Euler systems for GSp(4)

We construct an Euler system for Galois representations associated to cohomological cuspidal automorphic representations of the group GSp(4), using the pushforwards of Eisenstein classes for GL(2) x GL(2).


Introduction
The theory of Euler systems is one of the most powerful tools available for studying the arithmetic of global Galois representations. However, constructing Euler systems is a difficult problem, and the list of known constructions is accordingly rather short. In this paper, we construct a new example of an Euler system, for the four-dimensional Galois representations associated to cohomological cuspidal automorphic representations of GSp 4 /Q, and apply this to studying the Selmer groups of these Galois representations. Our construction relies crucially on an unexpected relation with branching problems in smooth representation theory, which is the key input in proving the norm-compatibility relations for our Euler system classes.
We construct this Euler system in theétale cohomology of the Shimura variety of GSp 4 . The strategy that we use for this construction is also applicable to many other examples of Shimura varieties, including those associated to the groups GU(2, 1), GSp 6 , and GSp 4 × GL 2 , which will be explored in forthcoming work.
The starting-point for our construction is a family of motivic cohomology classes for Siegel threefolds, which were introduced and studied by Francesco Lemma in the papers [Lem10,Lem15,Lem17]. Lemma's classes are constructed by using the subgroup H = GL 2 × GL1 GL 2 inside GSp 4 . Beilinson's Eisenstein symbol gives a supply of motivic cohomology classes for the Shimura varieties attached to H, and pushing these forward to GSp 4 gives motivic cohomology classes for the Siegel threefold. By applying theétale realisation map and projecting to an appropriate Hecke eigenspace, Lemma's motivic classes give rise to elements of the groups H 1 (Q, W * Π (−q)), where Π is a suitable automorphic representation of GSp 4 , W * Π the dual of the associated p-adic Galois representation, and q is an integer in a certain range depending on the weight of Π.
To build an Euler system for these representations W * Π , we need to modify this construction in order to obtain classes defined over cyclotomic fields Q(ζ m ). These classes are required to satisfy an appropriate Instead, we prove a version of this result after mapping to Galois cohomology. We choose Π a suitably nice cohomological automorphic representation of GSp 4 such that Π K f = 0. (We need Π to be generic for almost all , which excludes certain "endoscopic" representations such as Saito-Kurokawa lifts.) Then Π * f ⊗ W * Π appears with multiplicity 1 as a direct summand of lim − →U H 3 et (Y G (U ) Q , D), and does not contribute to cohomology outside degree 3. Choosing a vector ϕ ∈ Π f thus gives a homomorphism of Galois representations Combining this with the Hochschild-Serre spectral sequence gives a map of vector spaces . We thus obtain a collection of cohomology classes z Π M,m ∈ H 1 (Q(µ M p m ), W * Π ), depending on the choice of ϕ, and we shall prove the norm-compatibility relations for these instead.
For simplicity, assume that M = 1 and m = 0, so we are trying to compare z Π 1,0 with norm(z Π ,0 ) (the general case can be reduced to this by twisting). We have constructed a G(A f )-equivariant bilinear pairing • We write T for the diagonal torus of G, which is equal to the product A × T , where A, T are the tori defined by A = x y x y , T = x x 1 1 .
• Let H = GL 2 × GL1 GL 2 (fibre product over the determinant map), and let ι denote the embedding H → G given by f ( a b 0 d g) = χ(a)ψ(d)|a/d| 1/2 f (g), equipped with a GL 2 (Q )-action via right translation of the argument.
We will frequently need to use analytic continuation in an auxiliary parameter s. The following construction will be helpful: Definition 3.1.3. A polynomial section of the family of representations I(χ| · | s , ψ| · | −s ) is a function on GL 2 (Q ) × C, (g, s) → f s (g), such that g → f s (g) is in I(χ| · | s , ψ| · | −s ) for each s ∈ C, and s → f s (g) lies in C[ s , −s ] for every g ∈ GL 2 (Q ). A section is flat if its restriction to GL 2 (Z ) is independent of s.
If χ/ψ = |·| −1 we interpret the right-hand side as 0, so the elements f φ,χ,ψ all land in the 1-dimensional subrepresentation. Let us evaluate these integrals explicitly for some specific choices of φ, assuming now that χ and ψ are unramified characters.
Note that φ t is preserved by the action of the group K 0 ( t ) = a b c d ∈ GL 2 (Z ) : c = 0 mod t .
Proof. The computation of the value at the identity is immediate. The assertion regarding the support of the function is vacuous for t = 0, and for t > 1 we have φ t = 1−t 0 0 1 φ 1 , so in fact it suffices to prove the assertion for t = 1; in this case, we simply observe that the function f φ1,χ,ψ vanishes on the long Weyl element w = 0 1 −1 0 , since φ t (0, x)w = φ t (−x, 0) = 0 for all x.
3.3. Notation: subgroups of G and H . We now define an assortment of open compact subgroups of G and H . We represent elements of G in block form ( A B C D ), where A, B, C, D are 2 × 2 matrices. • K G = G(Z ).
(The subgroups K G ( m , n ) will not be used until §8.4 below.) We define subgroups K H , K H ,0 ( n ), etc of H as the preimages (via ι) of the corresponding subgroups of G . We write dg and dh for the Haar measures on G and H normalised such that K G (resp. K H ) has volume 1.
3.5.1. Principal series representations of G . We follow the notations of [RS07] for representations of G . See op.cit. for further details (in particular §2.2 and the tables in Appendix A).
Definition 3.5.1. Let χ 1 , χ 2 , ρ be smooth characters of Q × such that We let χ 1 × χ 2 ρ denote the representation of G afforded by the space of smooth functions f : with G acting by right translation. We refer to such representations as irreducible principal series.
This representation has central character χ 1 χ 2 ρ 2 ; the condition (2) implies that it is irreducible and generic. If η is a smooth character of Q × , then twisting χ 1 × χ 2 ρ by η (regarded as a character of G via the multiplier map) gives the representation χ 1 × χ 2 ρη.
3.5.2. Hecke operators. Firstly, we consider the action of the spherical Hecke algebra H(K G \G /K G ) on σ K G when σ is an unramified principal series representation.
Secondly, we consider the larger space of invariants under the Siegel parahoric subgroup K G ,0 ( ). We let U ( ) denote the Hecke operator x.
3.6. Zeta integrals for G. In this section we isolate the key local zeta integral calculations used in our proofs of the tame norm relations.
for all ϕ ∈ σ, a ∈ A, and u, v, w ∈ Q .
If σ is irreducible, then the space of λ-Bessel functionals on σ has dimension 1, by [RS16, Theorem 6.3.2]. It is clearly zero unless λ| Z(G ) coincides with the central character of σ.
Theorem 3.6.3 (Roberts-Schmidt). If σ is an irreducible generic representation of G (such as an irreducible principal series representation), then σ admits a non-zero λ-Bessel functional µ λ for every λ whose restriction to Z(G ) agrees with the central character of σ. If both σ and λ are unramified, then we may normalise µ λ such that µ λ (ϕ 0 ) = 1, where ϕ 0 is the spherical vector of σ.
Proof. For the existence of the Bessel functional see [RS16,Proposition 3.4.2]. It is shown in op.cit. that the Bessel functional can be explicitly constructed by integrating functions in the Whittaker model of σ; and the assertion that in the unramified case the spherical vector maps to 1 under this functional follows, for example, from the computations of [RS07, §7.1].
Proposition 3.6.4. Let σ be an irreducible representation of G admitting a λ-Bessel model, and let ϕ ∈ σ be invariant under N S (Z ). Let us define Then we have Proof. We first note that the assumption that the Bessel function B ϕ,λ is fixed by right-translation by N S (Z ), and transforms on the left via (3), implies that B ϕ,λ ( ).
