, denote the Hecke algebra, whose underlying vector space is S(G ) and whose algebra structure is given by convolution with respect to some choice of Haar measure dx: Any smooth left representation of G can be regarded as a left H G, -module, via the action In particular, if ξ = ch(gK ) for some subgroup K , and g is K -invariant, then ξ v = vol(K )g · v. Similar constructions apply to right modules; and these constructions are The same constructions apply likewise with H in place of G . Since a smooth G -representation is in particular a smooth H -representation by restriction, we can regard such representations as modules over either H G, or H H , , and if necessary we write G or H to distinguish between the two convolution operations.
If ξ ∈ H G, , we write ξ for its pullback via the involution g → g −1 of G , and similarly for H H , .

Spherical Hecke algebras
These are hyperspecial maximal compacts of G and H , respectively. We suppose that the Haar measures on G , H are chosen such that G 0 and H 0 have volume 1. The associated spherical Hecke algebras are commutative rings, and can be described (via the Satake isomorphism) as Weylgroup invariant polynomials in the Satake parameters.

Equivariant maps
We write [−] for the quotient map from S(Q 2 ) ⊗ H G, to its H -coinvariants I(G ), with the actions as given in Definition 3.1.1. An easy unravelling of definitions shows that

Cyclicity
We can consider the space of smooth, compactly supported functions G → C that are left H 0 -invariant and right G 0 -invariant. This is evidently a (H 0 H, , H 0 G, )-bimodule, via the convolution operations H and G .

Theorem 4.2.1 H is cyclic as an
If is split, this can be deduced from Corollary 8.0.4 of [27], applied to the group G = G × H , acting by right-translation on the quotient . It follows easily from Lemma 2.5.1 that X is spherical as a G -variety, i.e. the Borel subgroup B G = B G × B H has an open orbit on X . Sakellaridis' result shows that for any split reductive group G over Z and spherical G -variety X satisfying a certain list of conditions, the space of G (Z )invariant Schwartz functions on X (Q ) is cyclic as a module over the unramified Hecke algebra of G , generated by the characteristic function of X (Z ); applying this to our G and X gives the theorem.
However, since the hypotheses of Sakellaridis' general result are not entirely straightforward to verify in our setting, and Sakellaridis' argument does not cover the non-split case, we shall give a direct proof in an appendix; see Theorem A.1.1.

Remark 4.2.2
This theorem implies, in particular, that if π and σ are irreducible unramified representations of G and H respectively, then any element of Hom H (π ⊗ σ , C) is uniquely determined by its value on the spherical vectors, so the Hom-space has dimension ≤ 1. This relates our present approach to that of [16], where a "multiplicity ≤ 1" statement of this kind was taken as a starting-point for proving norm relations.

Hecke action on Schwartz functions
where T and S are the double cosets of 0 0 1 and 0 0 . We define ζ by This is surely well-known, but we give a sketch proof for completeness. It suffices to show that the C[GL 2 gives a (continuous) bijection between X n and X n+1 , we are reduced to showing that S(X 0 ) A 0 = S(P 1 (Z )) is contained in the GL 2 (Q )-span of φ 0 . However, for any t ≥ 1 this span contains the vector and these are the characteristic functions of a basis of neighbourhoods of (0 : 1) in P 1 (Z ). As GL 2 (Z ) acts transitively on P 1 (Z ), the translates of the φ t span S(P 1 (Z )).
Since H surjects onto GL 2 (Q ) for split, this shows that s(Q 2 ) = S(Q 2 ) A 0 in this case. In the inert case, if we write GL 2 (Q ) = GL 2 (Q ) + GL 2 (Q ) − according to the parity of the valuation of det g, then the image of H is GL 2 (Q ) + . By the preceding paragraph, we can write and both ξ + H (z A ( ) + ) and ξ − T are supported on GL 2 (Q ) + and hence in the image of H H , .

Remark 4.3.5 This result is essentially best possible, since the quotient S(Q
) is isomorphic to the induced representation I (| · | −1/2 , | · | −1/2 ). This is irreducible as a GL 2 (Q )-representation, but splits into two direct summands as a representation of GL 2 (Q ) + , and the spherical vector is contained in one of the summands. So s(Q 2 ) consists precisely of the vectors whose projection to the non-spherical summand of Since θ is invariant under right-translation by H 0 , and ξ under right-translation by G 0 , we conclude that σ ∈ H. By Theorem 4.2.1, we can express σ (possibly non-uniquely) as a finite sum If is inert, then we can still find θ such that θ H φ 0 = H (z A ( ) + )φ 0 , and the same argument as above produces a such that showing that G (z A ( )+ ) annihilates the class of φ ⊗ξ in I( If is inert, then we can find an element Proof Replacing δ with the sum of its translates by U /G 0 , we may assume U = G 0 , and the result is now obvious from the preceding theorem.

