Rankin--Eisenstein classes in Coleman families

We show that the Euler system associated to Rankin--Selberg convolutions of modular forms, introduced in our earlier works with Lei and Kings, varies analytically as the modular forms vary in $p$-adic Coleman families. We prove an explicit reciprocity law for these families, and use this to prove cases of the Bloch--Kato conjecture for Rankin--Selberg convolutions.


Introduction
Let p > 2 be a prime. The purpose of this paper is to study the p-adic interpolation ofétale Rankin-Eisenstein classes, which are Galois cohomology classes attached to pairs of modular forms f, g of weights 2, forming a "cohomological avatar" of the Rankin-Selberg L-function L(f, g, s).
In a previous work with Guido Kings [KLZ15b], we showed that these Rankin-Eisenstein classes for ordinary modular forms f, g interpolate in 3-parameter p-adic families, with f and g varying in Hida families and a third variable for twists by characters. We also proved an "explicit reciprocity law" relating certain specialisations of these families to critical values of Rankin-Selberg L-functions, with applications to the Birch-Swinnerton-Dyer conjecture for Artin twists of p-ordinary elliptic curves, extending earlier works of Bertolini-Darmon-Rotger [BDR15a,BDR15b].
In this paper, we generalise these results to non-ordinary modular forms f, g, replacing the Hida families by Coleman families: Theorem A. Let f, g be eigenforms of weights 2 and levels N f , N g coprime to p whose Hecke polynomials at p have distinct roots; and let f α , g α be non-critical p-stabilisations of f, g. Let F , G be Coleman families through f α , g α (over some sufficiently small affinoid discs V 1 , V 2 in weight space).
Then there exist classes c BF [F ,G] m 2.1. Continuous cohomology. We first collect some properties of Galois cohomology of profinite groups acting on "large" topological Z p -modules (not necessarily finitely generated over Z p ). A very rich theory is available for groups G satisfying some mild finiteness hypotheses (see e.g. [Pot13, §1.1]); but we will need to consider the Galois groups of infinite p-adic Lie extensions, which do not have good finiteness properties, so we shall proceed on a somewhat ad hoc basis, concentrating on H 0 and H 1 . With these definitions, the groups C * (G, −) define a functor from topological G-modules to complexes of topological groups (i.e. the topology is functorial in M , and the differentials C i (G, M ) → C i+1 (G, M ) are continuous). Hence the cocycles Z i (G, M ) are closed in C i (G, M ). However, the cochains B i (G, M ) need not be closed in general, so the quotient topology on the cohomology groups H i (G, M ) may fail to be Hausdorff; and the subspace and quotient topologies on B i (G, M ) may not agree. Our next goal is to show that these pathologies can be avoided for i = 1 and some special classes of modules M .
Let A be a Noetherian Banach algebra over Q p . Then any finitely-generated A-module has a unique Banach space structure making it into a Banach A-module [BGR84, Proposition 3.7.3/3]. as a topological A-module. Since C 1 (G, Q p ) ⊕d is orthonormalizable as a Q p -Banach space (every Q p -Banach space has this property), it follows that C 1 (G, M ) is orthonormalizable as an A-Banach module, as orthonormalizability is preserved by base extension. However, B 1 (G, M ) is manifestly finitelygenerated as an A-module, and any finitely-generated submodule of an orthonormalizable A-Banach module is closed [Buz07, Lemma 2.8]. This proves (1). Parts (2) and (3) now follow from the open image theorem [Col98,Proposition I.1.3], which shows that any continuous surjective map between Q p -Banach spaces has a continuous section (and, in particular, a continuous bijection between Q p -Banach spaces must be a homeomorphism).

2.2.
Distributions. For λ ∈ R 0 , we define the Banach space C λ (Z p , Q p ) of order λ functions on Z p as in [Col10a]. This has a Banach basis consisting of the functions p ⌊λℓ(n)⌋ x n for n 0, where ℓ(n) denotes the smallest integer L 0 such that p L > n. We define D λ (Z p , Q p ) as the continuous dual of C λ (Z p , Q p ); for f ∈ C λ (Z p , Q p ) and µ ∈ D λ (Z p , Q p ) we shall sometimes write f dµ for the evaluation µ(f ). The space D λ (Z p , Q p ) has a standard norm defined by Proposition 2.2.1. For any integer h ⌊λ⌋, the standard norm on D λ (Z p , Q p ) is equivalent to the norm defined by Proof. See [Col98], Lemma II.2.5.
As well as the Banach topology induced by the above norms (the so-called strong topology), the space D λ (Z p , Q p ) also has a weak topology 1 , which can be defined as the weakest topology making the evaluation maps µ → f dµ continuous for all f ∈ C λ (Z p , Q p ).
Remark 2.2.2. The weak topology is much more useful for our purposes than the strong topology, since the natural map Z p ֒→ D 0 (Z p , Q p ) given by mapping a ∈ Z p to the linear functional f → f (a) is not continuous in the strong topology, while it is obviously continuous in the weak topology.
More generally, if M is a Q p -Banach space, we define D λ (Z p , M ) = Hom cts (C λ (Z p , Q p ), M ); as before, this has a strong topology induced by the operator norm (which we write as − λ ), and a weak topology given by pointwise convergence on C λ (Z p , Q p ).
Proposition 2.2.3. Let X be a compact Hausdorff space, and M a Banach space, and let σ : X → D λ (Z p , M ) be a continuous map (with respect to the weak topology on D λ (Z p , M )). Then sup{ σ(x) λ : x ∈ X} < ∞.
Proof. For each f ∈ C λ (Z p , Q p ), the map X → M given by x → σ(x)(f ) is continuous, and hence bounded. By the Banach-Steinhaus theorem, this implies that the collection of linear maps {σ(x) : x ∈ X} is bounded in the uniform norm. Lemma 2.2.6. Let (µ n ) n 1 be a sequence of elements of D λ (Z p , M ) which is uniformly bounded (i.e. there is a constant C such that µ n λ C for all n), let µ ∈ D λ (Z p , M ), and let h ⌊λ⌋ be an integer. If we have f dµ n → f dµ as n → ∞ for all f ∈ LP [0,h] (Z p , Q p ), then µ n → µ in the weak topology of D λ (Z p , M ).
Proof. This is immediate from the density of LP [0,h] (Z p , Q p ) in C λ (Z p , Q p ).
Finally, if U is an open subset of Z p , we define D λ (U, M ) as the subspace of D λ (Z p , M ) consisting of distributions supported in U ; this is closed (in both weak and strong topology).
2.3. Cohomology of distribution modules. We now apply the theory of the preceding sections in the context of representations of Galois groups. Our arguments are closely based on those used by Colmez [Col98] for local Galois representations, but also incorporating some ideas from Appendix A.2 of [LLZ14].
We consider either of the two following settings: either K is a finite extension of Q p and G = Gal(K/K); or K is a finite extension of Q and G = Gal(K S /K), where K S is the maximal extension of K unramified outside some finite set of places S including all infinite places and all places above p. In both cases we write H * (K, −) for H * (G, −); this notation is a little abusive in the global setting, but this should not cause any major confusion.
We set K ∞ = K(µ p ∞ ), and H = Gal(K/K ∞ ) (resp. Gal(K S /K ∞ ) in the global case). Thus H is closed in G and the cyclotomic character identifies Γ = G/H with an open subset of Z × p . Remark 2.3.1. More generally, one may take for K ∞ any abelian p-adic Lie extension of K of dimension 1; see forthcoming work of Francesc Castella and Ming-Lun Hsieh for an application of this theory in the context of anticyclotomic extensions of imaginary quadratic fields.
As in section 2.1 above, we let A be a a Noetherian Q p -Banach algebra, and M a finite free A-module with a continuous A-linear action of H; and we fix a choice of norm · M on M making it into a Banach A-module. We shall be concerned with the continuous cohomology H 1 (K ∞ , D λ (Γ, M )), where D λ (Γ, M ) is equipped with the weak topology. Note that this cohomology group is endowed with a supremum seminorm, since every continuous cocycle H → D λ (Γ, M ) is bounded by Proposition 2.2.3.
where µ λ is the supremum seminorm on H 1 (K ∞ , D λ (Γ, M )) and D is a constant independent of K and M .
Proof. For the injectivity, see Proposition II.2.1 of [Col98], where this result is proved for arbitrary It remains to check that the cocycle g → µ(g) is continuous (for the weak topology of D λ (Γ, M )). This is asserted without proof loc.cit., and we are grateful to Pierre Colmez for explaining the argument. Since H is a compact Hausdorff space, it suffices to show that for every convergent sequence g n → g, the sequence µ n := µ(g n ) converges to µ(g) in D λ (Γ, M ). However, by construction we know that f dµ n converges to f dµ for each f ∈ LP [0,h] (Γ, Q p ). Since the µ n are uniformly bounded, Lemma 2.2.6 shows that they converge weakly to µ(g) as required.