If |x| > k then the Bessel function is zero; and if k |x| > 1, then the sum of the e terms vanishes. This leaves the cases |x| 1, in which case the terms e (xv) are all equal to 1 and we obtain the result.
3.6.2. Novodvorsky's integral. In order to construct an intertwining operator between σ and a principalseries H-representation, we shall use an integral involving a choice of vector in the Bessel model of σ, for some choice of character λ as above. For ϕ ∈ σ, η an unramified character of Q × , and s ∈ C, we define Here L(σ ⊗ η, s) is the spin L-factor of σ ⊗ η, as in Definition 3.5.2.
Remark 3.6.5. This integral apparently first appears in [Nov79,Equation 2.7]. In an earlier draft of the present paper, we mistakenly ascribed this construction to Sugano; in fact Sugano's paper [Sug85] considers a related but slightly different integral -see Remark 3.7.4 below.
Proposition 3.6.6. Suppose σ is an irreducible unramified principal series representation, with central character χ σ , and let η be an unramified character. Let λ be given by The integral defining Z(ϕ, η, λ, s) is absolutely convergent for (s) 0, and it has analytic continuation to all s ∈ C as an element of C[ s , −s ]. (b) If ϕ 0 is the spherical vector (normalised so that B ϕ0,λ (1) = 1) then we have for any v, w ∈ Q and a, b, t ∈ Q × .
Proof. Replacing σ with σ ⊗ η, and (λ 1 , λ 2 ) with (ηλ 1 , ηλ 2 ), we may assume η is trivial. It suffices to prove (a) under the assumption that ϕ = g · ϕ 0 for some g ∈ G (since these vectors span σ). The validity of (a) for this vector will only depend on the class of g in the double coset space R(Q )\G /K G . A set of coset representatives for this double quotient, and a formula for the values of B ϕ0,λ on these representatives, is given in [Sug85, Proposition 2-5]; see also [BFF97, Corollary 1.9] for an alternative, slightly more concrete formulation. The result now follows by an explicit calculation, which also gives (b) as a special case (compare also [PS09, §3.2]). Finally, part (c) is obvious from the integral formula if (s) 0, and follows for all s by analytic continuation.
From Proposition 3.6.4 above, we see that This formula will be fundamental to the proof of our Euler system norm relations later in the paper.
Lemma 3.7.2. The image of the homomorphism z is contained in the unique irreducible subrepresentation of I H (ψ −1 , χ −1 ).
Proof. If L(ψ 1 /χ 1 , s + 1 2 )L(ψ 2 /χ 2 , s + 1 2 ) is finite at s = 1 2 , then I H (ψ −1 , χ −1 ) is irreducible and there is nothing to prove. So it suffices to treat the case when one or both of χ i /ψ i is | · |, assuming that L(σ ⊗ ψ 1 ψ 2 , s) has no pole at s = 1 2 . We shall not give the details of this computation, as it is somewhat technical, and it will only be relevant in a few boundary cases. We write σ as an induced representation from the Siegel parabolic P S (Q ). There are exactly two orbits of H on the flag variety G /P S (Q ), and an application of Mackey theory allows us to compute Hom H (σ, τ ) for each non-generic quotient τ of I H (ψ −1 , χ −1 ) in terms of the inducing data for σ. These Hom-spaces all turn out to be zero unless L(σ ⊗ ψ 1 ψ 2 , s) has a pole at s = 1 2 .
Corollary 3.7.3. Let ·, · denote the canonical duality pairing Then the bilinear form z χ,ψ ∈ Hom H I(χ, ψ) ⊗ σ, C defined by for f s any polynomial section of I(χ s , ψ s ) passing through f , is well-defined and non-zero.

Proof.
We have M (f s ), z s (ϕ) = f s , M (z s (ϕ)) by Proposition 3.1.5. From the previous lemma, M (z s (ϕ)) vanishes at s = 0 to order equal to the order of the pole of the Euler factor, so the limit is well-defined and depends only on f .
If η and λ are chosen as above, then one checks that z χ,ψ fφ ,χ,ψ ⊗ ϕ is equal to the leading term of Z(ϕ, φ, Λ, η, s) at s = 1 2 , up to a non-zero scalar factor. However, we cannot simply take this as the definition of z χ,ψ , since it is not a priori clear that this leading term depends only on the vector fφ ,χ,ψ ∈ I H (χ, ψ) rather than on φ itself.
Proof. Let τ be the representation I H (χ, ψ). Since the group of unramified characters of Q × is 2-divisible, we may replace σ and τ with σ ⊗ ω −1 and τ ⊗ ω, where ω is any square root of χ σ , and therefore assume that both σ and τ are trivial on Z(G ). Thus σ factors through G = PGSp 4 (Q ) = SO 5 (Q ); and τ factors through the image H of H in SO 5 (Q ), which is a copy of SO 4 (Q ), embedded as the stabiliser of an anisotropic vector in the defining 5-dimensional representation. We now apply the main theorem of [KMS03], which shows that for any representations σ of SO 5 and τ of SO 4 which are generated by a spherical vector, the Hom-space Hom(τ ⊗ σ, C) has dimension 1.
Remark 3.7.6. Alternatively, it follows from the proof of Lemma 3.7.2 that this Hom-space injects into Hom H (σ ⊗ τ 0 , C) where τ 0 is the unique irreducible subrepresentation of I H (χ, ψ). We can now invoke a very general result, which forms part of the Gan-Gross-Prasad conjecture for special orthogonal groups: for any n 0 and any irreducible smooth representations σ of SO n+1 (Q ) and ρ of SO n (Q ), one has dim Hom SOn (σ ⊗ ρ, C) 1, by [Wal12, Théorème 1].
Corollary 3.7.7. In the situation of Theorem 3.7.5, the bilinear form z χ,ψ is a basis of Hom H I H (χ, ψ)⊗ σ, C .
3.8. Explicit formulae for the unramified local pairing. We record the following formulae for the values of z χ,ψ . We assume, as before, that σ is an irreducible unramified principal series representation of G . We choose our characters χ and ψ as follows: is a pair of finite-order unramified characters, and k i 0 are integers. If one or both of the k i is zero, we also assume that σ is essentially tempered (a twist of a tempered representation); since χ σ is | · | −(k1+k2) up to a finite-order character, all poles of L(σ, s) therefore have real part k1+k2 2 0, so that L(σ ⊗ ψ 1 ψ 2 , 1 2 ) = L(σ, − 1 2 ) is finite and the assumptions of the previous section are satisfied.
Theorem 3.8.1. Let z ∈ Hom H (I(χ, ψ) ⊗ σ, C). Then, for any t 1, we have Proof. We know that Hom H I(χ, ψ) ⊗ σ, C is 1-dimensional and spanned by the specific bilinear form z χ,ψ constructed above, so it suffices to assume that z = z χ,ψ . By construction F φ for any ϕ ∈ σ invariant under K H ,0 ( t ). In particular, if ϕ = ϕ 0 then the bracketed term is identically 1, and from the formula for f φ t ,χ,ψ (1) given in Lemma 3.2.5 we see that for t 1 we have which is the first formula claimed. The second formula is similar, using the formula for z s (U ( )ϕ 0 )(1) given in Proposition 3.7.1.
3.9. An application of Frobenius reciprocity.
Proposition 3.9.1. Let τ (resp. σ) be smooth representations of H and G respectively. Then there are canonical bijections of C-vector spaces Proof. The first isomorphism is standard, and interchanging the roles of σ and the compactly-induced representation also shows that

One has a canonical isomorphism (c-Ind
(Care must be taken here since the contragredient on the left-hand side denotes G -smooth vectors in the abstract vector-space dual, while on the right-hand side it denotes H -smooth vectors.) We then apply Frobenius reciprocity for the non-compact induction [op.cit, §III.2.5] to obtain as required.