Characterising P ı
Let π be an irreducible spherical representation of G . Then the Hecke algebra acts Proof As usual, we may assume U = G 0 . The homomorphism z determines a linear map Z : If is split, then we have , which is exactly the formula claimed in the proposition. If is inert, then we replace I(G /G 0 ) with its localisation

Choice of the data
Let D be prime, and w a prime of E above . Let q := Nm(w) = or 2 .

The operator P w
If π is an irreducible unramified representation of G , we write π for the associated character of the Hecke algebra H 0 G, , as in §4.4 above.
Proof This is immediate from the Satake isomorphism, since the coefficients of the L-factor are Weyl-group-invariant polynomials in the Satake parameters.
Remark 5.1.2 One can check that P w (X ) has the form 1 − 1 q ch G 0 t( w ) G 0 X + higher order terms, where w is a uniformizer at w; however, for our arguments it is actually not necessary to write down P w explicitly.

The element ı w
Note that φ 1,t is fixed by the action of the group Definition 5.2. 2 We define an element ξ w ∈ H(G /G 0 [w]), and an integer n w , as follows: , where a ∈ E ⊗ Q has valuation −1 at w and ≥ 1 atw; and we set has valuation −1; and we take With these notations, in both cases we define Proof A tedious explicit computation shows that the subgroup = n w in the former case, and 3 ( 2 − 1) 2 = 3 n w in the latter case. Thus Outline of proof. We need to show that if δ = δ w , then the operator P δ of Corollary 4.3.7 is P w (1). We will do this using Proposition 4.4.1 to compare the images of P w (1) and P δ under π , for a sufficiently dense set of unramified representations π . More precisely, for all unramified representations π which are generic (admit a Whittaker model), we shall construct below a non-zero, H (Q )-equivariant bilinear form z ∈ Hom H π ⊗ S(Q 2 ), C using zeta integrals, and show that for this z we have The left-hand side of this equality is Z(δ w ) in the notation of Proposition 4.4.1, so we must have π (P δ w ) = L w (π , 0) −1 . Thus P δ w = P w (1) modulo the kernel of π . Since the characters π for which this construction applies are dense in the spectrum of the Hecke algebra, we must in fact have P δ w = P w (1) as required. It remains only to construct the homomorphism z and prove Eq. 5.2.a; this will be carried out in the next section.

The zeta integral
Let be a rational prime (for now we do not need to assume D). If e is an additive character E ⊗ Q → C × , we can extend it to a character of N (Q ) via n(s, t) → e(s). We fix a choice of e whose restriction to E w is non-trivial for all w | , and denote the resulting character of N (Q ) by e N . Definition 6.1.1 An irreducible representation π of G is said to be generic if it is isomorphic to a space of functions on G transforming by e N under left-translation by N (Q ). If such a subspace exists, it is unique, and we call it the Whittaker model W(π ). Definition 6.1.2 Let π be a generic representation of G . For every W ∈ W(π ), and s ∈ C, define where t(z) = (diag(zz,z, 1), zz) as above.
where χ = χ π | Q × . In particular this is independent of z.
We expect that for any generic π , the "common denominator" of the z(W , φ, s) should coincide with the L-factor L(π , s) defined using the local base-change lifting as in §2.6. However, in the present work we only need this when and π are unramified. Some ramified cases are established in [23, §3.6 & §8.3].

Explicit formulae in the unramified case
We suppose henceforth that 2D, that π is an irreducible unramified principal series, and that the additive character e has conductor 1. Then π is generic, and its Whittaker model W(π ) has a unique spherical vector W π ,0 such that W π ,0 (1) = 1. 2s) , where χ = χ π | Q × as above, and L(π , s) is as in Section 2.6.
Proof The values of W π ,0 along the torus T are given by an explicit formula in terms of the Satake parameters; see [28] for split, and [4, §4.7] for inert. The result follows from these formulae by an explicit computation. exists for all W ∈ W(π ) and φ ∈ S(Q 2 ), and defines a non-zero element of the space Hom H (S(Q 2 ) ⊗ π , C) satisfying z(W π ,0 , φ 0 ) = 1.

