Rankin–Eisenstein classes in Coleman families
 David Loeffler^{1}Email authorView ORCID ID profile and
 Sarah Livia Zerbes^{2}
https://doi.org/10.1186/s4068701600776
© The Author(s) 2016
Received: 29 June 2015
Accepted: 19 April 2016
Published: 1 October 2016
Abstract
We show that the Euler system associated with Rankin–Selberg convolutions of modular forms, introduced in our earlier works with Lei and Kings, varies analytically as the modular forms vary in padic 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.
1 Background
Let \(p > 2\) be a prime. The purpose of this paper is to study the padic interpolation of étale Rankin–Eisenstein classes, which are Galois cohomology classes attached to pairs of modular forms f, g of weights \(\geqslant 2\), forming a “cohomological avatar” of the Rankin–Selberg Lfunction L(f, g, s).
In a previous work with Kings [19], we showed that these Rankin–Eisenstein classes for ordinary modular forms f, g interpolate in 3parameter padic 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 Lfunctions, with applications to the Birch–SwinnertonDyer conjecture for Artin twists of pordinary elliptic curves, extending earlier works of Bertolini–Darmon–Rotger [5, 6].
In this paper, we generalise these results to nonordinary modular forms f, g, replacing the Hida families by Coleman families:
Theorem A
Let f, g be eigenforms of weights \(\geqslant 2\) and levels \(N_f, N_g\) coprime to p whose Hecke polynomials at p have distinct roots, and let \(f_\alpha , g_\alpha \) be noncritical pstabilisations of f, g. Let \(\mathcal {F}, \mathcal {G}\) be Coleman families through \(f_\alpha , g_\alpha \) (over some sufficiently small affinoid discs \(V_1, V_2\) in weight space).
Here \(M_{V_1}(\mathcal {F})^*\) and \(M_{V_2}(\mathcal {G})^*\) are families of Galois representations over \(\mathcal {O}(V_1)\) and \(\mathcal {O}(V_2)\) attached to \(\mathcal {F}\) and \(\mathcal {G}\), and \(D^{\mathrm {la}}(\varGamma )\) is the algebra of distributions on the cyclotomic Galois group \(\varGamma \). A slightly modified version of this theorem holds for weight 1 forms as well. For a precise statement, see Theorem 5.4.2 below.
The proof of Theorem 5.4.2 reveals some new phenomena which may be of independent interest; the Galois modules in which these classes lie are, in a natural way, étale counterparts of the modules of “nearly overconvergent modular forms” introduced by Urban [32].
Theorem B
The image of the class \({}_c \mathcal {BF}^{[\mathcal {F}, \mathcal {G}]}_{1}\) under an appropriately defined PerrinRiou “big logarithm” map is Urban’s 3variable padic Rankin–Selberg Lfunction for \(\mathcal {F}\) and \(\mathcal {G}\).
See Theorem 7.1.5 for a precise statement. In order to define the PerrinRiou logarithm in this context, one needs to work with triangulations of \((\varphi , \varGamma )\)modules over the Robba ring; we use here results of Liu [21], showing that the \((\varphi , \varGamma )\)modules of the Galois representations \(M_{V_1}(\mathcal {F})^*\) and \(M_{V_2}(\mathcal {G})^*\) admit canonical triangulations.
Specialising this result at a point corresponding to a critical value of the Rankin–Selberg Lfunction, and applying the Euler system machine of Kolyvagin and Rubin, we obtain a case of the Bloch–Kato conjecture for Rankin convolutions:
Theorem C
(Theorem 8.2.1, Corollary 8.3.2) Let f, g be eigenforms of levels coprime to p and weights \(r, r'\), respectively, with \(1 \leqslant r' < r\), and let s be an integer such that \(r' \leqslant s \leqslant r1\) (equivalently, such that L(f, g, s) is a critical value of the Rankin–Selberg Lfunction). Suppose \(L(f, g, s) \ne 0\). Then, under certain technical hypotheses, the Bloch–Kato Selmer groups \(H^1_{\mathrm {f}}(\mathbf {Q}, M(f) \otimes M(g)(s))\) and \(H^1_{\mathrm {f}}(\mathbf {Q}, M(f)^* \otimes M(g)^*(1s))\) are both zero, where M(f) and M(g) are the padic representations attached to f and g.
One particularly interesting case is when \(f = f_E\) is the modular form attached to an elliptic curve E, and g is a weight 1 form corresponding to a 2dimensional odd irreducible Artin representation \(\rho \). In this case, the Bloch–Kato Selmer group \(H^1_{\mathrm {f}}(\mathbf {Q}, M(f) \otimes M(g)(1))\) is essentially the \(\rho \)isotypical part of the pSelmer group of E over the splitting field of \(\rho \), so we obtain new cases of the finiteness of Selmer (and hence Tate–Shafarevich) groups. See Theorem 8.4.1 for the precise statement.
Remark
Since this paper was originally submitted, it has come to light that there are some unresolved technical issues in the paper [32] upon which Theorem B, and hence Theorem C, relies. We hope that these issues will be resolved in the near future; as a temporary expedient, we have given in Sect. 9 below an alternate proof of a weaker form of Theorem B which avoids these problems and thus suffices to give an unconditional proof of Theorem C.
This paper could not have existed without the tremendous legacy of mathematical ideas left by the late Robert Coleman. We use Coleman’s work in three vital ways: firstly, Coleman was the first to construct the padic families of modular forms along which we interpolate; secondly, the PerrinRiou big logarithm map is a generalisation of Coleman power series in classical Iwasawa theory (introduced in Coleman’s Cambridge Part III dissertation); and finally, the results of [18] giving the link to values of padic Lfunctions, which are the main input to Theorem B, are proved using Coleman’s padic integration theory. We are happy to dedicate this paper to the memory of Robert Coleman, and we hope that his work continues to inspire other mathematicians as it has inspired us.
2 Analytic preliminaries
The aim of this section is to extend some of the results of Appendix A.2 of [20], by giving a criterion for a collection of cohomology classes to be interpolated by a distributionvalued cohomology class.
2.1 Continuous cohomology
We first collect some properties of Galois cohomology of profinite groups acting on “large” topological \(\mathbf {Z}_p\)modules (not necessarily finitely generated over \(\mathbf {Z}_p\)). A very rich theory is available for groups G satisfying some mild finiteness hypotheses (see e.g. [29, §1.1]), but we will need to consider the Galois groups of infinite padic Lie extensions, which do not have good finiteness properties, so we shall proceed on a somewhat ad hoc basis, concentrating on \(H^0\) and \(H^1\).
Definition 2.1.1
 (i)
If G is a profinite group, a topological Gmodule is an abelian topological group M endowed with an action of G which is (jointly) continuous as a map \(G \times M \rightarrow M\).
 (ii)
For G and M as in (i), we define the cohomology groups \(H^*(G, M)\) as the cohomology of the usual complex of continuous cochains \(C^\bullet (G, M)\).
 (iii)
We equip the groups \(C^i(G, M) = \mathrm{Maps}(G^i, M)\) with the compactopen topology (equivalently, the topology of uniform convergence).
With these definitions, the groups \(C^*(G, )\) define a functor from topological Gmodules to complexes of topological groups (i.e. the topology is functorial in M, and the differentials \(C^i(G, M) \rightarrow C^{i+1}(G, M)\) 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 \({\mathbf {Q}_p}\). Then any finitely generated Amodule has a unique Banach space structure making it into a Banach Amodule [7, Proposition 3.7.3/3].
Proposition 2.1.2
 (1)
the space \(B^1(G, M)\) is closed in \(Z^1(G, M)\);
 (2)
the subspace topology induced by \(B^1(G, M) \hookrightarrow Z^1(G, M)\) coincides with the quotient topology induced by \(M \twoheadrightarrow B^1(G, M)\);
 (3)
the quotient map \(M \twoheadrightarrow B^1(G, M)\) has a continuous section (not necessarily Alinear or Gequivariant).
Proof
Parts (2) and (3) now follow from the open image theorem [10, Proposition I.1.3], which shows that any continuous surjective map between \({\mathbf {Q}_p}\)Banach spaces has a continuous section (and, in particular, a continuous bijection between \({\mathbf {Q}_p}\)Banach spaces must be a homeomorphism). \(\square \)
Remark 2.1.3
It seems likely that this result is true for any finitely generated Amodule M with Gaction (without assuming that M be free), but we do not know how to prove this.
Definition 2.1.4
If X and Y are two \({\mathbf {Q}_p}\)Banach spaces, let \(\mathcal {L}_w(X, Y)\) denote the space of continuous linear maps \(X \rightarrow Y\) equipped with the weak topology (the topology of pointwise convergence).
Now if M is a \({\mathbf {Q}_p}\)Banach space with a continuous action of a profinite group G, then \(\mathcal {L}_w(X, M)\) also acquires a continuous Gaction by composition, for any Banach space X.
Proposition 2.1.5
Proof
Proposition 2.1.6
Proof
The injectivity of the first map, and the exactness at \(H^1(G, M)\), is easily seen by a direct cocycle computation (which is valid for arbitrary topological Gmodules).
Exactness at \(H^1(H, M)^{G/H}\) is much more subtle. Let \(\sigma {:}\,H \rightarrow M\) be a continuous cocycle whose class \([\sigma ] \in H^1(H, M)\) is Ginvariant. Then, for any \(g \in G\), the element \(\sigma ^g  \sigma \) lies in \(B^1(H, M)\), where \(\sigma ^g\) is the cocycle \(h \mapsto g \sigma (g^{1} h g)\). This defines a continuous map \(G \rightarrow B^1(H, M)\).