We now consider a special case of this statement. We impose the stronger assumption that M is a continuous representation of the larger group G = Gal(K/K) (resp. Gal(K S /K) in the global case), rather than just of H. We equip D λ (Γ, M ) with an action of G by Proposition 2.3.3. Let λ ∈ R 0 , h ⌊λ⌋ an integer, and suppose we are given elements x n,j ∈ H 1 (K ∞ , M ) Γn=χ j , for all n 0 and 0 j h, satisfying the following conditions: • For all n 0, we have γ∈Γn/Γn+1 χ(γ) −j γ · x n+1,j = x n,j .
• There is a constant C such that Then there is a unique element µ ∈ H 1 (K ∞ , D λ (Γ, M )) Γ satisfying x n,j = Γn χ j µ for all n 0 and 0 j h; and there is a constant D independent of K and of M such that where µ λ is the seminorm on H 1 (K ∞ , D λ (Γ, M )) induced by the norm of D λ (Γ, M ).
Proof. We claim first that there is a unique µ alg ∈ H 1 (K ∞ , D This follows from the fact that the functions φ n,j (x) := x j 1 1+p n Zp (x) for n 0 and 0 j h, and their translates under Γ, span the space LP [0,h] (Γ, Q p ).
By Proposition 2.3.2, the existence of the constant C implies that µ alg is the image of a class µ ∈ H 1 (K ∞ , D λ (Γ, M )), which must itself be Γ-invariant since the injection alg (Γ, M )) commutes with the action of Γ. This proposition also shows that µ λ is bounded above by CD.
Using the inflation-restriction exact sequence (and the fact that Γ has cohomological dimension 1) we see that µ lifts to a class in H 1 (K, D λ (Γ, M )). This lift is not necessarily unique, but it is unique modulo H 1 (Γ, D λ (Γ, M GK ∞ )) (and thus genuinely unique if M GK ∞ = 0).
2.4. Iwasawa cohomology. We now show that there is an interpretation of the module H 1 (K, D λ (Γ, M )) in terms of Iwasawa cohomology. Since the group G has excellent finiteness properties (unlike its subgroup H), we have the general finite-generation and base-change results of [Pot13] at our disposal.
We now assume that A is a reduced affinoid algebra over Q p . By a theorem of Chenevier (see [Che,Lemma 3.18]) we may find a Banach-algebra norm on A, with associated unit ball A • = {a ∈ A : a 1}, and a compatible Banach A-module norm on M with unit ball M • ⊂ M , such that G preserves M • and M • is locally free as an A • -module.
Definition 2.4.1. We set This is evidently independent of the choice of lattice M • .
Proposition 2.4.2. The module H 1 Iw (K ∞ , M ) is finitely-generated over D 0 (Γ, A), and there are isomorphisms is Noetherian, and it is complete and separated with respect to the ideal I = (p, [γ] − 1), where γ is a topological generator of Γ/Γ tors ; moreover, D 0 (Γ, Hence [Pot13, Theorem 1.1] applies. By part (4) of the theorem, we see that H 1 (K, D 0 (Γ, M • )) is finitely-generated over D 0 (Γ, A • ). Moreover, part (3) of the theorem shows that and every power I m contains the kernel of D 0 (Γ, A • ) → A[Γ/Γ n ] for all sufficiently large n, so we also have an isomorphism A very slightly finer statement is possible if we consider coefficients in a field: Proposition 2.4.5. Suppose V is a finite-dimensional p-adic representation of G. Then In the local case, this surprisingly nontrivial result is Proposition II.3.1 of [Col98]. The proof relies on local Tate duality at one point, so we shall explain briefly how this can be removed in order to obtain the result in the global case as well.
Firstly, from the finite generation of H 2 Iw (K ∞ , V ) as a Λ(Γ)-module, there exists a k such that H 2 Iw (K ∞ , V (k)) Γ = 0. We may suppose (by twisting) that we have, in fact, H 2 Iw (K ∞ , V ) Γ = 0. Let ν n = (γ − 1) n where γ is a topological generator of Γ, and let T be a lattice in V . Then the submodules H 2 Iw (K ∞ , T )[ν n ] are an ascending sequence of Λ(Γ)-submodules of the finitely-generated module H 2 Iw (K ∞ , T ). Since Λ(Γ) is Noetherian and H 2 Iw (K ∞ , T ) is finitely-generated, we conclude that this sequence of modules must eventually stabilize. But all the modules in this sequence are finite, since H 2 Iw (K ∞ , V ) Γ vanishes by assumption; this implies that there is a uniform power of p (independent of n) which annihilates H 2 Iw (K ∞ , T )[ν n ] for all n 1. (Compare the proof of [LLZ14, Proposition A.2.10], which is a similar argument with ν n = (γ − 1) n replaced by γ p n − 1.) With this in hand we may proceed as in [Col98].
Remark 2.4.6. We do not know if this result is valid for general p-adic Banach algebras (or even for affinoid algebras). It is also significant that the map is not an isometry with respect to the natural norms on either side; there is a denominator arising from the torsion in H 2 Iw (K ∞ , T ), which is difficult to control a priori (and, in particular, could potentially vary as we change the field K in an Euler system argument). We are grateful to Ming-Lun Hsieh for pointing this out. We shall instead control denominators by means of the proposition that follows, in which the denominator depends on an H 0 rather than an H 2 .
Let x n,j be a collection of elements, and C a constant, satisfying the hypotheses of Proposition 2.3.3; and let µ ∈ H 1 (K, D λ (Γ, V )) be the resulting distribution. Then for every character κ of Γ, we have where on the left-hand side · denotes the norm on H 1 (K, V (κ −1 )) for which the unit ball is the image of H 1 (K, T (κ −1 )) (and D is as in Proposition 2.3.2).
Proof. We know that µ λ CD as elements of By the definition of the supremum seminorm, this is equivalent to stating that the class CD κ λ · Γ κ dµ is the image of a class in H 1 (K ∞ , T (κ −1 )). This class is not uniquely determined, and hence not necessarily Γ-invariant; but the constant D ′ was chosen to annihilate the kernel of H 1 (K ∞ , T (κ −1 )) → H 1 (K ∞ , V (κ −1 )), so CDD ′ κ λ · Γ κ dµ lifts to a Γ-invariant class.

Cyclotomic compatibility congruences
In this section, we establish that the Beilinson-Flach cohomology classes constructed in [LLZ14] and [KLZ15b] satisfy the criteria of the previous section, allowing us to interpolate them by finite-order distributions.
3.1. Modular curves: notation and conventions. For N 4, we write Y 1 (N ) for the modular curve over Z[1/N ] parametrising elliptic curves with a point of order N . Note that the cusp ∞ is not defined over Q in this model, but rather over Q(µ N ).
More generally, for M, N integers with M + N 5, we write Y (M, N ) for the modular curve over Z[1/M N ] parametrising elliptic curves together with two sections (e 1 , e 2 ) which define an embedding of group schemes Z/M Z × Z/N Z ֒→ E (so that Y 1 (N ) = Y (1, N )). We shall only consider Y (M, N ) in the case M | N , in which case the Weil pairing defines a canonical map from Y (M, N ) to the scheme µ • M of primitive M -th roots of unity, whose fibres are geometrically connected.
If A is an integer prime to M N , we shall sometimes also consider the curve Y (M, N (A)) over Z[1/AM N ], parametrising elliptic curves with points e 1 , e 2 as above together with a cyclic subgroup of order A.
If Y is one of the curves Y (M, N ) or Y (M, N (A)), we write H Zp the relative Tate module of the universal elliptic curve over Y , which is anétale Z p -sheaf on Y [1/p]. If the prime p is clear from context we shall sometimes drop the subscript and write H for H Zp . We write H Qp for the associated Q psheaf. We write TSym k H Zp for the sheaf of degree k symmetric tensors over H Zp ; note that this is not isomorphic to the k-th symmetric power, although these coincide after inverting p.
Remark 3.1.1. In this paper we will frequently considerétale cohomology of modular curves Y (M, N (A)), or products of pairs of such curves. All the coefficient sheaves we consider will be inverse systems of finiteétale sheaves of p-power order, and we shall always work over bases on which p is invertible. To lighten the notation, the convention that if p is not invertible on Y , then Thirdly, for a ∈ Z/M Z, denote by u a the automorphism of Y (M, N ) 2 which is the identity on the first Y (M, N ) factor and is given by (E, e 1 , e 2 ) → E, e 1 + a N M e 2 , e 2 on the second factor. Definition 3.2.1. For integers M, N 1 with M | N and M + N 5, j 0, a ∈ Z/mZ, p a prime > 2, and c > 1 coprime to 6M N p, define the Rankin-Iwasawa class c RI The primary purpose of introducing the Rankin-Iwasawa class is that it is easy to prove normcompatibility relations for it. Our actual interest is in a second, related class, defined by pushing forward c RI [j] M,N,a via a degeneracy map.
Definition 3.2.2. For integers m 1 and N 4, j 0, a ∈ Z/mZ, and c > 1 coprime to 6mN p, define the Beilinson-Flach class to be the image of c RI which corresponds classically to restriction of cocycles in Galois cohomology. Note 3.3.2. The subvariety Z(m, mN ) is preserved by the action of Γ 1 (N ) × Γ 1 (N ), and in particular by the action of the element u a = (1, ( 1 a 0 1 )) for any a ∈ Z/mZ. Since u a is an automorphism, and its inverse is u −a , we have (u a ) * = (u −a ) * .