Remark 3.9.2. The Hom-spaces in Proposition 3.9.1 will not in general be isomorphic to Hom H τ, (σ ∨ )| H . The problem is that (σ ∨ )| H is in general much smaller than (σ| H ) ∨ , since the two notions of contragredient do not match -an H -smooth linear functional on σ may not be G -smooth.
For later use it will be important to have an explicit form for this bijection. Let H(G ) denote the Hecke algebra of locally-constant, compactly-supported C-valued functions on G , with the algebra structure defined by convolution (normalising Haar measure as in §3.3). We regard σ as a left H(G )-module via the usual formula Definition 3.9.3. For smooth representations τ , σ as above, let X(τ, σ ∨ ) denote the space of linear maps where the actions are defined as follows: • The H factor acts trivially on σ ∨ , and on τ ⊗ C H(G ) it acts via the formula • The G factor acts trivially on τ , and on H(G ) it acts via g · ξ = ξ((−)g).
Corollary 3.9.5. Suppose z ↔ Z as in the above proposition; and let U 0 U 1 be two open compact subgroups of G , f 0 , f 1 ∈ τ , and g 0 , g 1 ∈ G . Suppose that as elements of (σ ∨ ) U0 , where R (g) = R(g −1 ).
Proof. Since both sides of the desired equality are in (σ ∨ ) U0 = (σ U0 ) ∨ , it suffices to check that they pair to the same value with ϕ for every ϕ ∈ σ U0 . This now follows from the above description of Z(−)(ϕ).

3.10.
Results for deeper levels. In order to prove norm-compatibility relations in the "p-direction" for our Euler system, we shall also need a few supplementary results which are proved directly (rather than using the uniqueness result of Theorem 3.7.5). In this section, W denotes an arbitrary smooth complex representation of G (not necessarily irreducible or even admissible), and we let X(W ) denote the space of homomorphisms Z : S(Q 2 , C) ⊗2 ⊗ C H(G ) → W satisfying the same equivariance property under H × G as in Definition 3.9.3.
Lemma 3.10.2. Let ξ ∈ H(G ) be invariant under left-translation by the principal congruence subgroup of level T in H(Z ), for some T 1. Then, for any Z ∈ X(W ), the expression where Vol(−) denotes volume with respect to our fixed Haar measure on H .
Proof. For any integers t T 1, let J be a set of coset representatives for the quotient We can (and do) assume that J is a subset of the principal congruence subgroup of level T in H. By assumption, all such elements will act trivially on ξ from the left, so the above equality becomes Notation 3.10.3. We write Z φ 1,∞ ⊗ ξ for this limiting value.
The case that interests us is the following. Let m, n be integers with m 0 and n max(m, 1), and we consider the Hecke operator Proposition 3.10.4. For any Z ∈ X(W ) we have K .
There are now two cases to consider. If m 1 then all terms in this sum are actually equal, since the powers of η m+1 are conjugate via elements of the form a a 1 1 (with a ∈ 1 + m Z ), which are in K and act trivially on the Schwartz function φ 1,n ; so the sum is simply Z φ 1,∞ ⊗ ch(η m K) as required.
If m = 0, then − 1 of the terms are conjugate, but the term for u = −1 requires special consideration since 1 + m u = 0; thus we obtain which proves the formula in this case also.
We also have an analogous result for n = 0, under rather stricter hypotheses. We take for W the smooth dual σ ∨ of an essentially tempered, unramified principal series representation of G . We shall suppose that Z ∈ X(σ ∨ ) factors through a certain induced representation of H: more precisely, we shall take pairs of characters χ = (χ 1 , χ 2 ) and ψ = (ψ 1 , ψ 2 ) of Q × with ψ 1 = ψ 2 = | · | −1/2 and χ i = | · | ki+1/2 τ i for finite-order characters τ i and positive integers k i , so our setup is similar to Theorem 3.8.1 except that we do not assume the τ i to be unramified. We then have a natural map and we suppose that Z factors through this map.
Corollary 3.10.5. In this situation, we have . In particular, if the τ i are not both unramified, then this holds vacuously (both sides of the formula are zero).
Proof. Since K G ,0 ( ) fixes ch(K G ) on the left, we have On the other hand, taking m = 0 and n = 1 in Proposition 3.10.4, we have Since the action of the quotient K G ,0 ( )/K G ,1 ( ) ∼ = GL 2 (Z/ ) commutes with the Hecke operator U ( ), we can sum over representatives for the quotient to deduce that Combining these two formulae we have We can now quickly dispose of the ramified cases. The map φ → F φ is a morphism of (GL 2 × GL 2 )representations (not only of H-representations). Moreover, the elements φ 0 and φ 1 are the characteristic functions of subsets of Q 2 × Q 2 invariant under Z × × Z × ; hence their images in any representation of GL 2 × GL 2 with ramified central character must be zero. Hence F φ 0 and F φ 1 are both zero, and the desired formula becomes 0 = 0, if either of the characters τ i is ramified.
Let us now assume that the τ i are unramified, which means we may apply Theorem 3.8.1. Translating to the homomorphism Z from its corresponding bilinear form z, the first statement in the theorem (for t = 1) becomes . On the other hand, the second statement of Theorem 3.8.1 gives us Combining these two formulae, the "extra" Euler factors coming from the τ i cancel out, and we are left with the desired formula.

Preliminaries II: Algebraic representations and Lie theory
4.1. Representations of G. We recall the parametrization of algebraic representations of the group GSp 4 .
Notation 4.1.1. We write T for the diagonal torus of G (as in §2 above), and we write χ 1 , . . . , χ 4 for characters of T given by projection onto the four entries. Thus χ 1 + χ 4 = χ 2 + χ 3 is the restriction to T of the symplectic multiplier µ, and {χ 1 , χ 2 , µ} is a basis of the character group X • (T ).
Definition 4.1.2. Let a 0, b 0 be integers. We denote by V a,b the unique (up to isomorphism) irreducible algebraic representation of G whose highest weight, with respect to B, is the character (a + b)χ 1 + aχ 2 .
This representation has dimension 1 6 (a + 1) . Note that V 0,1 is the four-dimensional defining representation of GSp 4 , and V 1,0 is the 5-dimensional direct summand of 2 V 0,1 . The representation V 1,0 ⊗ µ −1 has trivial central character, and is the defining representation of G/Z G ∼ = SO 5 . 4.2. Integral models. Let λ = (a + b)χ 1 + aχ 2 + cµ, with a, b 0, be a dominant integral weight, V λ the corresponding representation, and v λ a highest weight vector in V λ . The pair (V λ , v λ ) is then unique up to unique isomorphism.
An admissible lattice in V λ is a Z-lattice L with the following properties: It is known that there are finitely many such lattices, each of which is the direct sum of its intersections with the weight spaces; and we set V λ,Z to be the maximal such lattice.
Proposition 4.2.1. Let λ, λ be dominant integral weights. Then there is a unique G-equivariant homomorphism, the Cartan product, Proof. After tensoring with Q the existence and uniqueness of this homomorphism is obvious from highest-weight theory. Hence the image of V λ,Z ⊗V λ ,Z is a Z-lattice in V λ+λ , which is clearly admissible; so it must be contained in the maximal one, which is V λ+λ ,Z .