Remark 6.3.2
Note that this is much stronger than we need for the proof of Theorem 5.2.4; it would suffice to know that there is some non-zero rational function P(s) such that lim s→0 z(W ,φ,s) P(s) is well-defined and not identically 0.

Unipotent twists
We want to evaluate the above integrals on certain ramified test data (still assuming π itself to be unramified). S )W π ,0 , s) is given by where n w is as in Definition 5.2.2.
Proof As in [16, §3.10], for any W ∈ W(π ), the values 2t−2 ( 2 − 1) · z(W , φ 1,t , s) are independent of t for t 0, and the limiting value is simply Z (W , s). In our case, it suffices to take t = 2 since both η (a) w and its inverse have matrix entries in O ⊗ Z , so the principal congruence subgroup modulo 2 fixes , the computation of the limiting value is immediate from Proposition 6.4.2.
This completes the proof of (5.2.a), and hence of Theorem 5.2.4.

Representations of G and H
Since G and H are split over E, their irreducible representations over E are parametrised by highest-weight theory.  2 The characters χ 1 and χ 2 are the highest weights (with respect to B G ) of the natural 3-dimensional representation V of G and its conjugateV . The characters χ 3 and χ 4 factor through the abelianisation of G: we have χ 3 = det ν =μ and χ 4 = μ, where μ = det/ν as above. Moreover, χ 3 χ 4 = ν.
Definition 7. 1.3 (1) For a 1 , a 2 ≥ 0, denote by V a 1 ,a 2  Thus every irreducible representation of G has the form V a 1 ,a 2 {a 3 , a 4 } for some a 1 , . . . , a 4 ∈ Z with a 1 , a 2 ≥ 0; and every irreducible representation of H has the form This representation will play an important role in the following, and we shall write it as D a 1 ,a 2 .

Branching laws
The restriction of G-representations to H is described by a branching law, which is equivalent to the usual branching law for GL 2 ⊂ GL 3 (see e.g.  Proof Let λ be the highest weight of D [a,b] {r , s}. We use the Borel-Weil presentation of D [a,b] {r , s}: it is isomorphic to the space of polynomial functions on G which transform via λ under left-translation byB G . This space has a canonical highest-weight vector f hw , whose restriction to the big Bruhat cell is given by f hw (ntn) = λ(t).
If f H denotes the polynomial corresponding to br [a,b,r ,s] , then f H must transform via λ under left-translation byB G , and trivially under right-translation by Q 0 H . Sincē Since projection to the highest-weight subspace is proportional to evaluation at the identity, and both u −1 f H and f hw take the value 1 at the identity, this shows that u −1 · f H has the same highest-weight projection as f hw .
For F an extension of E, we write D a,b F {r , s} for the base-extension of D a,b F {r , s} to F, which is an irreducible representation of G /F . If F = E w for a prime w | D, then G is a Chevalley group (a reductive group scheme) over O E,w , so we have the notion of admissible O E,w -lattices in the E w -vector space D a,b {r , s} ⊗ E E w ; see [13] for an overview. We are chiefly interested in the maximal admissible lattice, which we shall denote by D a,b O E,w {r , s}.

.3 lies in O E w [G].
Let F w be the residue field of E w . Then f H is regular on G /E w ; and it is also regular on a dense open subscheme of G /F w . So it is regular on a subset of G /O E,w of codimension ≥ 2. Since G /O E,w is smooth, it is a normal scheme. It follows that f H is regular everywhere on G /O E,w (see e.g. Stacks Project tag 031T).