There is a canonical section of the sheaf (H Zp ⊠ H Zp )(−1) over the subvariety Z(m, mN ), given by the Weil pairing (since along Z(m, mN ) the two universal elliptic curves coincide). We call this element CG (for "Clebsch-Gordan"), since the Clebsch-Gordan map CG [j] is given by cup-product with the j-th divided power CG [j] of this element. For t 1, we write CG t for the image of CG modulo p t . Note that we have u * a (CG) = CG for any a ∈ Z/mZ, since CG is independent of the level structure.
Let i be the inclusion of Z(m, mN ) into Y (m, mN ) 2 , so the diagonal embedding factors as By construction, the element c RI We now take integers r 1 and h 1 as above, and we assume p ∤ m. We also assume that the following condition is satisfied: Under this assumption, the moment map modulo p hr is given by cup-product with the element Y hr , so we obtain the following somewhat messy formula: Proposition 3.3.4. For any a ∈ Z/mp hr Z, we have the following equality modulo p hr : Theorem 3.3.5. Suppose that p | N . Then for any a ∈ Z/mp hr Z and any m coprime to p, we have Proof. It follows from [KLZ15b,Theorem 5.3.1] that if N ′ is any multiple of N with the same prime divisors as N , then c BF p r ,N ′ ,a under pushforward along the natural degeneracy map Y 1 (N ′ ) → Y 1 (N ). We can therefore assume without loss of generality that N satisfies Assumption 3.3.3.
We may factor the map (t mp r × t mp r ) * as the composite of a map on the coefficient sheaves, which is a morphism (t mp r × t mp r ) ♯ : H ⊠ H → t * mp r (H ) ⊠ t * mp r (H ) of sheaves on Y (mp r , mp r N ) 2 , followed by the pushforward via t mp r × t mp r on the underlying modular curve.
We claim that when restricted to the image of u a • ∆ : This follows from the fact that the map (t mp r ×t mp r ) ♯ is given by quotienting out by the first component of the level structure in each factor: on the fibre at a point ( Since this element is annihilated by (t mp r × t mp r ) ♯ modulo p r , its h-th tensor power is annihilated by the same map modulo p hr . This gives the congruence stated above. The Beilinson-Flach elements for p | N are, by construction, the images of elements of the group and exactly the same argument as above shows that we have a congruence modulo p hr in this group. We will need this below, in order to interpolate our elements in Coleman families.
3.4. Galois representations: notation and conventions. In this section, we shall fix notations for Galois representations attached to modular forms. Let f be a normalised cuspidal Hecke eigenform of some weight k + 2 2 and level N f 4, and let L be a number field containing the q-expansion coefficients of f .
which the Hecke operator T ℓ , for every prime ℓ, acts as multiplication by a ℓ (f ). Dually, we write M L P (f ) * for the maximal quotient of the space (1) ⊗ Qp L P on which the dual Hecke operators T ′ ℓ act as a ℓ (f ). Both spaces M L P (f ) and M L P (f ) * are 2-dimensional L P -vector spaces with continuous actions of Gal(Q/Q), unramified outside S, where S is the finite set of primes dividing pN f . The twist by 1 implies that the Poincaré duality pairing is well-defined (and perfect), justifying the notation. If f is new and f * is the eigenform conjugate to f , then the natural map M L P (f * )(1) → M L P (f ) * is an isomorphism of L P -vector spaces, although we shall rarely use this.
If f , g are two eigenforms (of some levels N f , N g and weights k +2, k ′ +2 2) with coefficients in L, we write M L P (f ⊗g) for the tensor product M L P (f )⊗ L P M L P (g), and similarly for the dual M L P (f ⊗g) * . Via the Künneth formula, we may regard 3.5. Consequences for pairs of newforms. We now use the congruences of Theorem 3.3.5, together with the p-adic analytic machinery of Section 2, in order to define "unbounded Iwasawa cohomology classes" interpolating the Beilinson-Flach elements for a given pair (f, g) of eigenforms.
Remark 3.5.1. We shall prove a considerably stronger result below (incorporating variation in Coleman families) which will mostly supersede Theorem 3.5.9: see Theorem 5.4.2. However, the proof of the stronger result is much more involved, so for the reader's convenience we have given this more direct argument.
Let us choose two normalised cuspidal eigenforms f , g, of weights k + 2, k ′ + 2 and levels N f , N g respectively, with k, k ′ 0. Let L be a number field containing the coefficients of f and g, and P a prime of L above p, so that the Galois representation M L P (f ⊗ g) * of §3.4 is defined. Assume that 0 ≤ j ≤ min{k, k ′ }, and let N be an integer divisible by N f and N g and having the same prime factors , which can be constructed using Beilinson's Eisenstein symbol (and in particular is the image of a class in motivic cohomology). By abuse of notation, we also denote by Eis m,a to be the image of (u a ) * Eis et,1,mN under the following composition of maps: This is independent of the choice of N . For c > 1 coprime to 6mpN f N g , we define Remark 3.5.3. Note that for m = 1 the class m,a is the Eisenstein class AJ f,g,ét Eis Let us recall the connection between these classes and the Iwasawa-theoretic classes of the previous sections. Recall that we have maps for each k j.
We now consider "p-stabilised" versions of these objects. If p ∤ N f , we choose a root α f ∈ L of the Hecke polynomial of f (after extending L if necessary); and we let f α be the corresponding pstabilisation of f , so f α is a normalised eigenform of level N fα = pN f , with U p -eigenvalue α f and the same T ℓ -eigenvalues as f for all ℓ = p. If p | N f , then we assume that a p (f ) = 0, and we set α f = a p (f ) and (for consistency) f α = f and N fα = N f . We define α g and g α similarly.
If p ∤ N f N g , then the class c BF [fα,gα,j] m,a for m coprime to p is related to the Eisenstein class for the forms f, g as follows. There is a correspondence Pr α f : Y 1 (pN f ) → Y 1 (N f ) given by pr 1 − β p k+1 pr 2 , and (Pr α f ) * gives an isomorphism M L P (f α ) * → M L P (f ) * , and similarly for g.
If p | N f but p ∤ mN g , then we have Proof. This is a restatement of Lemma 5.6.4 and Remark 5.6.5 of [KLZ15b].
We shall now interpolate the c BF [fα,gα,j] m,a for varying m and j, under the following assumption: Assumption 3.5.6. The automorphic representations π f and π g corresponding to f and g are not twists of each other.
Note 3.5.7. Assumption 3.5.6 is automatically satisfied if k = k ′ .
Let m be coprime to p and r ≥ 1. Then Assumption 3.5.6 implies that H 0 (Q(µ mp ∞ ), M L P (f ⊗g)) = 0, so the restriction map induces an isomorphism Convention. By abuse of notation, we write c BF for the image of the Beilinson-Flach element in These elements satisfy the following compatibility: Lemma 3.5.8. Let m ≥ 1 be coprime to p, and let r ≥ 0. Then Proof. This follows from the second norm relation for the Rankin-Iwasawa classes (c.f. [KLZ15b, Theorem 5.4.4]).
We impose the following "small slope" assumption: Theorem 3.5.9. If the small slope assumption (3.5.1) holds, then for any integers m 1 coprime to p and a ∈ (Z/mp ∞ Z) × , there exists a unique element Remark 3.5.10. Compare Theorem 6.8.4 of [LLZ14], which is the case k = k ′ = 0.
Proof. This amounts to reorganizing the output of Theorem 3.3.5 and Proposition 2.3.3. Let h = min(k, k ′ ). Consider the composition of maps where e h is the canonical basis of Z p (h) over Q(µ p ∞ ), and the third map is given by (mom k−h · id) ⊠ (mom k ′ −h · id). An unpleasant manipulation of factorials shows that the image of the expression in Theorem 3.3.5 under this composition of maps is equal to where we write y r,j for the quantity where the implied constant in the O() term depends on k, k ′ , h but not on r. Combining this fact with Lemma 3.5.8, we deduce that the quantities ) and has the required interpolation properties, which finishes the proof.
We now note, for future use, the following vital property of the classes c BF [fα,gα] m,a . Denote by Perrin-Riou's regulator map (c.f. [PR95] and [LZ14, Appendix B]).
Proposition 3.5.11. If the stronger inequality Proof. Let W be this eigenspace. It is well known that the projection of L M(f ⊗g) * to W gives a map However, for any character of Γ of the form z → z j χ(z), with 0 j min(k, k ′ ) and χ of finite order, to W is an element of D 2λ (Γ, Q p )⊗ W which vanishes at all but finitely many characters of the form j + χ with j ∈ {0, . . . , min(k, k ′ )} and χ of finite order. Since 2λ < 1 + min(k, k ′ ), this projection must be zero as required.