By hypothesis, the differential \(M \rightarrow B^1(H, M)\) has a continuous section. Composing this with the above map, we obtain a continuous map \(\phi {:}\,G \rightarrow M\) such that \(g\sigma (g^{1}hg)  \sigma (h) = (h1) \phi (g)\) for all \(h\in H\) and \(g \in G\). We may now argue as in the usual proof of the exactness of the inflationrestriction exact sequence for discrete modules [25, Proposition 1.6.5] to define a continuous 1cochain \(\tilde{\sigma }{:}\,G \rightarrow M\) such that \(\tilde{\sigma } _{H} = \sigma \) and \(d\tilde{\sigma } \in Z^2(G/H, M^H)\), which gives exactness at \(H^1(H, M)^{G/H}\). \(\square \)
Remark 2.1.7
The hypotheses of this proposition are satisfied, in particular, for any module of the form \(M = \mathcal {L}_w(X, N)\) where X is any Banach space, N is finitely generated and free over a Noetherian Banach algebra A, and the group H acts Alinearly on N and trivially on X. This covers all the cases we shall need below.
2.2 Distributions
Proposition 2.2.1
Proof
See [10], Lemma II.2.5. \(\square \)
As well as the Banach topology induced by the above norms (the socalled strong topology), the space \(D_\lambda (\mathbf {Z}_p, {\mathbf {Q}_p})\) also has a weak topology,^{1} which can be defined as the weakest topology making the evaluation maps \(\mu \mapsto \int {f}\,\mathrm {d}\mu \) continuous for all \(f \in C_\lambda (\mathbf {Z}_p, {\mathbf {Q}_p})\).
Remark 2.2.2
The weak topology is much more useful for our purposes than the strong topology, since the natural map \(\mathbf {Z}_p\hookrightarrow D_0(\mathbf {Z}_p, {\mathbf {Q}_p})\) given by mapping \(a \in \mathbf {Z}_p\) to the linear functional \(f \mapsto f(a)\) is not continuous in the strong topology, while it is obviously continuous in the weak topology.
More generally, if M is a \({\mathbf {Q}_p}\)Banach space, we define \(D_\lambda (\mathbf {Z}_p, M) = {{\mathrm{Hom}}}_{\mathrm {cts}}(C_\lambda (\mathbf {Z}_p, {\mathbf {Q}_p}), M)\); as before, this has a strong topology induced by the operator norm (which we write as \(\Vert \Vert _\lambda \)), and a weak topology given by pointwise convergence on \(C_\lambda (\mathbf {Z}_p, {\mathbf {Q}_p})\).
Proposition 2.2.3
Let X be a compact Hausdorff space, and M a Banach space, and let \(\sigma {:}\,X \rightarrow D_\lambda (\mathbf {Z}_p, M)\) be a continuous map (with respect to the weak topology on \(D_\lambda (\mathbf {Z}_p, M))\). Then \(\sup \{ \Vert \sigma (x)\Vert _{\lambda }{:}\,x \in X\} < \infty \).
Proof
For each \(f \in C_\lambda (\mathbf {Z}_p, {\mathbf {Q}_p})\), the map \(X \rightarrow M\) given by \(x \mapsto \sigma (x)(f)\) is continuous, and hence bounded. By the Banach–Steinhaus theorem, this implies that the collection of linear maps \(\{ \sigma (x){:}\,x \in X\}\) is bounded in the uniform norm.\(\square \)
Definition 2.2.4
For \(h\ge 0\), denote by \(LP^{[0, h]}(\mathbf {Z}_p, {\mathbf {Q}_p})\) the space of locally polynomial functions on \(\mathbf {Z}_p\) of degree \(\leqslant h\). If M is a \({\mathbf {Q}_p}\)vector space, write \(D_{\mathrm {alg}}^{[0, h]}(\mathbf {Z}_p,M)\) for the \({\mathbf {Q}_p}\)linear homomorphisms of \(LP^{[0, h]}(\mathbf {Z}_p, {\mathbf {Q}_p})\) into M.
Remark 2.2.5
Lemma 2.2.6
Let \((\mu _n)_{n \geqslant 1}\) be a sequence of elements of \(D_\lambda (\mathbf {Z}_p, M)\) which is uniformly bounded (i.e. there is a constant C such that \(\Vert \mu _n\Vert _{\lambda } \leqslant C\) for all n), let \(\mu \in D_\lambda (\mathbf {Z}_p, M)\), and let \(h \geqslant \lfloor \lambda \rfloor \) be an integer. If we have \(\int f \, \mathrm {d}\mu _n \rightarrow \int f \, \mathrm {d}\mu \) as \(n \rightarrow \infty \) for all \(f\in LP^{[0, h]}(\mathbf {Z}_p, {\mathbf {Q}_p})\), then \(\mu _n \rightarrow \mu \) in the weak topology of \(D_\lambda (\mathbf {Z}_p, M)\).
Proof
This is immediate from the density of \(LP^{[0, h]}(\mathbf {Z}_p, {\mathbf {Q}_p})\) in \(C_\lambda (\mathbf {Z}_p, {\mathbf {Q}_p})\). \(\square \)
Finally, if U is an open subset of \(\mathbf {Z}_p\), we define \(D_\lambda (U, M)\) as the subspace of \(D_\lambda (\mathbf {Z}_p, M)\) 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 [10] for local Galois representations, but also incorporating some ideas from Appendix A.2 of [20].
We consider either of the two following settings: either K is a finite extension of \({\mathbf {Q}_p}\) and \(G = {{\mathrm{Gal}}}(\overline{K}{/}K)\), or K is a finite extension of \(\mathbf {Q}\) and \(G = {{\mathrm{Gal}}}(K^S{/}K)\), 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 \(K_\infty = K(\mu _{p^\infty })\), and \(H = {{\mathrm{Gal}}}(\overline{K}{/}K_\infty )\) (resp. \({{\mathrm{Gal}}}(K^S{/}K_\infty )\) in the global case). Thus H is closed in G and the cyclotomic character identifies \(\varGamma = G{/}H\) with an open subset of \(\mathbf {Z}_p^\times \).
Remark 2.3.1
More generally, one may take for \(K_\infty \) any abelian padic Lie extension of K of dimension 1; see forthcoming work of Francesc Castella and MingLun Hsieh for an application of this theory in the context of anticyclotomic extensions of imaginary quadratic fields.
As in Sect. 2.1 above, we let A be a Noetherian \({\mathbf {Q}_p}\)Banach algebra, and M a finite free Amodule with a continuous Alinear action of H, and we fix a choice of norm \(\Vert \cdot \Vert _M\) on M making it into a Banach Amodule. We shall be concerned with the continuous cohomology \(H^1(K_\infty , D_{\lambda }(\varGamma , M))\), where \(D_{\lambda }(\varGamma , M)\) is equipped with the weak topology. Note that this cohomology group is endowed with a supremum seminorm, since every continuous cocycle \(H \rightarrow D_\lambda (\varGamma , M)\) is bounded by Proposition 2.2.3.
Proposition 2.3.2
Let \(\lambda \in \mathbf {R}_{\geqslant 0}\). Then \(H^1(K_\infty , D_\lambda (\varGamma , M))\) injects into \(H^1(K_\infty , D_{\mathrm {alg}}^{[0, h]}(\varGamma , M))\) for any integer \(h \geqslant \lfloor \lambda \rfloor \).
Proof
For the injectivity, see Proposition II.2.1 of [10], where this result is proved for arbitrary Banach representations M such that \(B^1(K_\infty , M)\) is closed in \(Z^1(K_\infty , M)\); Proposition 2.1.2 shows that this is automatic under our present hypotheses on M (the argument in op.cit. is given for K local, but it applies identically in the global case too).
To describe the image of this map, we follow the argument of Proposition II.2.3 of op.cit. in which the result is shown for \(A = {\mathbf {Q}_p}\) and K local. Exactly as in op.cit., given any class in \(H^1(K_\infty , D_{\mathrm {alg}}^{[0, h]}(\varGamma , M))\) satisfying \((\star )\), then we may represent it by a cocycle \(g \mapsto \mu (g)\) in \(Z^1(K_\infty , D_{\mathrm {alg}}^{[0, h]}(\varGamma , M))\) which also satisfies \((\star )\) in the supremum norm. For each \(h \in H\), we see that \(\mu (h)\) lies in the image of \(D_\lambda (\varGamma , M) \hookrightarrow D^{[0, h]}_{\mathrm {alg}}(\varGamma , M)\). Thus \(\mu \) defines a cocycle on H with values in \(D_\lambda (\varGamma , M)\). Moreover, the values \(\Vert \mu (h)\Vert _\lambda \) for \(h \in H\) are bounded above by a constant multiple of the supremum of the sequence in \((\star )\), by Proposition 2.2.1.
It remains to check that the cocycle \(g \mapsto \mu (g)\) is continuous (for the weak topology of \(D_\lambda (\varGamma , M)\)). 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 \(g_n \rightarrow g\), the sequence \(\mu _n := \mu (g_n)\) converges to \(\mu (g)\) in \(D_\lambda (\varGamma , M)\). However, by construction we know that \(\int f\, \mathrm {d}\mu _n\) converges to \(\int f\, \mathrm {d}\mu \) for each \(f \in LP^{[0,h]}(\varGamma , {\mathbf {Q}_p})\). Since the \(\mu _n\) are uniformly bounded, Lemma 2.2.6 shows that they converge weakly to \(\mu (g)\) as required. \(\square \)
Proposition 2.3.3

For all \(n \geqslant 0\), we have \(\sum _{\gamma \in \varGamma _n{/}\varGamma _{n + 1}} \chi (\gamma )^{j} \gamma \cdot x_{n+1, j} = x_{n, j}\).