This product gives the ring λ V λ,Z the structure of a graded ring. The Borel-Weil theorem shows that this ring injects into O(G), which is an integral domain; so the Cartan product of non-zero vectors is non-zero. 4.3. Branching laws. We are interested in the restriction of V a,b to H via the embedding ι : H → G, which we shall denote by ι * (V a,b ). Computing the weights of these representations (and their multiplicities), one deduces the following branching law describing ι * (V a,b ): For the constructions below it will be useful to fix choices of highest-weight vectors in each of these subrepresentations. For 0 q a and 0 r b we define a vector v a,b,q,r ∈ V a,b Z as follows: where the product operation is the Cartan product, and: • v ∈ V 0,1 is the highest-weight vector; • w is the highest-weight vector of V 1,0 .
Remark 4.3.3. We can identify V 0,1 with the standard representation of GSp 4 ⊆ GL 4 , with basis (e 1 , . . . , e 4 ), by choosing the highest-weight vector v = e 1 ; of course we then have v = e 2 . Moreover, we can identify V 1,0 with a subspace of 2 V 0,1 , by choosing e 1 ∧ e 2 for the highest-weight vector w; and it follows that w is the vector e 1 ∧ e 4 − e 2 ∧ e 3 . Proposition 4.3.4. For all integers 0 q a and 0 r b, the vector v a,b,q,r thus defined is a non-zero highest-weight vector for the unique irreducible Proof. Since v a,b,q,r is a Cartan product of non-zero H-highest-weight vectors (i.e. vectors fixed by the action of the unipotent radical of the Borel of H), it is itself a non-zero H-highest-weight vector, and thus generates an irreducible H-subrepresentation of V a,b . The result now follows by comparing weights.
Since the representation W c,d of H has a canonical highest-weight vector (namely e c 1 f d 1 , where (e 1 , e 2 ) and (f 1 , f 2 ) are bases of the standard representations of the two GL 2 factors), we therefore have a canonical homomorphism of H-representations mapping the highest-weight vector to v a,b,q,r . We refer to these homomorphisms as branching maps.
Proposition 4.3.5. The maps br [a,b,q,r] restrict to maps Z is a lattice in W a+b−q−r,a−q+r ⊗ det q stable under the action of H, and since v a,b,q,r ∈ V a,b Z , the intersection of this lattice with the highest-weight subspace contains the highest-weight vector e a+b−q−r 1 f a−q+r 1 . Hence this lattice must contain the minimal admissible lattice in W a+b−q−r,a−q+r .

4.4.
A Lie-theoretic computation. As in §2 above, let T ⊂ T be the rank-1 split torus and let u be the element Since T is split, the representations V a,b are the direct sums of their weight spaces relative to T , with weights between 0 and 2a + b; and the T -weight of v a,b,q,r is 2a + b − q. The purpose of this section is to prove the following result, which will be used in §9.5: The vectors v, v , and w all lie in the highest T -weight subspaces of their parent representations, so they are fixed by u. Hence it suffices to check that the projection of u h (w ) to the highest T -weight subspace of V 1,0 is non-trivial; and one computes easily that u h (w ) = w + 2hw.

Modular varieties
5.1. Modular curves. We fix conventions for modular curves.  We identify Y (N )(C) with the double quotient is the principal congruence subgroup of level N and H is the upper half-plane, in such a way that: • The double coset of (1, τ ), for τ ∈ H, corresponds to the triple • The right-translation action of g ∈ GL 2 (Ẑ) on the double quotient corresponds to the action on Y (N )(C) given by (E, e 1 , e 2 ) · g = (E, e 1 , e 2 ), e 1 e 2 = g −1 · e 1 e 2 .
If g ∈ SL 2 (Z/N Z) then the above action of g on Y (N )(C) coincides with the action of γ −1 on H, for any γ ∈ SL 2 (Z) congruent to g modulo N . The components of Y (N )(C) are indexed by the set µ • N of primitive N -th roots of unity, via the Weil pairing (E, e 1 , e 2 ) → e 1 , e 2 N ; and the induced action of g ∈ GL 2 (Z/N Z) on µ • N is given by g · ζ = ζ 1/ det(g) . Remark 5.1.2. Note that our model is not the Deligne-Shimura canonical model of the Shimura variety for GL 2 with its standard Shimura datum [Mil05, Example 5.6]. Rather, it is the canonical model for the twisted Shimura datum defined by which has the effect of flipping the sign of the Galois action on the connnected components.
By passage to the quotient, we define similarly algebraic varieties Y (U ) over Q, for every open compact The right-translation action gives isomorphisms η : Y (U ) → Y (η −1 U η) for every η ∈ GL 2 (A f ), which are compatible with the action of η −1 on H if η ∈ GL + 2 (Q); in particular, scalar matrices ( A 0 0 A ) with A ∈ Q × act trivially. This structure allows us to view the inverse limit Y = lim ← −U Y (U ) as a pro-variety over Q with a right action of GL 2 (A f ), whose C-points are GL + 2 (Q)\ (GL 2 (A f ) × H). Definition 5.1.3. We say U ⊂ GL 2 (A f ) is sufficiently small if every non-identity element of U acts without fixed points on the set GL + 2 (Q)\(GL 2 (A f ) × H). This condition is equivalent to requiring that for all g ∈ GL 2 (A f ), every non-identity element of the discrete group Γ = GL + 2 (Q) ∩ gU g −1 acts without fixed points on H. For instance, U (N ) is sufficiently small if N 3. If U is sufficiently small, then Y (U ) is the solution to a moduli problem (classifying elliptic curves with appropriate level structure), and therefore has an associated universal elliptic curve We define similarly algebraic surfaces Y H (U ), where U is an open compact subgroup of H(A f ), and by passage to the limit a pro-variety is the fibre product of the modular curves Y (U 1 ) and Y (U 2 ) over their common component set Z × / det(U 1 ) = Z × / det(U 2 ).
If U is sufficiently small (in the same sense as for GL 2 ), then Y G (U ) is smooth, and can be interpreted as a moduli space for abelian surfaces with level U structure. As before, the inverse limit lim ← −U Y G (U ) acquires a right action of G(A f ).

The embedding of
The map ι U is not always injective (even if U is sufficiently small). However, we have the following criterion: Since w is central in H, its action on Y G fixes Y H pointwise. Thus, if Q ∈ Y H and Qu ∈ Y H , we have Quw = Qu = Qwu, soũ = u · wu −1 w fixes Q. This elementũ lies inŨ , by hypothesis, and sinceŨ is sufficiently small, we conclude thatũ = 1. Thus u lies in the centraliser of w in G(A f ), which is exactly H(A f ).
We shall say a subgroup U is H-small if it satisfies the hypotheses of the above proposition. For instance, if U is contained in the principal congruence subgroup U G (N ) for some N 3, then U is H-small (since U G (N ) is normal in G( Z), and sufficiently small by [Pin90, §0.6]).

Component groups and base extension.
Via strong approximation for Sp 4 , we have an isomorphism of component sets Our moduli-space description of Y G determines a Galois action on these components as follows.
Definition 5.4.1. We write Art : Q × + \A × f → Gal(Q/Q) ab for the Artin reciprocity map of class field theory, normalised such that for x ∈ Z × ⊂ A × f , Art(x) acts on roots of unity as ζ → ζ x (and hence uniformizers map to geometric Frobenius elements).
Proposition 5.4.2. All components of Y G (C) are defined over the cyclotomic extension Q ab = Q(ζ n : n 1), and the right-translation action of u ∈ G(A f ) on π 0 (Y G (C)) coincides with the action of the Galois automorphism Art(µ(u) −1 ).
Proof. This is an instance of Deligne's reciprocity law for the action of Galois on the connected components of any Shimura variety; see e.g. [Mil05,§13].