The Shimura variety Y G
Let S = Res C/R G m , and consider the homomorphism We write X G for the space of G(R)-conjugates of h; we can identify X G as the unbounded Hermitian symmetric domain Then (G, h, X G ) is a Shimura datum. The reflex field of this Shimura datum is E (viewed as a subfield of C via our chosen identification of E ⊗ R with C). We let Y G be the canonical model over E of the Shimura variety associated with this datum. For any open compact subgroup K ⊂ G(A f ) we let Y G (K ) = Y G /K be the quotient by K ; this is a quasi-projective variety over E. If K is sufficiently small, it is smooth (it suffices to take K to be neat in the sense of [22]; see [6, §2.3]). We recall that the C-points of Y G (K ) have a natural description as

The Shimura variety Y H
The homomorphism h factors as ι • h H , where h H : S → H /R is the Shimura datum We let X H be the H (R)-conjugacy class of h H . Then (H , h, X H ) is also a Shimura datum, and its reflex field is also E. We let Y H be the canonical model over E of the associated Shimura variety. For an open compact K ⊂ H (A f ), the C points of the quasi-projective variety Y H (K ) are naturally described as

Functoriality
We also have the projection map π : H → GL 2 (forgetting z). The composite π • h is a Shimura datum for GL 2 , which coincides with the one used in [16, §5.1]; again, this differs from the "standard" Shimura datum by an automorphism of GL 2 .

The component groups of Y G and Y H
The set π 0 (Y G ) of connected components of Y G can be described as follows. Let μ = det/ν : G → Res E/Q (G m ), so that the composite μ • h is given by z → z −1 .
Then the map identifies the set of geometrically connected components π 0 (Y G (K The action of Gal(Ē/E) on π 0 (Y G ) can be described by the reciprocity law: if is the Artin reciprocity map of class field theory, normalized so that geometric Frobenius elements are mapped to uniformizers, then the map π 0 (

Sheaves corresponding to algebraic representations
Let G temporarily denote any of the three groups is naturally a smooth F-linear (left) representation of G (A f ).

Theorem 8.3.1 ( [1, Theorem 8.6]) There is an additive functor
with the following properties: (i) Anc G preserves tensor products and duals.

regarded as a left G(Q p )representation where p is the prime below v.
We shall always take the coefficient field F to be E, and frequently drop it from the notation.

Proposition 8.3.2 ( [29, Corollary 9.8]) There is a commutative diagram of functors
where the left-hand ι * denotes restriction of representations, and the right-hand ι * denotes pullback of relative motives.

Pushforwards in motivic cohomology
Let 0 ≤ r ≤ a, 0 ≤ s ≤ b be integers. We use script letters

Remark 9.1.2
The proof that this map is well-defined ultimately reduces to the compatibility of pushforward and pullback in Cartesian diagrams; it therefore carries over to the general setting of Cartesian cohomology functors for G and H , in the sense of [14]. For a careful proof of the well-definedness using this formalism, see [7, Proposition 5.9].

Eisenstein classes and the unitary Eisenstein map
described in [16,Theorem 7.2.2]. Here S (0) signifies S if k ≥ 1 and S 0 if k = 0.

Remark 9.2.2
This map can be characterised via its residue at ∞, or via its composite with the de Rham realisation functor; see loc.cit. for explicit formulae. When k = 0 and φ is the characteristic function of (α, β) +Ẑ 2 , for α, β ∈ Q/Z not both zero, we and Eis k mot,φ is the Siegel unit g α,β in the notation of [9].
Composing the Eisenstein symbol with pullback along the projection

Choices of the local data
We shall now fix choices of the input data to the above map UE [a,b,q,r ] , in order to define a collection of motivic cohomology classes satisfying appropriate norm relations (a "motivic Euler system"). We shall work with arbitrary (but fixed) choices of local data at the bad primes; it is the local data at good primes which we shall vary, depending on a choice of a parameter m.  (2) . m and (a, b, r , s)). We shall frequently omit δ S from the notation.

Note that this depends (H S × G S )-equivariantly on δ S (for fixed
We note that this V does satisfy the auxiliary hypothesis on the action of the torus A: as a representation of A(Q ), V is a direct sum of eigenspaces associated to characters of Q × of the form x → |x| n χ(x) with χ of finite order and n = a + b − r − s ≥ 0. Thus z A ( ) + is bijective on V . The corollary now gives an equality between two values of this H × G -invariant map on different input data, and these are precisely the local input data used to define Z [a,b,r ,s] mot,m and the pushforward of Z [a,b,r ,s] mot,n . We can give an alternative interpretation of these classes via Eq. 8.2.a. We denote by