There is a constant C such thatfor all n.$$\begin{aligned} \left\ p^{hn} \sum _{j = 0}^h (1)^j \left( {\begin{array}{c}h\\ j\end{array}}\right) x_{n, j}\right\ \leqslant Cp^{\lfloor \lambda n \rfloor } \end{aligned}$$
Proof
By Proposition 2.3.2, the existence of the constant C implies that \(\mu ^{\mathrm {alg}}\) is the image of a class \(\mu \in H^1(K_\infty , D_\lambda (\varGamma , M))\), which must itself be \(\varGamma \)invariant since the injection \( H^1(K_\infty , D_\lambda (\varGamma , M)) \hookrightarrow H^1(K_\infty , D_{\mathrm {alg}}^{[0, h]}(\varGamma , M))\) commutes with the action of \(\varGamma \). This proposition also shows that \(\Vert \mu \Vert _\lambda \) is bounded above by CD. \(\square \)
Using the inflationrestriction exact sequence (and the fact that \(\varGamma \) has cohomological dimension 1) we see that \(\mu \) lifts to a class in \(H^1(K, D_{\lambda }(\varGamma , M))\). This lift is not necessarily unique, but it is unique modulo \(H^1(\varGamma , D_\lambda (\varGamma , M^{G_{K_\infty }}))\) (and thus genuinely unique if \(M^{G_{K_\infty }} = 0\)).
2.4 Iwasawa cohomology
We now show that there is an interpretation of the module \(H^1(K, D_\lambda (\varGamma , M))\) in terms of Iwasawa cohomology. Since the group G has excellent finiteness properties (unlike its subgroup H), we have the general finitegeneration and basechange results of [29] at our disposal.
We now assume that A is a reduced affinoid algebra over \({\mathbf {Q}_p}\). By a theorem of Chenevier (see [9, Lemma 3.18]) we may find a Banach algebra norm on A, with associated unit ball \(A^{\circ } = \{ a \in A{:}\,\Vert a\Vert \leqslant 1\}\), and a compatible Banach Amodule norm on M with unit ball \(M^{\circ } \subset M\), such that G preserves \(M^{\circ }\) and \(M^{\circ }\) is locally free as an \(A^{\circ }\)module.
Definition 2.4.1
This is evidently independent of the choice of lattice \(M^{\circ }\).
Proposition 2.4.2
Proof
Corollary 2.4.3
Proof
Proposition 2.4.4
Proof
The corresponding statement for Iwasawa cohomology is well known, and the result now follows by tensoring with \(D^{\mathrm {la}}(\varGamma , A)\). \(\square \)
A very slightly finer statement is possible if we consider coefficients in a field:
Proposition 2.4.5
Proof
In the local case, this surprisingly nontrivial result is Proposition II.3.1 of [10]. 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 \(H^2_{\mathrm {Iw}}(K_\infty , V)\) as a \(\varLambda (\varGamma )\)module, there exists a k such that \(H^2_{\mathrm {Iw}}(K_\infty , V(k))^{\varGamma } = 0\). We may suppose (by twisting) that we have, in fact, \(H^2_{\mathrm {Iw}}(K_\infty , V)^{\varGamma } = 0\).
Let \(\nu _n = (\gamma  1)^n\) where \(\gamma \) is a topological generator of \(\varGamma \), and let T be a lattice in V. Then the submodules \(H^2_{\mathrm {Iw}}(K_\infty , T)[\nu _n]\) are an ascending sequence of \(\varLambda (\varGamma )\)submodules of the finitely generated module \(H^2_{\mathrm {Iw}}(K_\infty , T)\). Since \(\varLambda (\varGamma )\) is Noetherian and \(H^2_{\mathrm {Iw}}(K_\infty , T)\) is finitely generated, we conclude that this sequence of modules must eventually stabilise. But all the modules in this sequence are finite, since \(H^2_{\mathrm {Iw}}(K_\infty , V)^{\varGamma }\) vanishes by assumption; this implies that there is a uniform power of p (independent of n) which annihilates \(H^2_{\mathrm {Iw}}(K_\infty , T)[\nu _n]\) for all \(n \geqslant 1\) (compare the proof of [20, Proposition A.2.10], which is a similar argument with \(\nu _n = (\gamma  1)^n\) replaced by \(\gamma ^{p^n}1\)). With this in hand we may proceed as in [10]. \(\square \)
Remark 2.4.6
We do not know whether this result is valid for general padic 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 \(H^2_{\mathrm {Iw}}(K_\infty , T)\), 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 MingLun 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 \(H^2\).
Proposition 2.4.7
Suppose that V is a finitedimensional \({\mathbf {Q}_p}\)linear representation of G such that \(H^0(K_\infty , V) = 0\), and let \(D'\) be a constant annihilating the finite group \(H^0(K_\infty , V/T)\), for T a Ginvariant \(\mathbf {Z}_p\)lattice in V.
Proof
We know that \(\Vert \mu \Vert _\lambda \leqslant CD \) as elements of \(H^1(K_\infty , D_\lambda (\varGamma , V))^\varGamma \). So \(\Vert \int _\varGamma \kappa \, \mathrm {d}\mu \Vert \leqslant CD \Vert \kappa \Vert _\lambda \) as elements of \(H^1(K_\infty , V(\kappa ^{1}))^\varGamma \).
By the definition of the supremum seminorm, this is equivalent to stating that the class \( CD \Vert \kappa \Vert _\lambda \cdot \int _\varGamma \kappa \, \mathrm {d}\mu \) is the image of a class in \(H^1(K_\infty , T(\kappa ^{1}))\). This class is not uniquely determined, and hence not necessarily \(\varGamma \)invariant, but the constant \(D'\) was chosen to annihilate the kernel of \(H^1(K_\infty , T(\kappa ^{1})) \rightarrow H^1(K_\infty , V(\kappa ^{1}))\), so \( CDD ' \Vert \kappa \Vert _\lambda \cdot \int _\varGamma \kappa \, \mathrm {d}\mu \) lifts to a \(\varGamma \)invariant class.
Since \(H^0(K_\infty , T) = 0\), we conclude that \(H^1(K, T(\kappa ^{1})) \rightarrow H^1(K_\infty , T(\kappa ^{1}))^\varGamma \) is an isomorphism; thus \( CDD ' \Vert \kappa \Vert _\lambda \cdot \int _\varGamma \kappa \, \mathrm {d}\mu \) is in the image of the map \(H^1(K, T(\kappa ^{1})) \rightarrow H^1(K, V(\kappa ^{1}))\) as required. \(\square \)
3 Cyclotomic compatibility congruences
In this section, we establish that the Beilinson–Flach cohomology classes constructed in [19, 20] satisfy the criteria of the previous section, allowing us to interpolate them by finiteorder distributions.
3.1 Modular curves: notation and conventions
For \(N \geqslant 4\), we write \(Y_1(N)\) for the modular curve over \(\mathbf {Z}[1/N]\) parametrising elliptic curves with a point of order N. Note that the cusp \(\infty \) is not defined over \(\mathbf {Q}\) in this model, but rather over \(\mathbf {Q}(\mu _N)\).
More generally, for M, N integers with \(M + N \geqslant 5\), we write Y(M, N) for the modular curve over \(\mathbf {Z}[1/MN]\) parametrising elliptic curves together with two sections \((e_1, e_2)\) which define an embedding of group schemes \(\mathbf {Z}/M\mathbf {Z}\times \mathbf {Z}{/}N\mathbf {Z}\hookrightarrow E\) (so that \(Y_1(N) = Y(1, N)\)). We shall only consider Y(M, N) in the case \(M \mid N\), in which case the Weil pairing defines a canonical map from Y(M, N) to the scheme \(\mu _M^{\circ }\) of primitive Mth roots of unity, whose fibres are geometrically connected.
If A is an integer prime to MN, we shall sometimes also consider the curve Y(M, N(A)) over \(\mathbf {Z}[1/AMN]\), 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 \(\mathscr {H}_{\mathbf {Z}_p}\) the relative Tate module of the universal elliptic curve over Y, which is an étale \(\mathbf {Z}_p\)sheaf on Y[1 / p]. If the prime p is clear from context, we shall sometimes drop the subscript and write \(\mathscr {H}\) for \(\mathscr {H}_{\mathbf {Z}_p}\). We write \(\mathscr {H}_{{\mathbf {Q}_p}}\) for the associated \({\mathbf {Q}_p}\)sheaf. We write \({{\mathrm{TSym}}}^k \mathscr {H}_{\mathbf {Z}_p}\) for the sheaf of degree k symmetric tensors over \(\mathscr {H}_{\mathbf {Z}_p}\); note that this is not isomorphic to the kth symmetric power, although these coincide after inverting p.
Remark 3.1.1
In this paper we will frequently consider étale 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 finite étale sheaves of ppower 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 \(H^*_{{\acute{\mathrm{e}}{\mathrm{t}}}}(Y, )\) is a shorthand for \(H^*_{{\acute{\mathrm{e}}{\mathrm{t}}}}(Y[1/p], )\).
3.2 Iwasawa sheaves
Definition 3.2.1
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 \({}_c\mathcal {RI}^{[j]}_{M, N, a}\) via a degeneracy map.
Definition 3.2.2
Remark 3.2.3
Note that \(t_m\) corresponds to \(z \mapsto z/m\) on the upper halfplane.
3.3 Compatibility congruences
Definition 3.3.1
For an arbitrary m, let \(Z(m,mN) \subseteq Y(m, m N)^2\) denote the preimage of the diagonal subvariety of \(Y_1(N)\) under the natural projection map \(Y(m, m N)^2 \rightarrow Y_1(N)^2\) (i.e. the map corresponding to the identity on the upper halfplane, not the map \(t_m\)).