We will be particularly interested in the following special case. If U ⊂ G(A f ) is an open compact subgroup, and V N is the subgroup of Z × defined by {x : x = 1 mod N } for some integer N , then there is an embedding of Q-varieties which is an isomorphism if µ(U ) surjects onto (Z/N Z) × . This map intertwines the action of g ∈ G(A f ) on the left-hand side with that of (g, σ) on the right-hand side, where σ is the image of Art(µ(g) −1 ) in Gal(Q(ζ N )/Q). If S is a Z[U ]-module, then S can be considered as a locally constantétale sheaf of abelian groups over Y (U ). Note that the sections of S over Y (V ), for any V U open, are canonically isomorphic to S V , and the pullback action of u ∈ U/V on H 0 (Y (V ), S ) is identified with the native left action of U/V on S V . This construction extends in the obvious fashion to profinite modules S, and in particular to continuous representations of U on finite-rank Z p -modules; via passage to the isogeny category we may also allow S to be a Q p -vector space.
If the action of U on S extends to some larger monoid M ⊆ GL 2 (A f ) containing U , then the sheaf S naturally becomes a M-equivariant sheaf. That is, for every σ ∈ M, giving a morphism of varieties Y (U ) σ -Y (σ −1 U σ), we have morphisms σ * (S ) → S , where S and S are the sheaves on Y (U ) and Y (σ −1 U σ), respectively, corresponding to S; and these morphisms satisfy an appropriate cocycle condition. This construction equips the cohomology groups H * (Y (U ), S ) with an action of the Hecke algebra H(U \M/U ). Exactly the same theory applies, of course, to the modular varieties Y G (U ) and Y H (U ), and these constructions are compatible via ι: the pullback functor ι * onétale sheaves corresponds to restriction of representations from G to H. 6.2. Sheaves corresponding to algebraic representations. As we have seen above, the modular curves Y (U ), for U sufficiently small, are moduli spaces: Y (U ) parametrises elliptic curves E equipped with a U -orbit of isomorphisms E tors ∼ = (Q/Z) 2 . Thus Y (U ) comes equipped with a universal elliptic curve E . From the description of the action of GL 2 (Z/N Z) on the moduli problem, one deduces the following compatibility: Lemma 6.2.1. Suppose U ⊆ GL 2 (Ẑ). For N 1, the sheaf E [N ] of N -torsion points of E is canonically isomorphic to the sheaf associated to the dual of the standard representation of GL 2 (Z/N Z).
Let p be prime. Taking N = p r and passing to the limit over r shows that the relative Tate module T p E corresponds to the dual of the standard representation of GL 2 (Z p ). On the other hand, T p E is a lattice in the p-adicétale realisation of a "motivic sheaf" -a relative Chow motive -over Y (U ), namely This is the first instance of a more general phenomenon. Let G temporarily denote any of the three groups {GL 2 , GL 2 × GL1 GL 2 , GSp 4 }; and let U be a sufficiently small open compact in G (A f ), so we have an associated Shimura variety Y G (U ). from the category of representations of G over Q to the category of relative Chow motives over Y G (U ) with the following properties: • Anc G ,U preserves tensor products and duals; • if µ denotes the multiplier map G → G m , then Anc G ,U (µ) is the Lefschetz motive Q(−1); • if V denotes the defining representation of G , then Anc G ,U (V ) = h 1 (A U ), where A U is the universal PEL abelian variety over Y G (U ); • for any prime p and G -representation V , the p-adic realisation of Anc G ,U (V ) is theétale sheaf associated to V ⊗ Q p , regarded as a left U -representation via U → G (A f ) G (Q p ).
Remark 6.2.3. In fact Ancona's construction is much more general, applying to arbitrary PEL Shimura varieties, but we shall only need the above three groups here. The theorem stated in op.cit. is slightly different from ours, since he normalises his functor to send the multiplier representation to Q(1), and the defining representation to h 1 (A ) ∨ ; our functor is obtained from his by composing with the automorphism of Rep(G ) induced by the map g → µ(g) −1 g on G .
We shall need some "naturality" properties of Ancona's construction, which we now recall. Proof. Since Anc G ,U (V ) for a general V is defined as a direct summand of a tensor power of h 1 (A U ), one reduces easily to checking this functoriality property for the specific relative motives h 1 (A U ). By a standard argument (see e.g. [Del69, Prop 3.3]) one can interpret Y G (U ) as a moduli space for abelian varieties up to isogeny, from which we can deduce that there is a canonical isomorphism in the isogeny category of abelian varieties over Y G (U ). The functor h 1 (−) extends to the isogeny category of abelian varieties, so λ σ induces an isomorphism of relative Chow motives. Moreover, the λ σ satisfy a cocycle condition for varying σ (which we leave it to the reader to formulate explicitly).
Remark 6.2.5. Note that λ σ comes from a "genuine" isogeny if and only if the matrix of σ (in the defining matrix representation of G ) has entries inẐ. To fix conventions, note that if U = U and σ = diag(x, . . . , x) for some x ∈ Q ×+ , then the map λ σ is given by multiplication by x.
An equivalent way of stating Proposition 6.2.4 is as follows. We define a G (A f )-equivariant relative Chow motive over Y G to be the data of a relative Chow motive V U over Y G (U ) for each sufficiently small open U ⊂ G(A f ), together with a collection of isomorphisms σ * (V U ) ∼ = V U for each inclusion σ −1 U σ ⊆ U , compatible with composition. These objects form a category CHM(Y G ) G (A f ) , and the proposition states that the functors Anc G ,U for varying U assemble into a functor (Note that the isomorphisms Anc G ,U (µ) ∼ = Q(−1) are not compatible with the equivariant structure, since the isogenies λ σ only preserve the polarisation up to a scalar; as equivariant motives we have Anc G (µ) = Q(−1)[−1], where the notation [−1] denotes that the equivariant structure is twisted by the character µ −1 of G(A f ).) If V is a G (A f )-equivariant relative Chow motive over Y G , we can define its motivic cohomology by and this is naturally a smooth representation of G (A f ). As motivic cohomology with rational coefficients satisfies Galois descent (see e.g. [DS91, §1.3]), for each sufficiently small U we can recover H * mot (Y G (U ), V U ) as the U -invariants of the direct limit (6). Finally, we shall need to show a compatibility with respect to changing G : Proposition 6.2.6 ("Branching" for motivic sheaves). Let G = GSp 4 and H = GL 2 × GL1 GL 2 , as in §2 above. Then 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.
Proof. This is an instance of a general theorem due to Torzewski [Tor18, Corollary 9.8], which verifies the above naturality property for a wide class of homomorphisms of PEL-type Shimura data. Proposition 7.1.1. There is a canonical,

Eisenstein classes for
with the following characterising property: if φ is the characteristic function of (a, b) + NẐ 2 , for some N 1 and (a, b) ∈ Q 2 − N Z 2 , then g φ is the Siegel unit g a/N,b/N in the notation of [Kat04, §1.4].
In order to work integrally, we need to modify the construction somewhat. Let c > 1 be an integer.
f ) 2 , and Z c = |c Z . Then we have the following refinement:

Higher Eisenstein classes.
Definition 7.2.1. For k 0, let H k Q denote the GL 2 (A f )-equivariant relative Chow motive over Y associated to the representation Sym k (std) ⊗ det −k of GL 2 /Q.
mot,φ , the motivic Eisenstein symbol, with the following property: the pullback of the de Rham realization r dR Eis k mot,φ to the upper half-plane is the H k -valued differential 1-form is the Eisenstein series defined by Remark 7.2.3. Note that if φ is the characteristic function of (0, b) + NẐ 2 , then Eis k mot,φ is the class Eis k mot,b,N defined in [KLZ19, Theorem 4.1.1]. If k = 0, then we need to assume φ ∈ S 0 (A 2 f , Q) in order for the series defining F (2) φ to be absolutely convergent. With this assumption, we may define Eis 0 mot,φ to be the unit g φ , since H 1 mot (Y, Q(1)) = O × (Y ) ⊗ Q; the de Rham realisation of this class is then dlog g φ = −F (2) φ · (2πi dτ ), so our statements are consistent. Since we lack a good theory of relative Chow motives with coefficients in Z, we do not have an integral version of the motivic Eisenstein classes for k > 0. However, we can obtain a Z p -structure (for a fixed p) usingétale cohomology instead. For each U we have anétale realisation map for any prime p, where H * et denotes continuousétale cohomology in the sense of [Jan88], and H k Qp is the lisseétale Q p -sheaf which is the p-adic realisation of H Q . This is naturally the base-extension to Q p of theétale Z p -sheaf H k Zp associated to the minimal admissible lattice in the GL 2 -representation Sym k (std) ⊗ det −k .