Étale realisation and integrality
It would be desirable to have an "integral" version of this theory, with coefficients in O-modules, but this appears to be difficult for general coefficients (we do not know if the functors Anc G (−) can be defined integrally). So we shall instead work with the p-adic étale realisation, for a fixed prime p. In this section, we will fix values of [a, b, r , s] and omit them from the notation. Let p be a (rational) prime, and p | p a prime of E. We define where D a,b E p is the étale sheaf of E p -vector spaces corresponding to D a,b ⊗ E E p , and similarly ét,m (δ S ).
For simplicity, we assume here that p / ∈ S (similar, but more complicated, statements can be formulated if p ∈ S). If c is a prime, coprime to 6m and not in S, we shall write c for the action of z A ( c ), where c is a uniformizer of Q c . We extend this multiplicatively to all integers c > 1 coprime to 6 Nm(m)S. Then we define where σ c in the latter formula is the arithmetic Frobenius.  (2)) lifts (canonically) to the cohomology of the integral coefficient sheaf. Since C −1 = vol H (V ) is the normalising factor in the definition of the unitary Eisenstein class, this shows that c Z ét,m (δ S ) lifts to the integral cohomology, as required.

Norm relations at p
We now consider norm-compatibility relations in the " p-direction". We let p and p be as in the previous section, and we add the additional assumption that c is coprime to p.

Choice of local data
where u is an element of G(Z p ) satisfying the conditions of Lemma 2.5.1.
given for t ≥ 1 by We then set δ p,t = n p,t φ p,t ⊗ ξ p,t ∈ I(G 0 p /K G p ( p t ), Z).

Remark 10.1.2 Explicitly, we have
(These conditions also entail ν = 1 mod p t .) The subgroup V p,t consists of all ( a b c d , z) ∈ H (Z p ) with c = 0, d = 1 mod p 2t , z = 1 mod p t , and b satisfying a certain somewhat messy congruence modulo p 2t (whose precise form depends on the choice of u). Now let us choose arbitrary δ S ∈ I(G/K G,S , E) as before. For t ≥ 0, and m ∈ R coprime to p, we can define δ[m, p t ] = δ S · δ p,t · / ∈S∪{ p} δ [m], so that ξ [m, p t ] is fixed by the right action of the group K G [m, p t ] = K G,S · K G p ( p t ) · {g ∈ G(Ẑ S ) : μ(g) = 1 mod m}.

Definition 10.1.3 With the above notations, we set
Since this definition is a special case of Definition 9.4.1, these elements satisfy the norm-compatibility in m of Theorem 9.4.3; and it also clearly depends (G(Q S ) × H (Q S ))-equivariantly on the test data δ S at the bad primes. For the rest of this section we regard δ S as fixed, and drop it from the notation.
Similarly, we can introduce p-level structure to the classes mot,m as follows. Let Y Ih denote the Shimura variety of level K G,S · Ih p · G(Ẑ S∪{ p} ), where Ih p = {g ∈ G(Z p ) : g mod p ∈ B G (F p )} is the upper-triangular Iwahori 3 at p. Then we have a natural map We let be the image of Z [a,b,r ,s] mot,m, p t under pushforward along this map.

Norm-compatibility in t
We now observe that these classes satisfy norm-compatibility in t.
This operator preserves the integral étale cohomology, because p r +s bounds the denominator of τ −1 on the integral lattice D a,b O E,p {r , s}; this is also the reason for the factor p (r +s)t in the definition of the element. Proof This is a consequence of the general machinery developed in the paper [14], which proves a general norm-compatibility statement for elements defined by means of a "pushforward map of Cartesian cohomology functors" in the sense of §2.3 of op.cit., which is a formalism designed specifically for applications to the cohomology of Shimura varieties and other symmetric spaces. More precisely, we take the groups G and H of op.cit. to be the Q p -points of the groups G and H of the present paper; then the motivic cohomology groups of the Shimura varieties for G and H , and the pushforward maps ι [a,b,r ,s] U , between them, described in §9.1 (for varying levels U ), satisfy the axioms for a pushforward map of the required type. (Compare the case of étale cohomology treated in [14, §3.4]).
So we may apply the machinery of §4 of op.cit., with the parabolic subgroups Q G and Q H taken to be the Borel subgroups B G and B H , and open-orbit representative u taken be the one denoted by the same letter in Lemma 2.5.1 above. Then the first assertion of the theorem is exactly Proposition 4.5.2 of op.cit.; and the second assertion of the theorem follows from the first using (8.2.a).