Remark 3.5.12. We shall in fact show below that the result of Proposition 3.5.11 is actually true whenever α f α g satisfies the weaker assumption (3.5.1) (i.e. whenever the class c BF fα,gα m,a is defined), by deforming Proposition 3.5.11 along a Coleman family.
This vanishing property is natural in the context of Conjecture 8.2.6 of [LLZ14], which predicts the existence of an element in from which the Beilinson-Flach elements (for all choices of α f and α g ) can be obtained by pairing with the map L ML P (f ⊗g) * and projecting to a ϕ-eigenspace. Clearly, pairing an element of 2 with the same linear functional twice will give zero.

Overconvergentétale cohomology and Coleman families
We now recall the construction of p-adic families of Galois representations attached to modular forms via "big"étale sheaves on modular curves. We follow the account of [AIS15, §3], but with somewhat altered conventions (for reasons which will become clear later). We also use some results of Hansen [Han15] (from whom we have also borrowed the terminology "overconvergentétale cohomology").
The ring Λ U is endowed with two topologies: the p-adic topology (which we shall not use) and the m U -adic topology, which is the topology induced by the ideals m n U , where m U is the maximal ideal of Λ U .
Definition 4.1.4. For m 0, we write LA m (Z p , Λ U ) for the space of functions Z p → Λ U such that for all a ∈ Z/p m Z, the function z → f (a + p m z) is given by a power series n 0 b n z n with b n → 0 in the m U -adic topology of Λ U .
Proof. This is a standard computation, but we have not been able to find a reference, so we shall give a brief sketch of the proof. Let us write X m for the affinoid rigid-analytic space over Q p defined by Firstly, the map x → log(1+px) log(1+p) is a bijection from Z p to Z p which extends to a rigid-analytic isomorphism from X m to itself for every m; so it suffices to show that x → κ U ((1 + p) x ) extends to a Λ U -valued rigid-analytic function on X m whenever U ⊆ W m . It suffices to consider the universal case and we have ε n x m ∈ LA m (Z p , Z p ) for any n, by [Col10a, Theorem 1.29].
Remark 4.1.6. It is important to use the right topology on Λ U , because if one takes U = W m and writes x → κ U (1 + p m+1 x) as a series c n x n with c n ∈ Λ U , the c n tend to zero m U -adically (the above argument shows in fact that c n ∈ m n U ), but they do not tend to zero p-adically.
In particular, both T 0 and T ′ 0 are preserved by scalar multiplication by Z × p . Remark 4.2.3. The definition of T 0 coincides with that used in [AIS15] (and our Σ 0 (p) is their Ξ(p)). The subspace T ′ 0 is the image of T 0 under right multiplication by 0 −1 p 0 , and conjugation by this element interchanges Σ 0 (p) and Σ ′ 0 (p).
We equip this module with the topology defined by the subgroups m n and similarly for A • U,m (T ′ 0 ). Proof. We give the proof for T ′ 0 ; the proof for T 0 is similar. Unravelling the definition of the actions, we must show that if γ = a b pc d ∈ Σ ′ 0 (p) and f ∈ A • U (T ′ 0 ), then the function is closed under multiplication, and contains Z p , it suffices to check that z → κ U (1 + pd −1 bz) and z → f p · c+az d+pbz , 1 are in this space. For the factor κ U (1 + pd −1 bz) this follows from Lemma 4.1.5.
For the factor f p · c+az d+pbz , 1 , we note that the map z → c+az d+pbz preserves all the rigid-analytic neighbourhoods X m of Z p , so it preserves the ring of rigid-analytic functions convergent and bounded by 1 on these spaces; thus z → g c+az For the rest of this section, let T denote either T 0 or T ′ 0 , and Σ either Σ 0 or Σ ′ 0 respectively. Note that as a topological Λ U -module, A • U,m (T ) is isomorphic to the space of countable sequences (c n ) ∞ n=1 with c n ∈ Λ U such that c n → 0 in the m U -adic topology.
Proof. We give the proof for T ′ 0 , the proof for T 0 being similar. Because of the homogeneity requirement, any function in A • U,m (T ′ 0 ) is uniquely determined by its restriction to pZ p × 1, and this gives Both results now follow by passing to the inverse limit. Now let k ∈ W be an integer weight (i.e. of the form z → z k with k 0); any such weight automatically lies in W 0 . As for U above, we may define a space A • k,m (T ) of m-analytic O E -valued functions on T homogenous of weight k, and its dual D • k,m (T ), for any m 0. Restriction to T gives a natural embedding P • k ֒→ A • k,m (T ), where P • k is the space of polynomial functions on Z 2 p , homogenous of degree k, with O E coefficients. Dually, we obtain a canonical, Σ 0 (p)equivariant projection ρ k : D • k,m → (P • k ) * = TSym k O 2 E . Proposition 4.2.10. The following diagram is commutative, for any U , any m sufficiently large that U ⊆ W m , and any k ∈ U : Here mom k is as defined in [Kin15], and the left vertical arrow is the natural inclusion T ֒→ Z ⊕2 p . Proof. This is clear by construction.   x 1 , y 1 ), (x 2 , y 2 )) = x 1 y 2 − x 2 y 1 .
This clearly restricts to a map T 0 ×T ′ 0 → Z × p ; so the Λ U -valued function Φ on T 0 ×T ′ 0 given by Φ(t, t ′ ) = κ U (φ(t, t ′ )) is well-defined, homogenous of weight κ U in either variable, and m-analytic whenever U ⊆ W m .
. This is evidently Λ U -bilinear, and it satisfies Remark 4.3.3. Let us describe the above map slightly more concretely. We take m = 0, for simplicity; then the functions f n ((x, y)) = κ U (x) · (y/x) n are an orthonormal basis of A • U,0 (T 0 ), so a distribution µ ∈ D • U,0 (T 0 ) is uniquely determined by its moments µ n = µ(f n ), which can be any sequence of elements of Λ U . Similarly, the functions g n ((px, y)) = κ U (y)(x/y) n are an orthonormal basis of A • U,0 (T ′ 0 ) and any µ ′ ∈ D • U,0 (T ′ 0 ) is uniquely determined by its moments µ ′ n = µ ′ (g n ). Given such µ, µ ′ , we define an element of Λ U as follows: the function Φ ((1, z), (pw, 1)) = κ U (1 − pzw) can be written as a power series a n (wz) n , with a n ∈ Λ U such that a n → 0 in the m U -adic topology, by Lemma 4.1.5; then {µ, µ ′ } is the value of the convergent sum n 0 a n µ n µ ′ n . 4.4. Sheaves on modular curves.    ; the argument is given there for the Kummerétale site on a log rigid space over Q p (with log-structure given by the cusps), but the argument works equally well in the much simpler case of affine modular curves over Q.
Proposition 4.4.5. For any k ∈ U we have commutative diagrams of pro-sheaves on Y Here mom k is as defined in [Kin15].
Proof. We have the diagram of proposition 4.2.10, which we may interpret as a diagram of constant pro-sheaves on Y (p ∞ , N p ∞ ); and the morphisms in the diagram are all equivariant for the action of the Iwahori subgroup, so they descend to morphisms of sheaves on Y .
We can similarly construct D • U,m (H 0 ) and D • U,m (H ′ 0 ) as sheaves on Y (U ), for any sufficiently small open compact subgroup U ⊆ GL 2 ( Z) whose image in GL 2 (Z p ) is contained in the Iwahori subgroup. Moreover, if g ∈ GL 2 (Q) ∩ Σ 0 (p), so there is a natural map corresponding to z → gz on the upper half-plane, then the action of g on D • U,m (H 0 ) gives a map of sheaves on Y D • U,m (H 0 ) → g * D • U,m (H 0 ) ; the same holds with H ′ 0 and Σ ′ 0 in place of H 0 and Σ 0 . Definition 4.4.6. We define . We also make the same definitions for compactly-supported and parabolic cohomology, which we write as M • U,m (H 0 ) c , M • U,m (H 0 ) par (and similarly for H ′ 0 ).
These are profinite topological Λ U -modules, equipped with continuous actions of Gal(Q/Q) unramified outside N p∞. As topological Λ U -modules (forgetting the Galois actions) they are isomorphic to more familiar objects: • The space M • U,m (H 0 ) is isomorphic to the group cohomology H 1 Γ, D • U,m (T 0 ) , where Γ = Γ 1 (N (p)) = Γ 1 (N ) ∩ Γ 0 (p) (since Y 1 (N (p))(C) has contractible universal cover and its fundamental group is Γ 1 (N ) ∩ Γ 0 (p)). We now state some properties of these modules: (1) (Compatibility with specialisation) Let ̟ k be the ideal of Λ U corresponding to the character z → z k . For any integer k 0 ∈ U , there is an isomorphism For compactly-supported cohomology this is true for k 1, while for k = 0 we have an injective map c whose cokernel has rank 1 over O E , with the Hecke operator U p acting as multiplication by p. Similar statements hold for H ′ 0 in place of H 0 . (2) (Control theorem) For any integer k 0, the map is an isomorphism on the U p = α eigenspace, for any α of valuation < k + 1. The same holds for compactly-supported and parabolic cohomology, and for H ′ 0 and U ′ p in place of H 0 and U p .
where ev k is evaluation at k, and on the right-hand side {−, −} k signifies the Poincaré duality pairing.
where κ U is the composite of the cyclotomic character with the canonical map Z × p → Λ × U . Hence we have a cup-product pairing and since there is a canonical isomorphism H 2 c (Y , Z p (1)) ∼ = Z p , this gives a pairing into Λ U (1 + κ U ) as claimed. It is clear by construction that this is compatible with the Poincaré duality pairings with TSym k coefficients for each k 0.