Proposition 7.2.4. Let k 0. If c is coprime to 6p, then for each sufficiently small open compact which is equivariant for the action of GL 2 A (pc) f × Z pc , and satisfies Note thatétale cohomology with Z p coefficients does not satisfy Galois descent, so it is important to formulate Proposition 7.2.4 for each individual level, rather than simply passing to the direct limit.
Proof. For levels of the form U (N ) this is explained in [Kin15], and the arguments apply without change to a general U .
regarded as a representation of GL 2 (A f ) by right translation. For k = 0 and η = 1, let I 0 0 (1) denote the subrepresentation which is the kernel of the natural map I 0 (1) → C given by integration over characterised by the statement that if g ∈ O × (Y ), then ∂ 0 (g)(1) is the order of vanishing of g at the cusp ∞.
such that ∂ k (x)(1) is the residue at ∞ of the 1-form r dR (x). This map is an isomorphism on the image of the Eisenstein symbol φ → Eis k mot,φ . Proof. It is shown in [SS91,Theorem 7.4] that the residue map ∂ k gives an isomorphism between the image of the Eisenstein symbol and a certain vector space denoted B k . For the description of this B k as a sum of induced representations, see [Lem17,Lemma 4.3].
For η a character of A × f /Q ×+ as above, let us write S(A 2 f , C) η for the subspace of S(A 2 f , C) on which Z × acts via the character η. Proposition 7.3.4. Let φ ∈ S(A 2 f , C) η be of the form prime φ . If k = 0 and η = 1 then assume we have φ(0, 0) = 0. Then we have Proof. This follows from a computation of the constant term of the Eisenstein series F : The actions of H(A f ) × G(A f ) for which this map is equivariant are given as follows: • The H(A f ) factor acts trivially on the right-hand side of (8), and on the left-hand side it acts via the formula h · (x ⊗ ξ) = (h · x) ⊗ ξ(h −1 (−)).
• On the left-hand side, the G(A f ) factor acts trivially on H 2 mot Y H , H c,d Q (2) , and on H(G(A f ); Z) it acts via g · ξ = ξ((−)g).
• On the right-hand side, G(A f ) via its natural action on H 4 mot Y G (U ), D a,b Q (3 − q) (deduced from Proposition 6.2.4) twisted by the character µ(−) −q . For ease of reading we shall drop the superscripts [a, b, q, r] for the rest of this section.
Proof. This is immediate from the fact that the principal congruence subgroup U G (N ), for any N 3, is H-small, and these are cofinal among open compact subgroups of G(A f ).
For an element x ⊗ ch(U g) as in Lemma 8.2.1, we have a closed immersion ι gU : Combining this with the morphism of sheaves br [a,b,q,r] gives a map We define ι * (x ⊗ ch(gU )) as the image of the element in the direct limit (6). (Here Vol(V ) is volume with respect to Haar measure, normalised such that Vol H(Ẑ) = 1.) Now, suppose U ⊂ U is another H-small open compact subgroup, so that ch(gU ) = γ∈U/U ch(gγU ). We want to show that ι * (x⊗ch(gU )) = γ∈U/U ι * (x⊗ch(gγU )) for any x invariant under V . It suffices to prove this when U U (since otherwise we may compare both U and U with a third open compact U normal in both U and U ); we may clearly also assume g = 1.
Let V = U ∩ H(A f ); we then have degeneracy maps pr U U : By the functoriality of the pushforward maps, we have Pulling back from level U to the direct limit over all levels, we can write this as It follows that ι * is well-defined on Definition 8.3.1. We define the Lemma-Eisenstein map is as in (7). (h) considered in [Lem15], for S = Y G (U ), (m, n) = (a − q + b − r, a − q + r), W the representation V a,b , and h an appropriate element of Lemma's space B m ⊗ B n depending on φ. In particular, it follows from the regulator computations of [Lem17, §7] that the Lemma-Eisenstein map is non-zero under fairly mild hypotheses on a, b, q, r.
8.4. Choices of the local data. We shall now fix choices of the input data to the above map LE [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, according to the values of three parameters M, m, n. 8.4.1. Subgroups of tame level 1. We fix a prime p, a finite set of primes S not containing p, and an arbitrary open compact subgroup K S ⊂ G(Q S ) = ∈S G(Q ). By enlarging S and shrinking K S if necessary, we may assume that the open compact subgroup is sufficiently small in the sense of §5.2. For each n 0, we define an compact subgroup of G(A f ) by where the subgroup K Gp,0 (p n ) of G p = G(Q p ) is as defined in §3.3. We define similarly subgroups K G,1 (p n ) for n 0, and K G (p m , p n ) for m, n 0, using the other local subgroups at p defined in §3.3. All of these groups are contained in K G , and hence are sufficiently small.
Notation 8.4.1. We adopt the notational convention that if K G, ( ) denotes some open subgroup of G(A f ), then Y G, ( ) denotes the corresponding Shimura variety, so e.g. Y G,1 (p n ) is an abbreviation for Y G (K G,1 (p n )).

8.4.2.
Local data at the bad primes. We choose the following "test data" at S: Whenever we deal with norm-compatibility relations we shall assume that the local data K S , W S , φ S remains fixed (i.e. we shall not attempt to formulate any non-trivial norm-compatibilities at the bad primes). Regarding the choice of φ S , see Remark 10.6.4 below. 8.4.3. Subgroups of higher tame level. Now let us choose a square-free integer M 1 coprime to S ∪ {p} (which we shall refer to as a "tame level"). For m 0 and n 1, we define a subgroup K G (M, p m , p n ) ⊆ K G (p m , p n ) by K G (M, p m , p n ) = {k ∈ K G (p m , p n ) : µ(k) = 1 mod M }.
As explained in §5.4, we have isomorphisms Assuming n m, we also define K G (M, p m , p n ) = {k ∈ K G (p m , p n ) : µ(k) = 1 mod M }; note that the difference between this and K G (M, p m , p n ) is only at p -we do not impose stronger congruences at M . 8.4.4. Test data of higher level. Let (M, m, n) be integers as above. For each such triple, we shall define the following data: • an element ξ M,m,n ∈ H(G(A f ), Z), fixed by the right-translation action of K G (M, p m , p n ); • a subgroup W of H(A f ), such that for all x in the support of ξ M,m,n , we have We shall define these as products where the local data K S , W S , φ S at the bad places are the ones chosen above (independently of M, m, n), and the local data at primes / ∈ S are as follows. As in §3, we let η ,r ∈ G(Q ) denote the element • If M p, we set ξ = ch (G(Z )), W = H(Z ), and φ = ch(Z 2 ) ⊗2 . • If | M , we set ξ = ch(K G ( , 1)) − ch (η ,1 · K G ( , 1)), and W = K H ( , 2 ). We take φ = ch 2 Z × (1 + 2 Z ) ⊗2 .