Note 3.3.2
The subvariety Z(m, mN) is preserved by the action of \(\varGamma _1(N) \times \varGamma _1(N)\) and in particular by the action of the element \(u_a =\left( 1, \left( {\begin{matrix}1 &{} a \\ 0 &{} 1\end{matrix}}\right) \right) \) for any \(a \in \mathbf {Z}{/}m \mathbf {Z}\). Since \(u_a\) is an automorphism, and its inverse is \(u_{a}\), we have \((u_a)_* = (u_{a})^*\).
Assumption 3.3.3
We have \(p^{(h1)r} \mid N\), so there is a canonical section \(Y_{hr}\) of \(\mathscr {H}_{hr}\) over \(Y(mp^r, mp^r N)\).
Under this assumption, the moment map modulo \(p^{hr}\) is given by cupproduct with the element \(Y_{hr}\), so we obtain the following somewhat messy formula:
Proposition 3.3.4
Proof
This is a straightforward exercise from the definition of multiplication in the algebra \({{\mathrm{TSym}}}^\bullet \) (the factor of \((hj)!\) appears because \((Y \boxtimes Y)^{[hj]} = (h  j)! Y^{[hj]} \boxtimes Y^{[hj]}\)). \(\square \)
We can now prove the main theorem of this section:
Theorem 3.3.5
Proof
It follows from [19, Theorem 5.3.1] that if \(N'\) is any multiple of N with the same prime divisors as N, then \({}_c \mathcal {BF}^{[j]}_{p^{r}, N, a}\) is the image of \({}_c \mathcal {BF}^{[j]}_{p^{r}, N', a}\) under pushforward along the natural degeneracy map \(Y_1(N')\rightarrow Y_1(N)\). We can therefore assume without loss of generality that N satisfies Assumption 3.3.3.
We claim that when restricted to the image of \(u_a \circ \varDelta {:}\, Y(mp^r, mp^r N) \rightarrow Z(mp^r, mp^rN)\), the section \(a \cdot Y_{r} \boxtimes Y_{r} + \mathcal {CG}_{r} \otimes \zeta _{p^{r}}\) of \(\mathscr {H}_r \boxtimes \mathscr {H}_r\) is in the kernel of \((t_{mp^r} \times t_{mp^r})_\sharp \).
Since this element is annihilated by \((t_{mp^r} \times t_{mp^r})_\sharp \) modulo \(p^r\), its hth tensor power is annihilated by the same map modulo \(p^{hr}\). This gives the congruence stated above. \(\square \)
Remark 3.3.6
We shall in fact use a slight refinement of this theorem. Let \(\mathcal {E}\) be the universal elliptic curve over \(Y_1(N)\), and let \(D' = C  \{0\} \subset \mathcal {E}[p]\), where C is the universal level p subgroup. Then there is a subsheaf \(\mathscr {H}_{\mathbf {Z}_p} \langle D' \rangle \) of \(\mathscr {H}_{\mathbf {Z}_p}\), which is the preimage of \(D'\) under reduction modulo p, and a corresponding sheaf of Iwasawa modules \(\varLambda (\mathscr {H}_{\mathbf {Z}_p}\langle D'\rangle )\).
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 \(k+2 \geqslant 2\) and level \(N_f \geqslant 4\), and let L be a number field containing the qexpansion coefficients of f.
Definition 3.4.1
If f, g are two eigenforms (of some levels \(N_f, N_g\) and weights \(k+2, k' + 2 \geqslant 2\)) with coefficients in L, we write \(M_{L_{\mathfrak {P}}}(f \otimes g)\) for the tensor product \(M_{L_{\mathfrak {P}}}(f) \otimes _{L_{\mathfrak {P}}} M_{L_\mathfrak {P}}(g)\) and similarly for the dual \(M_{L_{\mathfrak {P}}}(f \otimes g)^*\). Via the Künneth formula, we may regard \(M_{L_{\mathfrak {P}}}(f \otimes g)^*\) as a quotient of \(H^2_{{\acute{\mathrm{e}}{\mathrm{t}}}}(Y_1(N)^2_{\overline{\mathbf {Q}}},{{\mathrm{TSym}}}^{[k,k']}(\mathscr {H}_{{\mathbf {Q}_p}})(2))\otimes _{{\mathbf {Q}_p}} L_{\mathfrak {P}}\), for any \(N \geqslant 4\) divisible by \(N_f\) and \(N_g\), where \({{\mathrm{TSym}}}^{[k,k']}(\mathscr {H}_{{\mathbf {Q}_p}})\) denotes the étale \({\mathbf {Q}_p}\)sheaf \({{\mathrm{TSym}}}^k \mathscr {H}_{{\mathbf {Q}_p}} \boxtimes {{\mathrm{TSym}}}^{k'} \mathscr {H}_{{\mathbf {Q}_p}}\).
3.5 Consequences for pairs of newforms
We now use the congruences of Theorem 3.3.5, together with the padic analytic machinery of Sect. 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.
Definition 3.5.2
Remark 3.5.3
Note that for \(m = 1\) the class \(\mathcal {BF}^{[f, g, j]}_{m,a}\) is the Eisenstein class \(\mathrm{AJ}_{f,g,{\acute{\mathrm{e}}{\mathrm{t}}}}\left( \mathrm {Eis}^{[k, k', j]}_{{\acute{\mathrm{e}}{\mathrm{t}}},1,N}\right) \) of [18, §5.4].
Proposition 3.5.4
We now consider “pstabilised” versions of these objects. If \(p \not \mid N_f\), we choose a root \(\alpha _f \in L\) of the Hecke polynomial of f (after extending L if necessary), and we let \(f_\alpha \) be the corresponding pstabilisation of f, so \(f_\alpha \) is a normalised eigenform of level \(N_{f_\alpha }=p N_f\), with \(U_p\)eigenvalue \(\alpha _f\) and the same \(T_\ell \)eigenvalues as f for all \(\ell \ne p\). If \(p \mid N_f\), then we assume that \(a_p(f) \ne 0\), and we set \(\alpha _f = a_p(f)\) and (for consistency) \(f_\alpha = f\) and \(N_{f_\alpha }=N_f\). We define \(\alpha _g\) and \(g_\alpha \) similarly.
Proposition 3.5.5
Proof
This is a restatement of Lemma 5.6.4 and Remark 5.6.5 of [19]. \(\square \)
We shall now interpolate the \({}_c \mathcal {BF}^{[f_\alpha , g_\alpha , j]}_{m, a}\) for varying m and j, under the following assumption:
Assumption 3.5.6
The automorphic representations \(\pi _f\) and \(\pi _g\) corresponding to f and g are not twists of each other.
Note 3.5.7
Assumption 3.5.6 is automatically satisfied if \(k \ne k'\).
Convention
By abuse of notation, we write \({}_c\mathcal {BF}^{[f_\alpha ,g_\alpha ,j]}_{mp^r,a}\) for the image of the Beilinson–Flach element in \(H^1(\mathbf {Q}({\mu _{mp^\infty }}),M_{L_{\mathfrak {P}}}(f_\alpha \otimes g_\alpha )^*)^{\varGamma _r=\chi ^{j}}\).
These elements satisfy the following compatibility:
Lemma 3.5.8
Proof
This follows from the second norm relation for the Rankin–Iwasawa classes (c.f. [19, Theorem 5.4.4]). \(\square \)
Theorem 3.5.9
Remark 3.5.10
Compare Theorem 6.8.4 of [20], which is the case \(k = k' = 0\).
Proof
Proposition 3.5.11
Proof
So the projection of \(\mathcal {L}_{M_{L_{\mathfrak {P}}}(f_\alpha \otimes g_\alpha )^*}\left( {}_c \mathcal {BF}^{[f_\alpha , g_\alpha ]}_{m, a}\right) \) to W is an element of \(D_{2\lambda }(\varGamma , {\mathbf {Q}_p}) \otimes W\) which vanishes at all but finitely many characters of the form \(j + \chi \) with \(j \in \{0, \ldots , \min (k, k')\}\) and \(\chi \) of finite order. Since \(2\lambda < 1 + \min (k, k')\), this projection must be zero as required. \(\square \)
Remark 3.5.12
We shall in fact show below that the result of Proposition 3.5.11 is actually true whenever \(\alpha _f \alpha _g\) satisfies the weaker assumption (3.5.1) (i.e. whenever the class \({}_c \mathcal {BF}^{f_\alpha , g_\alpha }_{m, a}\) 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 [20], which predicts the existence of an element in \(\bigwedge ^2 H^1_{\mathrm {Iw}}(\mathbf {Q}(\mu _{mp^\infty }), M_{L_{\mathfrak {P}}}(f \otimes g)^*)\) from which the Beilinson–Flach elements (for all choices of \(\alpha _f\) and \(\alpha _g\)) can be obtained by pairing with the map \(\mathcal {L}_{M_{L_{\mathfrak {P}}}(f \otimes g)^*}\) and projecting to a \(\varphi \)eigenspace. Clearly, pairing an element of \(\bigwedge ^2\) with the same linear functional twice will give zero.
4 Overconvergent étale cohomology and Coleman families
We now recall the construction of padic families of Galois representations attached to modular forms via “big” étale sheaves on modular curves. We follow the account of [1, §3], but with somewhat altered conventions (for reasons which will become clear later). We also use some results of Hansen [14] (from whom we have also borrowed the terminology “overconvergent étale cohomology”).