Remark 10.2.4
Since the operator U p is invertible in the Hecke algebra of level Ih p , this shows that the classes σ t p (U p ) −t [a,b,r ,s] mot,m, p t for varying t and m form a "motivic Euler system" over all the abelian extensions E[m p t ], for m ∈ R and t ≥ 1. However, these classes typically will not have bounded denominators with respect to t in the étale realisation, as will become clear from the analysis below.
As noted above, these classes extend naturally to the canonical integral model of , which we denote by Y p t . Their étale realisations are also integral in another, separate sense: namely, they arise from an integral lattice in the coefficient sheaf, as we now explain. We suppose δ S lies in I(G S /K G,S , O E,(p) ); and we choose an integer c > 1 coprime to 6 pS. (2) for all t ≥ 0 and m ∈ R coprime to c, such that:

Theorem 10.2.5 (Wild norm relation, integral étale form) There exists a collection of elements
(a) the image of z t after inverting p and restricting to the generic fibre is (

exactly, not just modulo torsion).
Proof The integrality of these classes follows by the same argument as Proposition 9.5.2, with a slight modification: we now need to consider ξ = ch(gK G ) where g is not a unit at p, so the pushforward g : O E,p . However, we are taking g p to be a unit multiple of τ t , and the denominator of (τ t ) (which corresponds to the action of τ −t on D a,b ) is bounded by p (r +s)t , which is exactly the normalising factor appearing in the definition of the classes. The fact that these classes are norm-compatible again follows from the normcompatibility machine developed in [14], applied to the integral étale cohomology of the two Shimura varieties, rather than motivic cohomology as in Theorem 10.2.2.
Note that the groups H 3 ét Y t , D a,b O E,p {r , s}(2) are finitely-generated over O E,p (this is an advantage of working with the integral model Y t ). In particular, the operator e p = lim k→∞ U p k! is defined on these spaces, and acts as an idempotent. So we

Automorphic Galois representations
We recall some results on automorphic Galois representations of GL 3 /E, following [2]. Let be a RAECSDC automorphic representation of GL 3 /E; and for each prime w of E such that w is unramified, let P w ( , X ) ∈ C[X ] denote the polynomial such that P w ( , Nm(w) −s ) −1 = L( w , s).
The coefficients of the polynomials P w ( , X ) lie in a finite extension F of E independent of w; and for each place P | p of F , there is a 3-dimensional F ,P -linear representation V P ( ) of Gal(Ē/E), uniquely determined up to semisimplification, with the property that if w is a prime not dividing p for which w is unramified, we have Remark 12.1.2 If we fix and let p vary, then [30,Theorem 2] shows that there is a density 1 set of rational primes p such that V P ( ) is irreducible for all P | p (and hence unique up to isomorphism).

Weights
Since is regular algebraic, it has a well-defined weight at each embedding τ : E → F , which is a triple of integers a τ,1 ≥ a τ,2 ≥ a τ,3 (see [2, §1]). Since c is a twist of ∨ , a τ,i + aτ ,4−i is independent of i. Thus, up to twisting by an algebraic Grössencharacter if necessary, we can (and do) assume that the weight of is (a+b, b, 0) at the identity embedding, and (a+b, a, 0) for the conjugate embedding, for some integers a, b ≥ 0. Definition 12.1. 4 We say is ordinary at the prime p | p (with respect to the prime P | p of F ) if the polynomial P p ( , q X) has a factor (1 − α p X ) with v P (α p ) = 0.
A standard argument using p-adic Hodge theory (see [2,Lemma 2.2]) shows that is ordinary at p if and only if V P ( ) has a 1-dimensional subspace invariant under Gal(E p /E p ) with the Galois group acting on this subspace by an unramified character. If this holds, then dually V P ( ) * has a codimension 1 subspace F 1 p V P ( ) * , such that V P ( ) * /F 1 p is unramified, with arithmetic Frobenius Frob p acting on this quotient by α p .

Remark 12.1.5
Since is conjugate self-dual up to a twist, one checks that V P ( ) has a 1-dimensional invariant subspace at p if and only if it has a 2-dimensional invariant subspace atp. So if is ordinary at all the primes above p, then V P ( ) and its dual preserve a full flag of invariant subspaces at each prime above p. (We will not use this fact directly in the present paper, but it may be relevant to future work relating the Euler system constructed here to Selmer groups and p-adic L-functions.)