• For = p, we set ξ p = ch η p,m · K Gp (p m , p n ) . We choose an integer t 1 sufficiently large 2 that K Hp (p m , p t ) is contained in η p,m · K Gp (p m , p n ) · η −1 p,m ; we let W p be this subgroup, and we define Note that φ M,m,n ∈ S 0 (A 2 f , Z) ⊗2 ⊂ S(A 2 f , Z) ⊗2 , since our local Schwartz functions at p vanish at (0, 0). Both the element φ M,m,n , and the group W , depend on the auxilliary choice of t; but if we let t • > t be another choice, and φ • M,m,n , W • the objects defined using t • in place of t, then we have We also define a version mildly modified at p, assuming that n 1 and m n. Recall the subgroup K Gp (p m , p n ) defined in §3.3. We define ξ p = ch(η p,0 K Gp (p m , p n )) = ch(K Gp (p m , p n )η p,0 ). Thus ξ p is preserved under left-translation by W p = K Hp (p m , p n ); and we choose φ p = ch (p n Z p × (1 + p n Z p )) ⊗2 .
We let K G (M, p m , p n ), ξ M,m,n , φ M,m,n and W be the adèlic objects defined using these modified choices at p, and the same choices as before at all other primes.
Remark 8.4.2. These alternative local choices will give elements related to the "non-dashed" versions in the same way as the elements Z ... relate to the elements Ξ ... in [LLZ14]. As in op.cit., it is the nondashed versions which are of interest for applications, but the dashed versions are convenient for certain calculations, in particular for studying p-adic integrality and interpolation properties.
We refer to these elements as Lemma-Eisenstein classes. A priori this element depends on the auxilliary integer t, but it follows readily from (10) that it is in fact independent of this choice (this is essentially the same computation as Lemma 3.10.2). It can be written concretely as follows: letting U be the subgroup K G (M, p m , p n ), we can write our Hecke-algebra element ξ M,m,n as a finite Z-linear combination of characteristic functions ch(x i U ). For each of these terms, if we set U i = x i U x −1 i , then by hypothesis we have W ⊆ V i := H ∩ U i , and we can consider the composition of maps (Note that U i may not be H-small, so ι Ui : Y H (V i ) → Y G (U i ) may not be a closed immersion, but it is still a finite morphism of smooth varieties and this suffices to define the pushforward map ι (ii) For m 0, we have M,m,n .
Here U (p) ∈ H(G p ) is given by the K Gp (p m , p n )-double coset of p −1 p −1 1 1 .
2 One can check that t = n + 2m suffices.
Proof. Part (i) of the theorem is immediate from the definition of the classes, since the sum of the translates of ξ M,m,n+1 over K G (M, p m , p n )/K G (M, p m , p n+1 ) is ξ M,m,n . For part (ii), we note that the Hecke-algebra elements ξ M,m,n and ξ M,m+1,n are identical outside p, as are the Schwartz functions φ M,m,n and φ M,m+1,n . So we need to compare two values of a map on S(Q 2 p , Q) ⊗2 ⊗ H(G(Q p )) (given by tensoring with the common away-from-p parts and applying LE). It clearly suffices to check the equality after tensoring with C, which puts us in a position where we may apply Proposition 3.10.4 (for = p). If we assume that the parameters t are chosen identically for the two elements, then the proposition shows that we have The factor of 1 p (resp. 1 p−1 ) is cancelled out by the factors 1 Vol(W ) , since the subgroups W corresponding to the classes at level M p m+1 and M p m differ in volume by exactly this factor. Finally, the twist [−q] gives a factor of p q . 8.4.6. Integral p-adicétale classes. We now treat questions of integrality. We choose integers c 1 , c 2 > 1 satisfying the following list of conditions: • The c i are coprime to 6p ∈S .
• Our chosen vector φ S ∈ S(Q 2 S , Z) ⊗2 is preserved by the action of the elements ( c1 1 ) , ( c1 1 ) −1 and ( c2 1 ) , ( c2 1 ) −1 of (GL 2 × GL 2 )(Q S ). (Note that these elements are not in H.) • For each ∈ S, the subgroup K is normalised by the elements To see that this is well-defined, we use the explicit description of the Lemma-Eisenstein map as a sum of pushforward maps as in (11). Since ξ M,m,n is a Z-linear combination of cosets x i U (where Cf. [KLZ17, §6.1]. The morphism s m is well-defined on the Shimura varieties, since we have To construct the morphism s m, of integral coefficient sheaves, we note that the representation D a,b of G has all weights 0 for the torus T of §2, so the action of G(Z p ) on D a,b Zp extends to an action of the monoid generated by G(Z p ) and whose image in the cohomology of D a,b (The factor p mq appears because we ignored the twist [−q] in the definition of s m, .) However, our assumptions on the c i imply that we have However, one can check that these norm relations actually hold integrally, without needing to quotient out by the torsion subgroup of theétale cohomology group. This is not obvious from the proofs we have given, but can easily be verified after carefully unwinding the normalisation factors.

Moment maps and p-adic interpolation
We now study the interpolation of theétale Euler system classes, for varying values of the parameters (a, b, q, r). Our goal is Theorem 9.6.4, which shows that these classes can all be obtained as specialisations of a single class "at infinite level". 9.1. Interpolation of the GL 2 Eisenstein classes. We begin by recalling a theorem of Kings [Kin15], which will be the fundamental input for our p-adic interpolation results. In this section, let us fix an arbitrary open compact subgroup K (p) ⊂ GL 2 (A (p) f ), and for n 1, write K n = K (p) × {g ∈ GL 2 (Z p ) : g ∼ = ( * * 0 1 ) mod p n }. Let us assume, by shrinking K (p) if necessary, that K 1 is sufficiently small (and hence so is K n for all n 1).
We also choose a finite set of primes Σ containing p and all primes where K (p) is ramified, so the modular curves have models Y (K n ) Σ over Z[1/Σ] for all n.
Remark 9.1.1. Working with integral models is necessary here, because continuousétale cohomology for Q-varieties does not necessarily commute with inverse limits, but this problem does not arise for finite-type Z-schemes such as the Y (K n ) Σ .
Definition 9.1.2. We define where the inverse limit is with respect to the pushforward maps.
If H k n denotes the mod p n reduction of the sheaf H k Zp on Y (K n ) (cf. §7.2), then we have a canonical section e k,n = (e 1 ) k ∈ H 0 et Y (K n ) Σ , H k n . Hence, for any n 1, we have a map Definition 9.1.3. Let φ be a Z p -valued Schwartz function on (A (p) f ) 2 , stable under K (p) ; and let φ s = φ ⊗ ch(p s Z p × (1 + p s Z p )). For n 1, and c > 1 coprime to 6p and to all primes where φ is ramified, we define The following theorem, which will be fundamental for our p-adic interpolation results later in this paper, shows that the Siegel units interpolate Eisenstein classes of all weights via these moment maps: Theorem 9.1.4 (Kings). For all integers k 0 and n 1, we have mom k n ( c EI φ,n ) = c Eis ḱ et,φn Proof. This is a generalisation of [KLZ17, Theorem 4.4.4 & Theorem 4.5.1], which is the case where φ is the characteristic function of (0, 1) + N ( Z (p) ) 2 for some integer N . The general case can be recovered from this using the action of the group J = ∈Σ−{p} GL 2 (Q ), since both the Siegel units and Eisenstein classes depend J-equivariantly on φ, and the moment map clearly commutes with the action of J (as it acts trivially on e k,s ).
f ) unramified outside Σ, and write K G,n = K (p) G × K Gp,1 (p n ). We assume that K G,n is sufficiently small for all n 1. , which is true by the construction of the branching maps.