4.1 Setup and notation
Definition 4.1.1
We write \(\mathcal {W}\) for the rigidanalytic space over \({\mathbf {Q}_p}\) parametrising continuous characters of the group \(\mathbf {Z}_p^\times \). For an integer \(m \geqslant 0\), we shall write \(\mathcal {W}_m\) for the wide open subspace parametrising “maccessible” weights, which are those satisfying \(v_p(\kappa (t)^{p1}  1) > \frac{1}{p^m(p1)}\) for all \(t \in \mathbf {Z}_p^\times \).
Remark 4.1.2
Note that \(\mathcal {W}\) is isomorphic to a disjoint union of \(p1\) open unit discs, and the boundedby1 rigidanalytic functions on \(\mathcal {W}\) are canonically \(\varLambda (\mathbf {Z}_p^\times )\); while \(\mathcal {W}_m\) is the union of the corresponding open subdiscs of radius \(p^{1/p^m(p1)}\) with centres in \(\mathbf {Z}_p^\times \). Thus \(\mathcal {W}_0\) (which is the space denoted by \(\mathcal {W}^*\) in [1]) contains every \({\mathbf {Q}_p}\)point of \(\mathcal {W}\), and in particular every weight of the form \(z \mapsto z^j\), \(j \in \mathbf {Z}\).
Now let us fix some coefficient field E (a finite extension of \({\mathbf {Q}_p}\)) with ring of integers \(\mathcal {O}_E\).
Definition 4.1.3
We let U denote a wide open disc defined over E, contained in \(\mathcal {W}_m\) for some \(m \geqslant 0\), and \(\varLambda _U\) the \(\mathcal {O}_E\)algebra of rigid functions on U bounded by 1 (so \(\varLambda _U \cong \mathcal {O}_E[[u]]\)). We write \(\kappa _U\) for the universal character \(\mathbf {Z}_p^\times \hookrightarrow \varLambda (\mathbf {Z}_p^\times )^\times \rightarrow \varLambda _U^\times \).
The ring \(\varLambda _U\) is endowed with two topologies: the padic topology (which we shall not use) and the \(m_U\)adic topology, which is the topology induced by the ideals \(m_U^n\), where \(m_U\) is the maximal ideal of \(\varLambda _U\).
Definition 4.1.4
For \(m \geqslant 0\), we write \(LA_m(\mathbf {Z}_p, \varLambda _U)\) for the space of functions \(\mathbf {Z}_p\rightarrow \varLambda _U\) such that for all \(a \in \mathbf {Z}{/}p^m \mathbf {Z}\), the function \(z \mapsto f(a + p^m z)\) is given by a power series \(\sum _{n \geqslant 0} b_n z^n\) with \(b_n \rightarrow 0\) in the \(m_U\)adic topology of \(\varLambda _U\).
Lemma 4.1.5
If \(U \subseteq \mathcal {W}_m\), then the function \(z \mapsto \kappa _U(1 + pz)\) is in \(LA_m(\mathbf {Z}_p, \varLambda _U)\).
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 rigidanalytic space over \({\mathbf {Q}_p}\) defined by \(\{ x{:}\,x  a \leqslant p^{m}\text { for some } a \in \mathbf {Z}_p\} \subseteq \mathbf {A}^1_\mathrm {rig}\). Then \(LA_m(\mathbf {Z}_p, \varLambda _U)\) is precisely the space of functions \(\mathbf {Z}_p\rightarrow \varLambda _U\) which extend to rigidanalytic \(\varLambda _U\)valued functions on \(X_m\).
Remark 4.1.6
It is important to use the right topology on \(\varLambda _U\), because if one takes \(U = \mathcal {W}_m\) and writes \(x \mapsto \kappa _U( 1 + p^{m+1}x )\) as a series \(\sum c_n x^n\) with \(c_n \in \varLambda _U\), the \(c_n\) tend to zero \(m_U\)adically (the above argument shows in fact that \(c_n \in m_U^n\)), but they do not tend to zero padically.
4.2 The spaces \(D_U^{\circ }(T_0)\) and \(D_U^{\circ }(T_0')\)
Definition 4.2.1
Proposition 4.2.2
The subset \(T_0\) is preserved by right multiplication by the monoid \(\varSigma _0(p) = \begin{pmatrix}{\mathbf {Z}_p^\times } &{}\quad {\mathbf {Z}_p} \\ {\mathbf {Z}_p} &{}\quad {p\mathbf {Z}_p} \end{pmatrix} \subset \mathrm{Mat}_{2 \times 2}(\mathbf {Z}_p)\), and \(T_0'\) by the monoid \(\varSigma _0'(p) = \begin{pmatrix}{\mathbf {Z}_p} &{}\quad {\mathbf {Z}_p} \\ {p\mathbf {Z}_p} &{}\quad {\mathbf {Z}_p^\times } \end{pmatrix}\). In particular, both \(T_0\) and \(T_0'\) are preserved by scalar multiplication by \(\mathbf {Z}_p^\times \). \(\square \)
Remark 4.2.3
The definition of \(T_0\) coincides with that used in [1] (and our \(\varSigma _0(p)\) is their \(\Xi (p)\)). The subspace \(T_0'\) is the image of \(T_0\) under right multiplication by \(\begin{pmatrix}{0} &{}\quad {1} \\ {p} &{}\quad {0} \end{pmatrix}\) and conjugation by this element interchanges \(\varSigma _0(p)\) and \(\varSigma _0'(p)\).
Definition 4.2.4
Similarly, we write \(A^{\circ }_{U, m}(T_0')\) for the space of functions \(T_0' \rightarrow \varLambda _U\) which are homogenous of weight \(\kappa _U\) and are such that \(z \mapsto f(pz, 1) \in LA_m(\mathbf {Z}_p, \varLambda _U)\), again endowed with the \(m_U\)adic topology.
Proposition 4.2.5
Proof
We give the proof for \(T_0'\); the proof for \(T_0\) is similar.
For the factor \(f\left( p \cdot \frac{c + az}{d + pbz}, 1\right) \), we note that the map \(z \mapsto \frac{c + az}{d + pbz}\) preserves all the rigidanalytic neighbourhoods \(X_m\) of \(\mathbf {Z}_p\), so it preserves the ring of rigidanalytic functions convergent and bounded by 1 on these spaces; thus, \(z \mapsto g\left( \frac{c + az}{d + pbz}\right) \) is in \(LA_m(\mathbf {Z}_p, \varLambda _U)\) if \(g \in LA_m(\mathbf {Z}_p, \varLambda _U)\). \(\square \)
For the rest of this section, let T denote either \(T_0\) or \(T_0'\), and \(\varSigma \) either \(\varSigma _0\) or \(\varSigma '_0\), respectively.
Note that as a topological \(\varLambda _U\)module, \(A^{\circ }_{U, m}(T)\) is isomorphic to the space of countable sequences \((c_n)_{n=1}^\infty \) with \(c_n \in \varLambda _U\) such that \(c_n \rightarrow 0\) in the \(m_U\)adic topology.
Definition 4.2.6
Note that any linear functional \(\mu \in D^{\circ }_{U, m}(T)\) is necessarily continuous (where we endow both \(A^{\circ }_{U, m}(T)\) and \(\varLambda _U\) with their \(m_U\)adic topologies). We equip \(D^{\circ }_{U, m}(T)\) with the weak (or more formally weakstar) topology, generated by sets of the form \(\{ \mu {:}\,\mu (f) \in m_U^n \}\) for \(f \in A^{\circ }_{U, m}(T)\) and \(n \geqslant 0\), i.e. the weakest topology such that all the evaluationatf morphisms are continuous (when the target \(\varLambda _U\) is equipped with the \(m_U\)adic topology).
In this topology \(D^{\circ }_{U, m}(T)\) becomes compact; indeed, we have a topological isomorphism \(D^{\circ }_U \rightarrow \prod _{n=0}^\infty \varLambda _U\), with the inverselimit topology.
Lemma 4.2.7
Proof
Clear by construction. \(\square \)
Lemma 4.2.8
Proof
For \(T = T_0\) and \(m = 0\) this is [1, Proposition 3.10], and the generalisation to \(m \geqslant 1\) is given in [14, §2.1]. The case of \(T = T_0'\) is proved similarly [or, alternatively, follows from the case of \(T = T_0\) via conjugation by \(\left( {\begin{matrix}0 &{} \quad \\ 1 &{} p\end{matrix}}\right) \quad 0\)]. \(\square \)
Proposition 4.2.9
Proof
We give the proof for \(T_0'\), the proof for \(T_0\) being similar. Because of the homogeneity requirement, any function in \(A^{\circ }_{U, m}(T_0')\) is uniquely determined by its restriction to \(p\mathbf {Z}_p\times 1\), and this gives an isomorphism \(D^{\circ }_{U, m}(T) \cong LA_m(\mathbf {Z}_p, \mathcal {O}_E)^* \mathop {\hat{\otimes }}_{\mathcal {O}_E} \varLambda _U\). Both results now follow by passing to the inverse limit. \(\square \)
Now let \(k \in \mathcal {W}\) be an integer weight (i.e. of the form \(z \mapsto z^k\) with \(k \geqslant 0\)); any such weight automatically lies in \(\mathcal {W}_0\). As for U above, we may define a space \(A^{\circ }_{k, m}(T)\) of manalytic \(\mathcal {O}_E\)valued functions on T homogenous of weight k, and its dual \(D^{\circ }_{k, m}(T)\), for any \(m \geqslant 0\).
Restriction to T gives a natural embedding \(P^{\circ }_k \hookrightarrow A^{\circ }_{k, m}(T)\), where \(P^{\circ }_k\) is the space of polynomial functions on \(\mathbf {Z}_p^2\), homogenous of degree k, with \(\mathcal {O}_E\) coefficients. Dually, we obtain a canonical, \(\varSigma _0(p)\)equivariant projection \(\rho _k{:}\,D^{\circ }_{k, m} \rightarrow (P^{\circ }_k)^* = {{\mathrm{TSym}}}^k \mathcal {O}_E^2\).