Part (4) follows from the fact that the action of the matrix 0 −1 N p 0 on H interchanges T 0 and T ′ 0 .
Remark 4.4.9. The pairing {−, −} (in any of its various incarnations) is far from perfect (since its specialisation at a classical weight k 0 factors through the maps ρ k , so any non-classical eigenclass of weight k must be in its kernel). Nonetheless, we shall see below that it induces a perfect pairing on small slope parts. Remark 4.5.2. There are several equivalent definitions of slope λ, see [AIS15] for further discussion. We shall use the following formulation: the endomorphism U p of M ( λ) U is invertible, and the sequence of endomorphisms p ⌊nλ⌋ · (U p ) −n n 0 is bounded in the operator norm. Note that the summands M ( λ) and M (>λ) must be stable under the actions of the prime-to-p Hecke operators, and of the Galois group G Q , since these commute with the action of U p . The same results hold mutatis mutandis for M = M U,0 (H ′ 0 ), using the Hecke operator U ′ p in place of U p ; this follows directly from the previous statement using the isomorphism between the two modules provided by the Atkin-Lehner involution. There are also corresponding statements for compactly-supported and parabolic cohomology. 4.6. Coleman families. A considerably finer statement is possible if we restrict to a "neighbourhood" of a classical modular form. We make the following definition: with a 1 (F ) = 1 and a p (F ) invertible in B U , such that for all but finitely many classical weights k ∈ U ∩ Z 0 , the series F k = n 1 a n (F )(k) ∈ O E [[q]] is the q-expansion of a classical modular form of weight k + 2 and level Γ 1 (N ) ∩ Γ 0 (p) which is a normalised eigenform for the Hecke operators.
Remark 4.6.2. This definition is somewhat crude, since for a more satisfying theory one should also consider more general classical weights of the form z → z k χ(z) for χ of finite order, and allow families indexed by a finite flat rigid-analytic cover of U rather than by U itself. This leads to the construction of the eigencurve. However, the above definition will suffice for our purposes, since we are only interested in small neighbourhoods in the eigencurve around a classical point.
Definition 4.6.3. A noble eigenform of tame level N is a normalised cuspidal Hecke eigenform f α of level Γ 1 (N ) ∩ Γ 0 (p) and some weight k + 2 2, with coefficients in E, having U p -eigenvalue α = a p (f α ), such that: • f α is a p-stabilisation of a newform f of level N whose Hecke polynomial X 2 −a p (f )X +p k+1 ε f (p) has distinct roots ("p-regularity"); • if v p (α) = k + 1, then the Galois representation M E (f )| GQ p is not a direct sum of two characters ("non-criticality").
Theorem 4.6.4. Suppose f α is a noble eigenform of weight k 0 + 2. Then there exists a disc U ∋ k 0 in W, and a unique Coleman family F over U , such that F k0 = f α .
Proof. This follows from the fact that the Coleman-Mazur-Buzzard eigencurve C (N ) of tame level N isétale over W (and, in particular, smooth) at the point corresponding to a noble eigenform f α . See [Bel12].
Remark 4.6.5. As remarked in [Han15], the condition that the Hecke polynomial of f has distinct roots is conjectured to be redundant, and known to be so when f has weight 2; and it is also conjectured that the only newforms f of weight 2 such that M E (f )| GQ p splits as a direct sum are those which are of CM type with p split in the CM field. • For each k 0 ∈ U , the form F k is a classical eigenform, and we have isomorphisms of E-linear G Qp -representations Proof. The finite-slope parts of all the various overconvergent cohomology groups can be glued into coherent sheaves on the eigencurve C (N ). In a neighbourhood of a noble point, the eigencurve isétale over weight space and these sheaves are all locally free of rank 2; and the map from H 1 c to H 1 is an isomorphism at the noble point, so it must be an isomorphism on some neighbourhood of it. See [Han15, Proposition 2.3.5] for further details. 4.7. Weight one forms. If f is a cuspidal newform of level N and weight 1, and f α is a p-stabilisation of f , then it is always the case that v p (α) = k 0 + 1 = 0 and M E (f )| GQ p splits as a direct sum (since M E (f ) is an Artin representation). Nonetheless, analogues of Theorem 4.6.4 and Theorem 4.6.6 do hold for these forms. Proof. Part (1) is exactly the statement that the eigencurve is smooth at the point corresponding to f α , and isétale over weight space except in the real-multiplication setting; see [BD16]. Part

Coefficient modules.
Let H be a group isomorphic to Z 2 p (but not necessarily canonically so), for p an odd prime. Then we can regard the modules TSym r H as representations of Aut(H) ≈ GL 2 (Z p ). In this section, we shall show that the Clebsch-Gordan decompositions of the groups TSym r H ⊗ TSym s H can themselves be interpolated as r varies (for fixed s), after passing to a suitable completion.
In this section we shall refer to morphisms as natural if they are functorial with respect to automorphisms of H.
Proposition 5.1.1. For A an open compact subset of H such that A ∩ pH = ∅, and any r 1, there is a short exact sequence This short exact sequence is natural, and split (but not naturally split).
Proof. Let us begin by defining the maps. The map β, which is the simpler of the two, is given by interpreting Sym j H ∨ as a subspace of C(A) (consisting of functions which are the restrictions to A of homogenous polynomial functions on H of degree j) and composing with the multiplication map The map α is more intricate: it is given by including 2 (H ∨ ) in H ∨ ⊗ H ∨ , and grouping the terms as As above, we have a canonical multiplication map C(A)⊗H ∨ → H ∨ , and multiplication in the symmetric algebra Sym • (H ∨ ) gives a map Sym j−1 (H ∨ ) ⊗ H ∨ → Sym j H ∨ , and this gives the first map in the sequence. The composite β • α is clearly 0, since it factors through the map ∧ 2 H ∨ → Sym 2 H ∨ .
Having defined the maps intrinsically, we may check the exactness of the sequence after fixing a basis of H. Let x, y be the corresponding coordinate functions, so that x j , x j−1 y, . . . , y j is a basis of Sym j H ∨ and x ⊗ y − y ⊗ x is a basis of ∧ 2 H ∨ . With these identifications we can write the sequence as with the maps being (f 0 , . . . , f j−1 ) → (−yf 0 , xf 0 − yf 1 , . . . , xf j−1 ) and (f 0 , . . . , f j ) → x j f 0 + · · · + y j f j . The injectivity of α is now clear, since multiplication by x (or by y) is injective in C(A).
To show that the map β is surjective, we write down a (non-canonical) section. We can decompose A as a union A 1 ⊔ A 2 where x is invertible on A 1 and y is invertible on A 2 . We define δ(f ) = (x −j f, 0, . . . , 0) on C(A 1 ) and δ(f ) = (0, . . . , 0, y −j f ) on the C(A 2 ) factor; then β • δ is clearly the identity, so β is surjective.
Finally, let (f 0 , . . . , f j ) ∈ ker(β). Choosing A = A 1 ⊔ A 2 as before, we may assume either x or y is invertible on A. We treat the first case, the second being similar. We define γ(f 1 , . . . , f j ) = (g 0 , . . . , g j−1 ) where g j−1 = x −1 f j , g j−2 = x −2 (xf j−1 + yf j ) etc, down to g 0 = x −j (x j−1 f 1 + · · · + y j−1 f j ). But then (α • γ) + (β • δ) = id, so we have exactness at the middle term. Now let C la (A) denote the space of locally analytic E-valued functions on A; exactly the same argument shows that we have an exact sequence analogous to (5.1.1), Proposition 5.1.2. Let δ : C la (A) → C la (A) ⊗ Sym j H ∨ be the morphism defined in a basis by Then δ is natural, and the composite β • δ is the endomorphism of C la (A) given by 1 Proof. The morphism δ is simply 1 j! times the j-th power of the total derivative map C la (A) → C la (A) ⊗ Tan(A) * , combined with the identification Tan(A) ∼ = Tan(H) ∼ = H. From this description the naturality is clear, and a computation shows that it agrees with the more concrete description above. The identity for β • δ is easily seen by induction on j.
It will be convenient to adopt the notation ∇ j for the endomorphism 1 . We may regard this as an element of the space D la (Z × p ) of locally analytic distributions on Z × p .
Proposition 5.1.3. For any k j, the restriction of δ to the space Sym k H ∨ of homogenous polynomials of degree k lands in the subspace and the resulting map Sym k H ∨ → Sym k−j H ∨ ⊗ Sym j H ∨ is the dual of the symmetrised tensor product map TSym k−j H ⊗ TSym j H → TSym k H. If k < j then the restriction of δ to Sym k H ∨ is the zero map.