Remark 9.3.2. In [KLZ17], the analogous statment for the GL 2 × GL 2 -moment maps (Lemma 6.3.1) gives rise to a binomial factor; so using the Cartan product simplifies matters considerably. 9.4. Application to Lemma-Eisenstein classes. We now return to the situation considered in §8.4. We can apply the machinery of the previous section with K (p) G taken to be the product K S × / ∈S∪{p} G(Z ), so the K G,n of the previous section is K G,1 (p n ), and we obtain moment maps mom [a,b,q,r] G,n for any Σ ⊇ S ∪ {p}. More generally, if we take any m 0 and any squarefree M 1 whose prime divisors lie in Σ − (S ∪ {p}), the same construction also gives maps Zp (3 − q) , for any n 1, and for any n max(m, 1), which we also denote by mom for all integers a, b, r, n with a q, 0 r b and n max(m, 1).
Remark 9.4.2. Strictly speaking the elements c1,c2 Z q Iw,M,m depend also on Σ, but they are easily seen to be compatible with the natural maps given by enlarging Σ, so shall suppress this from the notation.
Proof. We define c1,c2 Z q Iw,M,m to be the sequence c1,c2 Z We can therefore write Proof. The moment maps of (12a) and (12b) are compatible with respect to the pushforwards (s m ) * , because the action of diag(p −1 , p −1 , 1, 1) on D a−q,b Zp fixes the vector d [a−q,b,0,r] . So for n m the corollary follows from Proposition 9.4.1 by applying (s m ) * to both sides; and since both sides of the desired formula are norm-compatible in n, the result follows for n < m also. 9.5. Cyclotomic twists. We now consider the more difficult problem of interpolating the classes of the previous sections as the parameter q varies, analogously to [KLZ17, Theorem 6.2.4] in the Rankin-Selberg setting. Our main technical result will be the following: Proposition 9.5.1. For each m 1 and q 0, we have (1, p m , p m ), Z/p m (1)) is the canonical p m -th root of unity given by the isomorphism (9). In an attempt to restrain the excessive proliferation of indices in our notations, we shall give the arguments assuming M = 1, and drop M from the subscripts throughout the remainder of the section; the case of general M can be handled similarly (using the decomposition of ξ M,m,n as a finite sum of characteristic functions of cosets, as in the proof of Proposition 9.4.1). In this case, we have c1,c2 Z q Iw,m := η * • ι ∞, * • br [q,0,q,0] c1,c2 EI φ ∈ H 4 Iw Y G (p m , p ∞ ), D q,0 Zp (3 − q) .
The branching map br [q,0,q,0] appearing in the above constructions is given by mapping 1 ∈ Z p to the H(Z p )-invariant element d [q,0,q,0] ⊗ ζ −q ∈ D q,0 Zp (−q), where ζ denotes a basis of the multiplier representation µ of G. After reducing modulo p m , this element is invariant under a larger group: Proposition 9.5.2. The modulo p m reduction d [q,0,q,0] m is stable under K Gp (p m , p ∞ ) ⊂ G(Z p ).
Proof. Since K Gp (p m , p ∞ ) is contained in the principal congruence subgroup modulo p m , it acts trivially on D a,b m for any a, b. It follows that we may write where η * d Zp as a direct sum of its eigenspaces for the action of the torus T , which all have weights 0. On all eigenspaces other than the weight 0 eigenspace, the map s m, is zero, since diag(p, p, 1, 1) −m acts as a positive power of p m , which annihilates the module D q,0 m . Hence s m, factors through projection to the highest weight space relative to T . So we need to compute the projection of η * (d [q,0,q,0] m ) = (η −1 ) * (d [q,0,q,0] m ) to this weight space. This is precisely the situation of Lemma 4.4.1 (with h = −1 in the notation of the lemma), which gives the result above.
Proposition 9.5.1 follows immediately from this, by applying s m, * to both sides of (13). 9.6. Projection to the ordinary part. We now define a limiting element in which m (as well as n) goes to ∞. We set On this module, there is an action of the ordinary idempotent e ord = lim k→∞ U (p) k! .
Remark 9.6.1. The fact that this limit exists, and is an idempotent, follows from the corresponding statements forétale cohomology at finite levels, for which see [TU99]. Note that Tilouine and Urban define multiple ordinary idempotents, one for each standard parabolic subgroup; ours is the one associated to the Siegel parabolic P S .
10.1. Automorphic representations of GSp 4 . Let (k 1 , k 2 ) be integers with k 1 k 2 3, and let (a, b) = (k 2 − 3, k 1 − k 2 ). There are exactly two unitary discrete-series representations of G(R) which are cohomological with coefficients in the algebraic representation V a,b : the holomorphic discrete series Π H k1,k2 , and a non-holomorphic generic discrete series Π W k1,k2 . We refer to these as the discrete series representations of weight (k 1 , k 2 ). The cuspidal automorphic representations with infinite component Π H k1,k2 are precisely those generated by classical holomorphic Siegel modular forms of weight (k 1 , k 2 ). Remark 10.1.1. The pair {Π H k1,k2 , Π W k1,k2 } is an example of a local L-packet. Definition 10.1.2. Let Π = Π f ⊗ Π ∞ be a cuspidal automorphic representation of G(A f ) with Π ∞ discrete series of weight (k 1 , k 2 ).
• We say Π is of Saito-Kurokawa type if k 1 = k 2 and there exists a Dirichlet character χ such that χ 2 = ω Π , and a cuspidal automorphic representation of GL 2 (A) of central character ω Π attached to some holomorphic newform of weight 2k 1 − 2, such that for all but finitely many places v we have L(Π v , s) = L(π v , s)L(χ v , s − 1 2 )L(χ v , s + 1 2 ). • We say Π is of Yoshida type if there is a pair (π 1 , π 2 ) of cuspidal automorphic representations of GL 2 (A), both with central character ω Π , corresponding to two elliptic modular newforms of weights r 1 = k 1 + k 2 − 2 and r 2 = k 1 − k 2 + 2, such that for all but finitely many places v we have L(Π v , s) = L(π 1,v , s)L(π 2,v , s).
Theorem 10.1.3 (Taylor, Weissauer, Urban, Xu). Let Π be as in the previous definition, and suppose Π is non-endoscopic. Let S be the set of primes at which Π ramifies, and let w = k 1 + k 2 − 3.
(2) For any prime / ∈ S, the local representation Π is an unramified principal series representation.
(3) For / ∈ S, let P (X) ∈ C[X] denote the quartic polynomial such that L(Π , s − w 2 ) = P ( −s ) −1 . Then the subfield E ⊂ C generated by the coefficients of the P (X), for all / ∈ S, is a finite extension of Q.
(4) For any prime p and choice of embedding E → Q p , there is a semi-simple Galois representation ρ Π,p : Gal(Q/Q) → GL 4 (Q p ) characterised (up to isomorphism) by the property that, for all primes / ∈ S ∪ {p}, we have det 1 − Xρ Π,p (Frob −1 ) = P (X), where Frob is the arithmetic Frobenius. (5) The representation ρ Π,p is either irreducible, or is the direct sum of two distinct irreducible two-dimensional representations. In particular, we have H 0 Q ab , ρ Π,p = 0.
Proof. Parts (3) and (4) are [Wei05, Theorem I]. Part (2) is also implicit in this theorem, since the "purity" statement on ρ Π,p implies that the local L-factor L(Π , s) has the form III] under the assumption that Π is weakly equivalent to a globally generic representation; and in fact all such Π have this property by the main theorem of [Wei08]. The assertion regarding D cris is [Urb05, Theorem 1]. Finally, Xu has shown in [Xu18, §3.5] that the common multiplicity of Π H and Π W is equal to 1.
It is expected that ρ Π,p is always irreducible, but this is only known for large p: Theorem 10.1.5 (Ramakrishnan). If Π is unramified at p and p > 2w + 1, then the representation ρ Π,p is irreducible.