Proposition 4.2.10
Proof
This is clear by construction. \(\square \)
4.3 The Ohta pairing
We now define a pairing between distribution modules on \(T_0\) and \(T_0'\), following [26, §4].
Definition 4.3.1
This clearly restricts to a map \(T_0 \times T_0' \rightarrow \mathbf {Z}_p^\times \), so the \(\varLambda _U\)valued function \(\Phi \) on \(T_0 \times T_0'\) given by \(\Phi (t, t') = \kappa _U(\phi (t, t'))\) is well defined, homogenous of weight \(\kappa _U\) in either variable, and manalytic whenever \(U \subseteq \mathcal {W}_m\).
Definition 4.3.2
Remark 4.3.3
Let us describe the above map slightly more concretely. We take \(m = 0\), for simplicity; then, the functions \(f_n( (x, y) ) = \kappa _U(x) \cdot (y/x)^n\) are an orthonormal basis of \(A^{\circ }_{U, 0}(T_0)\), so a distribution \(\mu \in D^{\circ }_{U, 0}(T_0)\) is uniquely determined by its moments \(\mu _n = \mu (f_n)\), which can be any sequence of elements of \(\varLambda _U\). Similarly, the functions \(g_n( (px, y) ) = \kappa _U(y) (x/y)^n\) are an orthonormal basis of \(A^{\circ }_{U, 0}(T_0')\) and any \(\mu ' \in D^{\circ }_{U, 0}(T_0')\) is uniquely determined by its moments \(\mu _n' = \mu '(g_n)\).
Given such \(\mu , \mu '\), we define an element of \(\varLambda _U\) as follows: the function \(\Phi \left( (1, z), (pw, 1) \right) = \kappa _U(1  p z w)\) can be written as a power series \(\sum a_n (wz)^n\), with \(a_n \in \varLambda _U\) such that \(a_n \rightarrow 0\) in the \(m_U\)adic topology, by Lemma 4.1.5; then \(\{ \mu , \mu '\}\) is the value of the convergent sum \(\sum _{n \geqslant 0} a_n \mu _n \mu _n'\).
4.4 Sheaves on modular curves
Notation 4.4.1
Let M, N be integers \(\geqslant 1\) with \(M \mid N\) and \(M + N \geqslant 5\). We write Y(M, N) for the modular curve over \(\mathbf {Z}[1/N]\) defined in [15, §2.1].
We recall the construction of an étale sheaf of abelian groups \(\mathscr {H}_{\mathbf {Z}_p}\), and the corresponding sheaf of Iwasawa algebras \(\varLambda (\mathscr {H}_{\mathbf {Z}_p})\), associated with the universal elliptic curve \(\mathcal {E}\) over Y(M, N), and more generally the sheaf of sets \(\mathscr {H}_{\mathbf {Z}_p}\langle D \rangle \) and sheaf of \(\varLambda (\mathscr {H}_{\mathbf {Z}_p})\)modules \(\varLambda (\mathscr {H}_{\mathbf {Z}_p}\langle D \rangle )\), where D is a subscheme of \(\mathcal {E}\) finite étale over Y(M, N). Cf. [19, §4.1].
Proposition 4.4.2
The pullbacks of the sheaves \(\varLambda (\mathscr {H}_{\mathbf {Z}_p})\), and \(\varLambda (\mathscr {H}_{\mathbf {Z}_p}\langle D \rangle )\), and \(\varLambda (\mathscr {H}_{\mathbf {Z}_p}\langle D' \rangle )\) to the proscheme \(Y(p^\infty , Np^\infty )\) are isomorphic to the constant sheaves \(\varLambda (\mathbf {Z}_p^2)\), \(\varLambda (T_0)\), and \(\varLambda (T_0')\), respectively, and the maps \([p]_*\) are induced by the natural inclusions \(T_0 \hookrightarrow \mathbf {Z}_p^2\) and \(T_0' \hookrightarrow \mathbf {Z}_p^2\).
Proof
It suffices to check the corresponding statement for the inverse systems of sheaves of sets \(\mathscr {H}_{\mathbf {Z}_p}\), \(\mathscr {H}_{\mathbf {Z}_p}\langle D \rangle \) and \(\mathscr {H}_{\mathbf {Z}_p}\langle D' \rangle \). However, over \(Y(p^\infty , Np^\infty )\) we have two sections \(e_1, e_2\) of \(\mathscr {H}_{\mathbf {Z}_p}\) identifying it with the constant sheaf \(\mathbf {Z}_p^2\), and since the level p subgroup C is generated by \(e_2 \bmod p\), the sheaf \(\mathscr {H}_{\mathbf {Z}_p}\langle D \rangle \) is precisely the subset of linear combinations \(ae_1 + be_2\) such that \(a \ne 0 \bmod p\), which is \(T_0\), while \(\mathscr {H}_{\mathbf {Z}_p}\langle D' \rangle \) is similarly identified with \(T_0'\). \(\square \)
Now let \(m \geqslant 0\), and U a wide open disc contained in \(\mathcal {W}_m\), as before.
Proposition 4.4.3
There are prosheaves of \(\varLambda _U\)modules \(\mathcal {D}^{\circ }_{U, m}(\mathscr {H}_0)\) and \(\mathcal {D}^{\circ }_{U, m}(\mathscr {H}_0')\) on Y, whose pullbacks to \(Y(p^\infty , Np^\infty )\) are the constant prosheaves \(D^{\circ }_{U, m}(T_0)\) and \(D^{\circ }_{U, m}(T_0')\), respectively, and the Galois group of \(Y(p^\infty , Np^\infty ){/}Y\) acts on \(D^{\circ }_{U, m}(T_0)\) and \(D^{\circ }_{U, m}(T_0')\) via its natural identification with the Iwahori subgroup of \({{\mathrm{GL}}}_2(\mathbf {Z}_p)\).
Proof
The above trivialisation of \(\mathscr {H}_{\mathbf {Z}_p}\) over \(Y(p^\infty , Np^\infty )\) determines a homomorphism from the étale fundamental group \(\pi _1^{{\acute{\mathrm{e}}{\mathrm{t}}}}(Y)\) to the Iwahori subgroup \(U_0(p) \subseteq {{\mathrm{GL}}}_2(\mathbf {Z}_p)\). Since \(D^{\circ }_{U, m}(T_0)\) is an inverse limit of finite right modules for \(U_0(p)\), and any finite right \(\pi _1^{{\acute{\mathrm{e}}{\mathrm{t}}}}(Y)\)module defines an étale sheaf on Y, we obtain a prosheaf \(\mathcal {D}^{\circ }_{U, m}(\mathscr {H}_0)\), and similarly for \(D^{\circ }_{U, m}(T_0')\). These are sheaves of \(\varLambda _U\)modules since the action of \(U_0(p)\) on the modules \(D^{\circ }_{U, m}(T_0)\) and \(D^{\circ }_{U, m}(T_0')\) is \(\varLambda _U\)linear. \(\square \)
Remark 4.4.4
Compare [1, §3.3]; the argument is given there for the Kummer étale site on a log rigid space over \({\mathbf {Q}_p}\) (with log structure given by the cusps), but the argument works equally well in the much simpler case of affine modular curves over \(\mathbf {Q}\).
Proposition 4.4.5
Proof
We have the diagram of Proposition 4.2.10, which we may interpret as a diagram of constant prosheaves on \(Y(p^\infty , Np^\infty )\), 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. \(\square \)
Definition 4.4.6

The space \(M^{\circ }_{U, m}(\mathscr {H}_0)\) is isomorphic to the group cohomology \(H^1\left( \varGamma , \mathcal {D}_{U, m}^{\circ }(T_0)\right) \), where \(\varGamma = \varGamma _1(N(p)) = \varGamma _1(N) \cap \varGamma _0(p)\) (since \(Y_1(N(p))(\mathbf {C})\) has contractible universal cover and its fundamental group is \(\varGamma _1(N) \cap \varGamma _0(p)\)).

The space \(M^{\circ }_{U, m}(\mathscr {H}_0)_c\) is isomorphic to the space of modular symbols$$\begin{aligned} {{\mathrm{Hom}}}_{\varGamma }\left( \mathrm{Div}^0(\mathbf {P}^1_{\mathbf {Q}}), \mathcal {D}_{U, m}^{\circ }(T_0)\right) . \end{aligned}$$
Notation 4.4.7
We shall refer to \(M^{\circ }_{U, m}(\mathscr {H}_0)\) and \(M^{\circ }_{U, m}(\mathscr {H}_0')\) as étale overconvergent cohomology (of weight U, tame level N and degree of overconvergence m).
We now state some properties of these modules:
Proposition 4.4.8
 (1)(Compatibility with specialisation) Let \(\varpi _k\) be the ideal of \(\varLambda _U\) corresponding to the character \(z \mapsto z^k\). For any integer \(k \geqslant 0 \in U\), there is an isomorphismFor compactly supported cohomology this is true for \(k \geqslant 1\), while for \(k = 0\) we have an injective map$$\begin{aligned} M^{\circ }_{U, m}(\mathscr {H}_0){/}\varpi _k \cong M^{\circ }_{k, m}(\mathscr {H}_0).\end{aligned}$$whose cokernel has rank 1 over \(\mathcal {O}_E\), with the Hecke operator \(U_p\) acting as multiplication by p. Similar statements hold for \(\mathscr {H}_0'\) in place of \(\mathscr {H}_0\).$$\begin{aligned} M^{\circ }_{U, m}(\mathscr {H}_0)_c{/}\varpi _0 \hookrightarrow M^{\circ }_{0, m}(\mathscr {H}_0)_c \end{aligned}$$
 (2)(Control theorem) For any integer \(k \geqslant 0\), the map is an isomorphism on the \(U_p = \alpha \) eigenspace, for any \(\alpha \) of valuation \(< k + 1\). The same holds for compactly supported and parabolic cohomology, and for \(\mathscr {H}_0'\) and \(U_p'\) in place of \(\mathscr {H}_0\) and \(U_p\).