Proof. It is obvious that Sym k H ∨ embeds naturally into C la (A), and its image under δ is contained in A straightforward computation in coordinates shows that this map sends x a y b to s+t=j a s b t x a−s y b−t ⊗ x s y t , which coincides with the dual of the symmetrised tensor product. On the other hand it is obvious from equation (5.1.1) that δ vanishes on any polynomial of total degree < j.
Moreover, for any k 0 we have where mom k−j ·1 denotes the composition (where the second map is the symmetrized tensor product).
Proof. This follows by dualizing the previous proposition.
We now consider varying j, for which it is convenient to re-label the maps β * , δ * above as β * j and δ * j . Lemma 5.1.5. Let h j 0. Then the composition where the unlabelled arrow is given by the symmetrised tensor product, is given by Proof. Explicit computation.

5.2.
Nearly-overconvergentétale cohomology. We also have an analogue of the Clebsch-Gordan map for the distribution spaces D • U,m (T ′ 0 ) introduced above, which are completions of D la (T ′ 0 ). The rigid space W has a group structure, so we can make sense of U − j for any integer j.
Proposition 5.2.1. There are natural maps Proof. We simply transport the constructions of §5.1 to the present setting (taking A = T ′ 0 ). The naturality of these constructions precisely translates into the assertion that the resulting maps commute with the Σ 0 (p)-action. Since the functions in A U,m are homogenous of weight κ U (the canonical character where on the right-hand side ∇ is regarded as an element of Λ U [1/p]; that is, the two actions of ∇ on A U,m , as a differential operator and as an element of the coefficient ring, coincide. The maps of spaces β * j and δ * j induce maps ofétale sheaves on Y = Y 1 (N (p)) (for any N ), , which we denote by the same symbols.
Definition 5.2.3. We shall refer to the cohomology groups H * et (Y , D U−j,m (H ′ 0 ) ⊗ TSym j H ) as nearlyoverconvergentétale cohomology, and the map as the overconvergent projector.
Remark 5.2.4. The motivation for this terminology is that the sheaves D U−j,m (H ′ 0 ) ⊗ TSym j H , and the maps β * j and δ * j relating them to the overconvergent cohomology sheaves D U,m (H ′ 0 ), are anétale analogue of the coherent sheaves appearing in the theory of nearly-overconvergent p-adic modular forms (see [Urb14]).
Recall from Corollary 5.1.4 that the composite of δ * j with the moment map ρ k is zero if 0 k < j, which is somewhat undesirable. We can rectify this issue as follows. Recall that we have defined Proposition 5.2.5. Let U be an open disc contained in W 0 , and F a Coleman family defined over U . Suppose the following condition is satisfied: for any integer weight k 0 in U , the projection map M k (H ′ 0 ) → M k (F ) * factors through ρ k . Then, for any j 0, the composite map takes values in ∇(∇ − 1) . . . (∇ − j + 1)M U (F ) * , and hence the map Proof. Note that ∇, regarded as a rigid-analytic function on W, takes the value k at an integer weight k. So the only points in W 0 at which ∇(∇ − 1) . . . (∇ − j + 1) fails to be invertible are the positive integers {0, . . . , j − 1}, and it has simple zeroes at all of these points. If k is one of these integers, then we have . Hence it suffices to show that pr F •δ * j is zero on M k (H ′ 0 ); but this is immediate since the specialisation of pr F at k factors through ρ k , and ρ k • δ * j is zero for 0 k < j. This shows that pr F •δ * j lands in the stated submodule. Since M U (F ) * is a free Λ U [1/p]-module (and Λ U [1/p] is an integral domain), the map pr F is therefore well-defined.
Remark 5.2.6. This proposition can be interpreted as follows: we can renormalise δ * j to be an inverse to β * j , as long as we avoid points on the eigencurve which are non-classical but have classical weights.
By construction, the map pr F has the property that the following diagram commutes: p r F ✲ More generally, if 0 j h, then (as in Lemma 5.1.5) we can consider β * h−j · id as a map (H ), and from Lemma 5.1.5 one computes that 5.3. Two-parameter families of Beilinson-Flach elements. Let N 1 , N 2 be integers such that p ∤ N i and pN 1 , pN 2 4. We also choose two wide open discs U 1 and U 2 in W 0 , and consider the sheaf Definition 5.3.1. Let N be any integer divisible by N 1 and N 2 and with the same prime factors as N 1 N 2 . For any j 0 and m 1, we define the element under pushforward along Y 1 (N p) 2 → Y 1 (N 1 (p)) × Y 1 (N 2 (p)), composed with the map induced by the morphisms of sheaves Here, the first map is given by the natural maps Λ(H C ) → D • U , for U = U i − j, and the second map is the overconvergent projector δ * j of Proposition 5.2.1. Remark 5.3.2. We are using implicitly here the fact that the Beilinson-Flach elements can be lifted canonically to classes with coefficients in the sheaves Λ(H Zp D ′ ). Cf. Remark 3.3.6 above.
The Hochschild-Serre spectral sequence and the Künneth formula give a canonical surjection m,N p,a under the map (mom k−j · id) ⊠ (mom k ′ −j · id). Now let us choose newforms f, g, of levels N 1 , N 2 and weights k 1 + 2, k 2 + 2 2, and roots α 1 , α 2 of their Hecke polynomials, such that the p-stabilisations f i,αi both satisfy the hypotheses of Theorem 4.6.6. The theorem then gives us families of overconvergent eigenforms F 1 , F 2 passing through the p-stabilisations of f and g, defined over some discs U 1 ∋ k 1 , U 2 ∋ k 2 . Proposition 5.3.4. If the discs U i are sufficiently small, then there exist classes Proof. After shrinking the discs U i if necessary so that all integer-weight specialisations of F and G are classical, so that Proposition 5.  mp r ,a , for 0 j h and r 1, satisfy the following norm bound: Proof. We shall deduce this from Theorem 3.3.5 (and Remark 3.3.6). This theorem gives a bound for the classes mp r ,N p,a .
We apply to this the map pr mp r ,a , by (5.2.1). So the image of the expression of Theorem 3.3.5 is which is exactly a h h! times the quantity in the proposition. We may ignore the factor a h h!, since it is nonzero and independent of r.
We now choose affinoid discs V i contained in the U i (so the M Vi (F i ) * become Banach spaces).
Theorem 5.4.2. There is a element c BF [F ,G] m,a ∈ H 1 Q(µ m ), D λ1+λ2 (Γ, M V1 (F ) * ⊗ M V2 (G) * ) which enjoys the following interpolating property: for any integers (k 1 , k 2 , j) with k i ∈ V i and 0 j min(k, k ′ ), the image of c BF [F ,G] m,a at (k 1 , k 2 , j) is Proof. We choose an integer h ⌊λ 1 + λ 2 ⌋, and apply Proposition 2.3.3 with K = Q(µ m ), S the set of primes dividing (−a) j j! for 0 j h and n 1. These x n,j are norm-compatible for n 1, and we obtain norm-compatible elements for all n 0 by defining Moreover, the bound we have just established in Proposition 5.4.1 shows that p −nh h j=0 (−1) j h j x n,j Cp λn , which is exactly the growth bound required for Proposition 2.3.3. It is not difficult to see that H 0 (Q ∞ , M V1 (F ) * ⊗ M V2 (G) * ) = 0, so we obtain a class interpolating the classes x n,j for all n 0 and all j ∈ {0, . . . , h}. However, if we have two integers h ′ h ⌊λ 1 + λ 2 ⌋, then the element x[h ′ ] satisfies an interpolating property strictly stronger than that of x[h], so we deduce that x[h] is in fact independent of h and interpolates x n,j for all j 0. We define c BF [F ,G] m,a to be this element. The interpolating property is now immediate from the interpolating property of the 2-variable classes c BF [F ,G,j] m,a at integers k 1 , k 2 j.
6. Phi-Gamma modules and triangulations 6.1. Phi-Gamma modules in families. Let R denote 2 the Robba ring (of Q p ), which is the ring of formal Laurent series over Q p in a variable π, convergent on some annulus of the form {x : 0 < v p (x) < ε} ⊆ A 1 rig ; and let R + ⊆ Q p [[π]] be its subring of elements that are analytic on the whole disc {x : v p (x) > 0}. We endow these with their usual actions of Frobenius ϕ and the group Γ ∼ = Z × p . We define a left inverse ψ of ϕ by putting for any f (π) ∈ R + . As is well known, there is a functor D † rig mapping p-adic representations of G Qp to (ϕ, Γ)-modules over R (finitely-generated free R-modules with commuting R-semilinear operators ϕ and Γ), and this is a fully faithful functor whose essential image is the subcategory of (ϕ, Γ)-modules of slope 0.