Realisation via Shimura varieties
We add the further assumption that V P ( ) be irreducible. We now realise this representation in the étale cohomology (with compact support) of the infinite-level Shimura variety Let π be the automorphic representation of G corresponding to (and some choice of ω such that ( , ω) is RAECSDC) as in Theorem 2.6.3. Proof The computation of the intersection cohomology IH 2 ét of the Baily-Borel compactification of the Picard modular surface is the main result of the volume [12]; see in particular §4.3 of [25] for an overview. This computation shows that the intersection cohomology has a direct summand isomorphic to V P ( ) ⊗ π f . There is a natural map from H 2 ét,c of the open modular surface to IH 2 ét of the compactification; and the Hecke eigensystems appearing in the kernel and cokernel of this map are associated to non-cuspidal automorphic representations of GL 3 /E. So the map is an isomorphism on the generalised eigenspace for the spherical Hecke algebra associated to π f , which gives the result.
We can thus interpret any v ∈ π f as a homomorphism of Galois representations V P ( ) → lim − →K H 2 ét,c , or dually as a homomorphism which we can consider as a "modular parametrisation" of the Galois representation V P ( ) * . This homomorphism factors through projection to Y G (K ) for any level K which fixes v.

An Euler system for V P (5)
We now choose the following data: • A finite S of primes, an open compact K G,S ⊆ G(Q S ), and an element δ S ∈ I(G S /K G,S , Z), as in Section 9.3; • A non-zero vector v ∈ π f stable under the group K G,S · Ih p · G(Ẑ S∪{ p} ).
• An integer c coprime to 6 pS.
We suppose that is ordinary above p, and we let α p = p| p α p where α p is as in Definition 12.1.4. Then the generalised U p -eigenspace of (π p ) Ih p with eigenvalue α p is 1-dimensional, where U p denotes the double-coset operator [Ih p τ Ih p ] acting on the Ih p -invariants (this is easily checked from the explicit formulae for Whittaker functions in §6; compare [16, §3.5.5] in the GSp 4 case). We shall choose v to lie in this eigenspace. Then the projection map pr ,v factors through the U p = α p eigenspace, and hence through the ordinary idempotent e p of Sect. 10.2. m p ∞ , for all m ∈ R coprime to pc, taking values in the e p -ordinary part of ). Moreover, these classes all land in a lattice independent of m. The modular parametrisation map pr ,v sends this lattice in H 2 ét (Y Ih,Q , D a,b E p (2)) to a lattice in V P ( ) * , and we take T P ( ) * to be this lattice. Then we may define We now prove the properties (i)-(iii). Property (i) follows from the tame norm relation Eq. 9.4.a, but the argument is a little delicate. Since v ∈ π f is unramified outside S ∪ {p}, the homomorphism pr ,v factors through the eigenspace where the Hecke-algebra-valued polynomial P w (X ) acts as P w ( , X ) for all w pS. So for all t, and passing to the inverse limit, we deduce that h is annhilated by a finite power of p. Since the Iwasawa cohomology of an infinite p-adic Lie extension is p-torsion-free, we must have h = 0, which proves part (i) of the theorem. The remaining properties are somewhat simpler. For property (ii), we use the compatibility with moment maps (Corollary 11.3.1), and we note that for any η of ∞-type (s, r ) and conductor dividing m p t , the twist V P ( ) * ⊗ η −1 can be realised as a direct (2)), exactly as in the case of Heegner points described in §3.4 of [8]. (The switch in ordering of r and s arises because the character μ : G → Res E/Q GL 1 corresponds to μ 4 , not μ 3 , in our parametrisation of algebraic weights.) Finally, the local Selmer condition (iii) at the primes above p follows from part (ii), since any class in the image of motivic cohomology must lie in the Bloch-Kato H 1 g subspace at primes above p; and this subspace projects to 0 in the cohomology of the quotient (compare [16, Proposition 11.2.2]).

Concluding remarks
Remark 12.4.1 The Euler system of Theorem B depends on choices of local data at the primes in S: the vector v ∈ π f defining the modular parametrisation, and the element δ S ∈ I(G S /K G,S , Z). It should be possible to check that the Euler systems obtained for different choices of these data are proportional to each other, with the proportionality factor being essentially the local zeta integral of Sect. 6; compare [19, §6.6].

A.2.3. Second key lemma
The second key lemma is about the support of certain Hecke operators.
Proof Our proof is inspired by the proof of [10]. We proceed by considering the -adic valuations of values of various weight functions in Z[GL 3  .., j m }, i 1 < i 2 < · · · < i m , j 1 < · · · j m .