 (3)(Duality) There are \(\varLambda _U\)bilinear, \(G_{{\mathbf {Q}_p}}\)equivariant pairingswhich we denote by \(\{ , \}\). For integers \(k \geqslant 0\) we have$$\begin{aligned} M^{\circ }_{U, m}(\mathscr {H}_0)_c \times M^{\circ }_{U, m}(\mathscr {H}_0')&\rightarrow \varLambda _U, \\ M^{\circ }_{U, m}(\mathscr {H}_0) \times M^{\circ }_{U, m}(\mathscr {H}_0')_c&\rightarrow \varLambda _U,\\ M^{\circ }_{U, m}(\mathscr {H}_0)_{\mathrm {par}} \times M^{\circ }_{U, m}(\mathscr {H}_0')_{\mathrm {par}}&\rightarrow \varLambda _U, \end{aligned}$$where \(\mathrm{ev}_k\) is evaluation at k, and on the righthand side \(\{, \}_k\) signifies the Poincaré duality pairing.$$\begin{aligned} \mathrm{ev}_k\left( \{ x, x'\} \right) = \left\{ \rho _k(x), \rho _k(x') \right\} _k \end{aligned}$$
 (4)
There is an isomorphism \(W{:}\,M^{\circ }_{U, m}(\mathscr {H}_0)_{?} \rightarrow M^{\circ }_{U, m}(\mathscr {H}_0')_?\) (where \(? \in \{ \varnothing , c, \mathrm {par}\})\), intertwining the action of the Hecke operators \(T_n\) with the \(T_n'\) (including \(n = p)\); this is compatible via the maps \(\rho _k\) with the Atkin–Lehner operator \(W_{Np}\) (but not with the Galois action).
Proof
For part (1), see [1, Lemma 3.18]. For compactly supported cohomology see [2, Theorem 3.10]. (Bellaïche works with coefficients in an affinoid disc, rather than a wide open disc as we do, but the argument is the same.)
Part (2) is the celebrated Stevens control theorem; see [1, Theorem 3.16] for \(H^1\) and [28, Theorem 1.1] for \(H^1_c\).
Part (4) follows from the fact that the action of the matrix \(\left( {\begin{matrix}0 &{} 1 \\ Np &{} 0\end{matrix}}\right) \) on H interchanges \(T_0\) and \(T_0'\). \(\square \)
Remark 4.4.9
The pairing \(\{ , \}\) (in any of its various incarnations) is far from perfect (since its specialisation at a classical weight \(k \geqslant 0\) factors through the maps \(\rho _k\), so any nonclassical eigenclass of weight k must be in its kernel). Nonetheless, we shall see below that it induces a perfect pairing on small slope parts.
4.5 Slope decompositions
As before, let U be a wide open disc contained in \(\mathcal {W}_m\), for some m. Let \(B_U = \varLambda _U[1/p]\), and let M be one of the \(B_U\)modules \(M_{U, m}(\mathscr {H}_0)_?\), for \(? \in \{ \varnothing , c, \mathrm {par}\}\), and let \(\lambda \in \mathbf {R}_{\geqslant 0}\).
Definition 4.5.1

the action of the Hecke operator \(U_p\) preserves the two summands;

the module \(M_U^{(\leqslant \lambda )}\) is finitely generated over \(B_U\);

the restrictions of \(U_p\) to \( M_U^{(\leqslant \lambda )}\) and \(M_U^{(> \lambda )}\) have slope \(\leqslant \lambda \) and slope \(> \lambda \), respectively.
Remark 4.5.2
There are several equivalent definitions of slope \(\leqslant \lambda \), see [1] for further discussion. We shall use the following formulation: the endomorphism \(U_p\) of \(M_U^{(\leqslant \lambda )}\) is invertible, and the sequence of endomorphisms \(\left( p^{\lfloor n\lambda \rfloor } \cdot (U_p)^{n}\right) _{n \geqslant 0}\) is bounded in the operator norm.
Note that the summands \(M^{(\leqslant \lambda )}\) and \(M^{(> \lambda )}\) must be stable under the actions of the primetop Hecke operators, and of the Galois group \(G_{\mathbf {Q}}\), since these commute with the action of \(U_p\).
Theorem 4.5.3
([1, Theorem 3.17]) Let \(k \geqslant 0\) and \(0 \leqslant \lambda < k + 1\). Then there exists an open disc \(U \ni k\) in \(\mathcal {W}\), defined over E, such that the module \(M_{U, 0}(\mathscr {H}_0)\) has a slope \(\leqslant \lambda \) decomposition.
The same results hold mutatis mutandis for \(M = M_{U, 0}(\mathscr {H}_0')\), 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:
Definition 4.6.1
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 \(z \mapsto z^k \chi (z)\) for \(\chi \) of finite order and allow families indexed by a finite flat rigidanalytic 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

\(f_\alpha \) is a pstabilisation of a newform f of level N whose Hecke polynomial \(X^2  a_p(f) X + p^{k+1} \varepsilon _f(p)\) has distinct roots (“pregularity”);

if \(v_p(\alpha ) = k + 1\), then the Galois representation \(M_E(f) _{G_{{\mathbf {Q}_p}}}\) is not a direct sum of two characters (“noncriticality”).
Theorem 4.6.4
Suppose \(f_\alpha \) is a noble eigenform of weight \(k_0 + 2\). Then there exists a disc \(U \ni k_0\) in \(\mathcal {W}\), and a unique Coleman family \(\mathcal {F}\) over U, such that \(\mathcal {F}_{k_0} = f_\alpha \).
Proof
This follows from the fact that the Coleman–Mazur–Buzzard eigencurve \(\mathscr {C}(N)\) of tame level N is étale over \(\mathcal {W}\) (and, in particular, smooth) at the point corresponding to a noble eigenform \(f_\alpha \). See [2].
Remark 4.6.5
As remarked in [14], 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 \({\geqslant }2\) such that \(M_E(f)_{G_{{\mathbf {Q}_p}}}\) splits as a direct sum are those which are of CM type with p split in the CM field.
Theorem 4.6.6

The moduleis a direct summand of \(M_{U, 0}(\mathscr {H}_0)\) as a \(B_U\)module, free of rank 2 over \(B_U\), and lifts canonically to \(M_{U, 0}(\mathscr {H}_0)_c\).$$\begin{aligned} M_U(\mathcal {F}) :=M_{U, 0}(\mathscr {H}_0) \Big [T_n = a_n(\mathcal {F})\ \forall n \geqslant 1\Big ] \end{aligned}$$

The same is true of the module$$\begin{aligned} M_U(\mathcal {F})^* :=M_{U, 0}(\mathscr {H}_0') \Big [T_n' = a_n(\mathcal {F})\ \forall n \geqslant 1\Big ]. \end{aligned}$$

The pairing \(\{, \}\) induces an isomorphism of \(B_U[G_{{\mathbf {Q}_p}}]\)modules$$\begin{aligned} M_U(\mathcal {F})^* \cong {{\mathrm{Hom}}}_{B_U}(M_U(\mathcal {F}), B_U). \end{aligned}$$

For each \(k \geqslant 0 \in U\), the form \(\mathcal {F}_k\) is a classical eigenform, and we have isomorphisms of Elinear \(G_{{\mathbf {Q}_p}}\)representations$$\begin{aligned} M_U(\mathcal {F}){/}\varpi _k M_U(\mathcal {F}) = M_E(\mathcal {F}_k) \quad \text {and}\quad M_U(\mathcal {F})^*{/}\varpi _k M_U(\mathcal {F})^* = M_E(\mathcal {F}_k)^*. \end{aligned}$$
Proof
The finiteslope parts of all the various overconvergent cohomology groups can be glued into coherent sheaves on the eigencurve \(\mathscr {C}(N)\). In a neighbourhood of a noble point, the eigencurve is étale 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 [14, Proposition 2.3.5] for further details. \(\square \)
4.7 Weight one forms
If f is a cuspidal newform of level N and weight 1, and \(f_\alpha \) is a pstabilisation of f, then it is always the case that \(v_p(\alpha ) = k_0 + 1 = 0\) and \(M_E(f)_{G_{{\mathbf {Q}_p}}}\) splits as a direct sum (since \(M_E(f)\) is an Artin representation). Nonetheless, analogues of Theorems 4.6.4 and 4.6.6 do hold for these forms.
Notation 4.7.1
We say that f has real multiplication by a real quadratic field K if there is a Hecke character \(\psi \) of K such that \(M_E(f) \cong \mathrm{Ind}_{G_K}^{G_\mathbf {Q}}(\psi )\).
Theorem 4.7.2
 (1)
There is an open disc \(U \ni 1\) in \(\mathcal {W}\), a finite flat rigidanalytic covering unramified away from \(1\) and totally ramified at \(1\), and a family of eigenforms \(\mathcal {F}\in B_{\tilde{U}}[[q]]\), whose specialisation at \(\kappa ^{1}(1)\) is \(f_\alpha \). We may take \(\tilde{U} = U\) if (and only if) f does not have real multiplication by a quadratic field in which p is split.