Remark 6.1.1. Strictly speaking, the definition of the functor D † rig depends on the auxilliary choice of a compatible system of p-power roots of unity (ζ p n ) n 0 in Q p . We shall fix, once and for all, such a choice, and in applications to global problems we shall often assume that ζ p n corresponds to e 2πi/p n ∈ C. Now let A be a reduced affinoid algebra over Q p , and write R A = R⊗ A and similarly for R + A . We define an A-representation of G Qp to be a finitely-generated locally free A-module endowed with an A-linear action of G Qp (continuous with respect to the canonical Banach topology of M ).  Proof. Let us choose an increasing sequence of affinoid discs X n ⊆ W whose union is W. Since we have D la (Γ, Q p ) = O(W) = lim ← −n O(X n ), we can regard D la (Γ, M ) as a locally free sheaf of G Qprepresentations on W × Max A, and we deduce that by [Pot13,Theorem 1.7]. For each n, X n × Max A is an affinoid space, so we obtain by [Pot13,Proposition 2.7]. Finally, the inverse limit of the modules D † rig (O(X n )⊗ M ) is the module Dfm(D † rig (M )) considered in [KPX14,Theorem 4.4.8], where it is shown that Finally, if the base A is a finite field extension of Q p , then the functors D cris (−) and D dR (−) can be extended from A-linear representations of G Qp to the larger category of (ϕ, Γ)-modules over R A , and one has the following fact: Theorem 6.1.5 (Nakamura, see [Nak14]). If A is a finite extension of Q p , there exist Bloch-Kato exponential and dual-exponential maps Definition 6.2.1. We write R A (α −1 ) for the free rank 1 (ϕ, Γ)-module over R A with basis vector e such that ϕ(e) = α −1 e and γe = e for all γ ∈ Γ. We write R + A (α −1 ) for the submodule R + A · e of R A (α −1 ). Lemma 6.2.2. Suppose α 1 and α − 1 is not a zero-divisor in A. Then . Proof. This follows from Lemma 1.11 of [Col10b]. Cf. [Han15, §4.1].
We use this lemma to define a Perrin-Riou big logarithm map for R A (α −1 ) when α − 1 is not a zero-divisor, following closely the construction in [Han15, §4.2], as the composition (6.2.1) where the third arrow is the base-extension to A of the Mellin transform (and W is weight space). Note that our assumption that α − 1 is not a zero-divisor in A implies that R A (α −1 ) ϕ=1 = 0, and hence that L RA(α −1 ) is injective.
Definition 6.3.1. Let D be a (ϕ, Γ)-module over R⊗ A which is locally free of rank 2. A triangulation of D is a short exact sequence of (ϕ, Γ)-modules over R⊗ A, where the modules F ± D are locally free of rank 1 over R⊗ A.
Theorem 6.3.2 (Ruochuan Liu, [Liu14]). Let (f, α) be as in Theorem 4.6.6. Then one can find an affinoid disc V ⊂ W containing k such that the (ϕ, Γ)-module 6.4. Eichler-Shimura isomorphisms. The last technical ingredient needed to proceed to the proof of our explicit reciprocity law is the following: Theorem 6.4.1 (Eichler-Shimura relation in families). In the setting of Theorem 6.3.2, after possibly shrinking V , there is a canonical O(V )-basis vector such that for every integer weight t 0 in V , the specialisation of ω F at t coincides with the image of the differential form ω ft attached to the normalised eigenform f t . This is a minor modification of results of Ruochuan Liu (in preparation); we outline the proof below. The starting point is the following theorem: Theorem 6.4.2 (Andreatta-Iovita-Stevens, [AIS15]). For any integer k 0 0, and real λ < k 0 + 1, we can find an open disc V ⊂ W containing k 0 and a Hecke-equivariant isomorphism interpolating Faltings' Hodge-Tate comparison isomorphisms for each k ∈ V . Here X(w) is a rigidanalytic neighbourhood of the component of ∞ in the ordinary locus of the compactification X of Y ; and ω †,κV +2 V is a certain sheaf of O(V )-modules on X(w), whose specialisation at any integer k 0 ∈ V is the (k + 2)-th power of the Hodge bundle for every k ∈ V .
Proof of Theorem 6.4.1. We translate the statement of the above theorem into the language of (ϕ, Γ)modules. For any family of G Qp -representations M over an affinoid algebra A, we have a canonical isomorphism where D Sen (M ) is defined in terms of the (ϕ, Γ)-module D † rig (M ). Moreover, D Sen (F + D V (F )(1 + κ V )) Γ is zero. Hence, by composing comp V with the projection to F − , we have an isomorphism The left-hand side is free of rank 1, spanned by τ ·F where τ is the Gauss sum of ε with the property that for every classical specialisation F t of F , the specialisation of η F at t is the unique vector whose cup product with the differential ω Ft attached to the complex conjugate F t of F t is given by where α and β are the roots of the Hecke polynomial of F t , and λ N (F t ) is its Atkin-Lehner pseudoeigenvalue.
Proof. This follows by dualising ω F using the Ohta and similarly for F −+ , F +− and F ++ . We also define Theorem 7.1.2. If V 1 and V 2 are sufficiently small, then (for any m coprime to p) the image of c BF [F ,G] m,a under projection to the module H 1 By taking the V i sufficiently small, we may assume that F −− D V1×V2 (F ⊗ G) * is actually isomorphic to R A (α −1 ), where α = α F α G and A = O(V 1 × V 2 ), and that α −1 < p 1+h and α − 1 is not a zero-divisor. It suffices, therefore, to show that L RA(α −1 ) maps the image of c BF [F ,G] m,a to zero. However, for each pair of integers (ℓ, ℓ ′ ) ∈ V 1 × V 2 with ℓ, ℓ ′ 1 + 2h and such that F ℓ and G ℓ ′ are not twists of each other, we know that the image of L RA(α −1 ) ( c BF [F ,G] m,a ) vanishes when restricted to (ℓ, ℓ ′ ) × W ⊆ Max(A) × W, by Proposition 3.5.11. Since such pairs (ℓ, ℓ ′ ) are Zariski-dense in Max(A), the result follows.
Remark 7.1.3. Cf. [KLZ15b, Lemma 8.1.5], which is an analogous (but rather stronger) statement in the ordinary case.
Hence the projection of c BF [F ,G] m,a to F −• is in the image of the injection . Since F + D V2 (G) * is isomorphic to an unramified module twisted by an A × -valued character of the cyclotomic Galois group Γ, we may define a Perrin-Riou logarithm map for F −+ D V1×V2 (F ⊗ G) * by reparametrising the corresponding map for its unramified twist, exactly as in Theorem 8.2.8 of [KLZ15b]. That is, if we define which is free of rank 1 over O(V 1 × V 2 ), then we obtain the following theorem: Theorem 7.1.4. There is an injective morphism of O(V 1 × V 2 × W )-modules , with the following property: for all classical specialisations f, g of F , G, and all characters of Γ of the form τ = j + η with η of finite order and j ∈ Z, we have a commutative diagram in which the bottom horizontal map is given by where exp * and log are the Bloch-Kato dual-exponential and logarithm maps, ε is the finite-order character ε g,p · η −1 of Γ, r 0 is the conductor of ε, and G(ε) = a∈(Z/p r Z) × ε(a)ζ a p r is the Gauss sum.
Proof. The construction of the map L is immediate from (6.2.1). The content of the theorem is that the map L recovers the maps exp * and log for the specialisations of F and G; this follows from Nakamura's construction of exp * and log for (ϕ, Γ)-modules.
Theorem 7.1.5 (Explicit reciprocity law). If the V i are sufficiently small, then we have Here, L p (F , G, 1 + j) denotes Urban's 3-variable p-adic L-function as constructed in [Urb14], and ε F and ε G are the characters by which the prime-to-p diamond operators act on F and G.
Proof. The two sides of the desired formula agree at every (k, k ′ , j) with k ∈ V 1 , k ′ ∈ V 2 and 0 j min(k, k ′ ), by [KLZ15a, Theorem 6.5.9]. These points are manifestly Zariski-dense, and the result follows.
Remark 7.1.6. The construction of ω G , and the proof of the explicit reciprocity law, are also valid if G is a Coleman family passing through a p-stabilisation g α of a p-regular weight 1 form, as in Theorem 4.7.2; the only difference is that one may need to replace V 2 with a finite flat coveringṼ 2 . In this setting, g α is automatically ordinary, so G is in fact a Hida family, and one can use the construction of ω G given in [KLZ15b, Proposition 10.12.2]. 8. Bounding Selmer groups 8.1. Notation and hypotheses. Let f, g be cuspidal modular newforms of weights k + 2, k ′ + 2 respectively, and levels N f , N g prime to p. We do permit here the case k ′ = −1. We suppose, however, that k > k ′ , so in particular k 0; and we choose an integer j such that k ′ + 1 j k. If j = k+k ′ 2 + 1, then we assume that ε f ε g is not trivial, where ε f and ε g are the characters of f and g.
As usual, we let E be a finite extension of Q p with ring of integers O, containing the coefficients of f and g. Our goal will be to bound the Selmer group associated to the Galois representation M O (f ⊗ g)(1 + j), in terms of the L-value L(f, g, 1 + j); our hypotheses on (k, k ′ , j) are precisely those required to ensure that this L-value is a critical value.
It will be convenient to impose the following local assumptions at p: • (p-regularity) We have α f = β f and α g = β g , where α f , β f are the roots of the Hecke polynomial of f at p, and similarly for g. • (no local zero) None of the pairwise products {α f α g , α f β g , β f α g , β f β g } is equal to p j or p 1+j , so the Euler factor of L(f, g, s) at p does not vanish at s = j or s = 1 + j.
• (nobility of f α ) If f is ordinary, then either α f is the unit root of the Hecke polynomial, or M E (f )| GQ p is not the direct sum of two characters (so the eigenform f α is noble in the sense of 4.6.3). • (nobility of g α and g β ) If k ′ 0, then M E (g)| GQ p does not split as a direct sum of characters, so both p-stabilisations g α and g β are noble.
(1) In our arguments we will use both p-stabilisations g α and g β of g, but only the one p-stabilisation f α of f ; in particular, we do not require that the other p-stabilisation f β be noble.
(2) Note that the "no local zero" hypothesis is automatic, for weight reasons, unless k + k ′ is even and j = k+k ′ 2 or j = k+k ′ 2 + 1 (so the L-value L(f, g, 1 + j) is a "near-central" value). The p-regularity hypothesis implies that we have direct sum decompositions where ϕ acts on the two direct summands as multiplication by α −1 f , β −1 f respectively, and similarly for g. This induces a decomposition of D cris (M E (f ⊗ g) * ) into four direct summands D cris (M E (f ⊗ g) * ) α f αg etc.
Definition 8.1.2. We write We write pr α f for the projection and c α f βg 1 are a basis of H 1 s (Q p , V * (1)), so these two classes must be a basis of H 1 relaxed (Q, V * (1)). Corollary 8.3.3. Let L S (f, g, s) = ℓ / ∈S P ℓ (ℓ −s ) −1 be the L-function without its local factors at places in S. If the hypotheses of Theorem 8.2.1 are satisfied and L S (f, g, 1 + j) = 0, then H 2 (Q S /Q, M E (f ⊗ g) * (−j)) = 0.
Remark 8.3.4. One can check that the only values of s at which the Euler factors P ℓ (ℓ −s ) may vanish for some ℓ ∈ S are s ∈ k + k ′ 2 , k + k ′ + 1 2 , k + k ′ + 2 2 .
Note that the centre of the functional equation, with our normalisations, is at s = k+k ′ +3 2 . 8.4. Application to elliptic curves. Theorem 8.2.1 above allows us to strengthen one of the results of [KLZ15b] to cover elliptic curves which are not necessarily ordinary at p: Theorem 8.4.1. Let E/Q be an elliptic curve without complex multiplication, and ρ a 2-dimensional odd irreducible Artin representation of G Q (with values in some finite extension L/Q). Let p be a prime. Suppose that the following hypotheses are satisfied: (i) The conductors N E and N ρ are coprime; (ii) p 5; (iii) p ∤ N E N ρ ; (iv) the map G Q → Aut Zp (T p E) is surjective; (v) ρ(Frob p ) has distinct eigenvalues.
Proof. This is exactly Theorem 8.2.1 applied with f = f E , the weight 2 form attached to E, and g = g ρ , the weight 1 form attached to ρ. Compare Theorem 11.7.4 of [KLZ15b], which is exactly the same theorem under the additional hypotheses that E is ordinary at p and ρ(Frob p ) has distinct eigenvalues modulo a prime of L above p.

Addendum: Remarks on the proof of the reciprocity law
In order to formulate the explicit reciprocity law of Theorem 7.1.5, one needs to invoke the main theorem of [Urb14]: the construction of a 3-variable p-adic Rankin-Selberg L-function as a rigid-analytic function on V 1 × V 2 × W, where V i are small discs in the Coleman-Mazur eigencurve surrounding classical p-stabilised eigenforms, and W is weight space.
Unfortunately, since the present paper was submitted, it has emerged that there are some unresolved technical issues in the paper [Urb14], so the existence of this p-adic L-function is not at present on a firm footing. We hope that this issue will be resolved in the near future; but as a temporary expedient we explain here an unconditional proof of a weaker form of explicit reciprocity law which suffices for the arithmetic applications in the present paper. 9.1. A three-variable geometric p-adic L-function. We place ourselves in the situation of §7.1, so f α , g α are noble eigenforms, obtained as p-stabilisations of newforms f, g of weights k 0 + 2, k ′ 0 + 2 and levels prime to p; and V 1 , V 2 are small enough affinoid discs in weight space around k 0 and k ′ 0 , over which there are Coleman families F , G passing through f α , g α . We also allow the possibility that k ′ 0 = −1, g is a p-regular weight 1 newform, and g does not have real multiplication by a field in which p splits. (The exceptional real-multiplication case can be handled similarly by replacing V 2 with a ramified covering; we leave the details to the reader.) For notational simplicity, we shall suppose that ε F ε G is nontrivial, and is not of p-power order. Thus there is a c > 1 coprime to 6pN f N g for which the factor c 2 − c 2j−k−k ′ ε F (c) −1 ε G (c) −1 is a unit in O(V 1 × V 2 × W); and we may define BF [F ,G] 1,1 (without c) by dividing out by this factor. We shall begin by turning Theorem C on its head, and defining a p-adic L-function to be the output of this theorem: Our goal is now to show that this geometrically-defined p-adic L-function is related to critical values of complex L-functions.

9.2.
Values in the geometric range. By construction, for integer points of V 1 × V 2 × W in the "geometric range" -that is, the points (k, k ′ , j) with 0 j min(k, k ′ ) -the geometric p-adic Lfunction interpolates the syntomic regulators of the Rankin-Eisenstein classes. From the computations of [KLZ15a], we have the following explicit formula for these syntomic regulators.
Let f k,α be the p-stabilised eigenform that is the specialisation of F in weight k + 2, and let λ f k,α be the unique linear functional on the space S oc k+2 (N f , E) of overconvergent cusp forms that factors through projection to the f k,α -isotypical subspace and sends f k,α to 1. We view λ f k,α as a linear functional on S oc k+2 (N, E), where N = lcm(N f , N g ), by composing with the trace map from level N to level N f .
Theorem 9.2.1 ([KLZ15a, Theorem 6.5.9]). For (k, k ′ , j) in the geometric range, with j > k 2 − 1, we have L geom Here F [p] k−k ′ ,k ′ −j+1 is a nearly-overconvergent p-adic Eisenstein series of weight k − k ′ and degree of near-overconvergence k − j, whose p-adic q-expansion (image under the unit-root splitting) is given by Note that we have F where θ = q d dq and E Since E [p] r is overconvergent of weight r, it follows that g k ′ ,α ·θ k−j E [p] 2j−k−k ′ lies in the space S n−oc,k−j k+2 (N ) of nearly-overconvergent cusp forms of weight k + 2 and degree of near-overconvergence k − j. The condition j > k 2 − 1 implies that k + 2 > 2(k − j), so Urban's overconvergent projector Π oc is defined on S n−oc,k−j k+2 (N ) [Urb14, §3.3.3]. Thus the right-hand side of the formula in the theorem is defined. 9.3. Two-variable analytic L-functions. Let us now pick an integer t 0, and set j = k − t in the above formulae. Then, for varying k and k ′ (but t fixed), the forms g k ′ ,α · θ t E [p] k−k ′ −2t interpolate to a 2-parameter family of nearly-overconvergent cusp forms over V 1 × V 2 (of weight k + 2 and degree t, where k is the universal weight of V 1 ). Hence we may make sense of as a meromorphic rigid-analytic function on V 1 × V 2 , analytic except possibly for simple poles along V 1 ∩ {0, . . . , 2t − 2} [Urb14, §3.3.4].
Remark 9.3.1. The important point here is that the power of the differential operator appearing is constant in the family; this circumvents the technical issues in [Urb14], which concern interpolation of families where the degree of near-overconvergence is unbounded.
We have the following special sets of integer points (k, k ′ ) ∈ V 1 × V 2 : (i) If k max(t, 2t − 1) and k ′ k − t, then the "geometric" interpolating property above applies, showing that for these values of (k, k ′ ) we have Since such (k, k ′ ) are manifestly Zariski-dense in V 1 × V 2 , this relation must in fact hold for all points (κ, κ ′ ) ∈ V 1 × V 2 . (ii) If k ′ 0 and k − k ′ 2t + 1, then both g k ′ ,α and E [p] k−k ′ −2t are classical modular forms (since, after possibly shrinking V 2 , we may arrange that the specialisations of the family G at classical weights are classical). Thus the product g k ′ ,α · θ t E [p] k−k ′ −2t is a classical nearly-holomorphic form, and on such forms Urban's overconvergent projector coincides with the holomorphic projector Π hol . This shows that the values of L (t) p (F , G)(k, k ′ ) for (k, k ′ ) in this range are algebraic, and they compute the values of the Rankin-Selberg L-function in the usual way. This also holds for k ′ = −1, as long as we assume that the weight 1 specialisation g k ′ ,α is classical (which is no longer automatic).