 (2)The moduleis a direct summand of \(\kappa ^* M_{U, 0}(\mathscr {H}_0)\), free of rank 2 as a \(B_{\tilde{U}}\)module, and lifts canonically to \(\kappa ^* M_{U, 0}(\mathscr {H}_0)_c\).$$\begin{aligned} M_{\tilde{U}}(\mathcal {F}) = \left( \kappa ^* M_{U, 0}(\mathscr {H}_0)\right) \left[ T_n = a_n(\mathcal {F}) \, \forall n \geqslant 1\right] \end{aligned}$$
 (3)The same is true ofand the pairing \(\{, \}\) induces an isomorphism \(M_{\tilde{U}}(\mathcal {F})^* \cong {{\mathrm{Hom}}}_{B_{\tilde{U}}}(M_{\tilde{U}}(\mathcal {F}), B_{\tilde{U}})\).$$\begin{aligned} M_{\tilde{U}}(\mathcal {F})^* = \left( \kappa ^* M_{U, 0}(\mathscr {H}_0')\right) \left[ T_n' = a_n(\mathcal {F}) \, \forall n \geqslant 1\right] , \end{aligned}$$
Proof
Part (1) is exactly the statement that the eigencurve is smooth at the point corresponding to \(f_\alpha \), and is étale over weight space except in the realmultiplication setting, see [3].
Part (2) for compactly supported cohomology is an instance of [2, Proposition 4.3]. However, the kernel and cokernel of the map \( M_{U, 0}(\mathscr {H}_0)_c \rightarrow M_{U, 0}(\mathscr {H}_0)\) are supported on the Eisenstein component of the eigencurve, and since \(f_\alpha \) is a smooth point on the cuspidal eigencurve \(\mathscr {C}^0(N) \subset \mathscr {C}(N)\), it does not lie on the Eisenstein component. Hence the kernel and cokernel localise to 0 at \(f_\alpha \), implying that for small enough U the \(\mathcal {F}\)eigenspaces of \(M_U(\mathscr {H}_0)_c\) and \(M_U(\mathscr {H}_0)\) coincide.
For part (3) we use the fact that the Ohta pairings induce perfect dualities on the ordinary parts of the modules \(M_U(\mathscr {H}_0)_c\) and \(M_U(\mathscr {H}_0')\) (cf. [26]). \(\square \)
5 Rankin–Eisenstein classes in Coleman families
5.1 Coefficient modules
Let H be a group isomorphic to \(\mathbf {Z}_p^2\) (but not necessarily canonically so), for p an odd prime. Then we can regard the modules \({{\mathrm{TSym}}}^r H\) as representations of \(\mathrm{Aut}(H) \approx \mathrm{GL}_2(\mathbf {Z}_p)\). In this section, we shall show that the Clebsch–Gordan decompositions of the groups \({{\mathrm{TSym}}}^r H \otimes {{\mathrm{TSym}}}^s H\) 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
Proof
Let us begin by defining the maps. The map \(\beta \), which is the simpler of the two, is given by interpreting \({{\mathrm{Sym}}}^j H^\vee \) 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 \(C(A) \otimes C(A) \rightarrow C(A)\).
To show that the map \(\beta \) is surjective, we write down a (noncanonical) section. We can decompose A as a union \(A_1 \sqcup A_2\) where x is invertible on \(A_1\) and y is invertible on \(A_2\). We define \(\delta (f) = (x^{j} f, 0, \ldots , 0)\) on \(C(A_1)\) and \(\delta (f) = (0, \ldots , 0, y^{j} f)\) on the \(C(A_2)\) factor; then \(\beta \circ \delta \) is clearly the identity, so \(\beta \) is surjective.
Finally, let \((f_0, \ldots , f_j) \in \ker (\beta )\). Choosing \(A = A_1 \sqcup A_2\) as before, we may assume either x or y is invertible on A. We treat the first case, the second being similar. We define \(\gamma (f_1, \ldots , f_j) = (g_0, \ldots , g_{j1})\) where \(g_{j1} = x^{1} f_j\), \(g_{j2} = x^{2}(x f_{j1} + y f_j)\), etc., down to \(g_0 = x^{j}(x^{j1} f_{1} + \cdots + y^{j1} f_j)\). But then \((\alpha \circ \gamma ) + (\beta \circ \delta ) = {{\mathrm{id}}}\), so we have exactness at the middle term. \(\square \)
Proposition 5.1.2
Proof
The morphism \(\delta \) is simply \(\tfrac{1}{j!}\) times the jth power of the total derivative map \(C^{\mathrm {la}}(A) \rightarrow C^{\mathrm {la}}(A) \otimes \mathrm{Tan}(A)^*\), combined with the identification \(\mathrm{Tan}(A) \cong \mathrm{Tan}(H) \cong H\). From this description the naturality is clear, and a computation shows that it agrees with the more concrete description above. The identity for \(\beta \circ \delta \) is easily seen by induction on j.
\(\square \)
It will be convenient to adopt the notation \(\left( {\begin{array}{c}\nabla \\ j\end{array}}\right) \) for the endomorphism \(\tfrac{1}{j!} \prod _{i=0}^{j1}(\nabla i)\). We may regard this as an element of the space \(D^{\mathrm {la}}(\mathbf {Z}_p^\times )\) of locally analytic distributions on \(\mathbf {Z}_p^\times \).
Proposition 5.1.3
If \(k < j\), then the restriction of \(\delta \) to \({{\mathrm{Sym}}}^k H^\vee \) is the zero map.
Proof
It is obvious that \({{\mathrm{Sym}}}^k H^\vee \) embeds naturally into \(C^{\mathrm {la}}(A)\), and its image under \(\delta \) is contained in \({{\mathrm{Sym}}}^{kj} H^\vee \otimes {{\mathrm{Sym}}}^j H^\vee \). A straightforward computation in coordinates shows that this map sends \(x^a y^b\) to \(\sum _{s + t = j} \left( {\begin{array}{c}a\\ s\end{array}}\right) \left( {\begin{array}{c}b\\ t\end{array}}\right) \left( x^{as} y^{bt} \otimes x^s y^t\right) \), which coincides with the dual of the symmetrised tensor product.
On the other hand it is obvious from Eq. (5.1.1) that \(\delta \) vanishes on any polynomial of total degree \(< j\). \(\square \)
Corollary 5.1.4
Proof
This follows by dualising the previous proposition. \(\square \)
We now consider varying j, for which it is convenient to relabel the maps \(\beta ^*, \delta ^*\) above as \(\beta _j^*\) and \(\delta _j^*\).
Lemma 5.1.5
Proof
Explicit computation. \(\square \)
5.2 Nearly overconvergent étale cohomology
We also have an analogue of the Clebsch–Gordan map for the distribution spaces \(D^{\circ }_{U, m}(T_0')\) introduced above, which are completions of \(D^{\mathrm {la}}(T_0')\). The rigid space \(\mathcal {W}\) has a group structure, so we can make sense of \(U  j\) for any integer j.
Proposition 5.2.1
Proof
We simply transport the constructions of Sect. 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 \(\varSigma _0(p)\)action. Since the functions in \(A_{U, m}\) are homogenous of weight \(\kappa _U\) (the canonical character \(\mathbf {Z}_p^\times \rightarrow \varLambda _U^\times \)), we have \(\frac{\mathrm {d}}{\mathrm {d}t} f(th)_{t = 1} = \nabla \cdot f(h)\) for all \(f \in A_{U, m}\), where on the righthand side \(\nabla \) is regarded as an element of \(\varLambda _U[1/p]\); that is, the two actions of \(\nabla \) on \(A_{U, m}\), as a differential operator and as an element of the coefficient ring, coincide. \(\square \)
Remark 5.2.2
Note that \(\delta ^*_j\) takes values in \(D_{U, m} = D^{\circ }_{U, m}[1/p]\), not in \(D^{\circ }_{U, m}\) itself; the denominator arises from the fact that the map \(\delta _j\) on \(A^{\circ }_{U, m}\) does not preserve the \(\varLambda _U\)lattice \(A^{\circ }_{U, m}\), but rather maps \(A^{\circ }_{U, m}\) to \(\frac{1}{j! p^{1+m}} A^{\circ }_{U, m}\). Note also that if \(U \subset \mathcal {W}_0\) and U contains none of the integers \(\{0, \ldots , j1\}\), then \(\left( {\begin{array}{c}\nabla \\ j\end{array}}\right) \) is invertible in \(\varLambda _U[1/p]\).
The maps of spaces \(\beta _j^*\) and \(\delta _j^*\) induce maps of étale sheaves on \(Y = Y_1(N(p))\) (for any N), \(\mathcal {D}^{\circ }_{U, m}(\mathscr {H}_0') \rightarrow \mathcal {D}^{\circ }_{Uj, m}(\mathscr {H}_0') \otimes {{\mathrm{TSym}}}^j \mathscr {H}\) and \(\mathcal {D}^{\circ }_{Uj, m}(\mathscr {H}_0') \otimes {{\mathrm{TSym}}}^j \mathscr {H}\rightarrow \mathcal {D}_{U, m}(\mathscr {H}_0')\), which we denote by the same symbols.
Definition 5.2.3
Remark 5.2.4
The motivation for this terminology is that the sheaves \(\mathcal {D}_{Uj, m}(\mathscr {H}_0') \otimes {{\mathrm{TSym}}}^j \mathscr {H}\), and the maps \(\beta _j^*\) and \(\delta _j^*\) relating them to the overconvergent cohomology sheaves \(\mathcal {D}_{U, m}(\mathscr {H}_0')\), are an étale analogue of the coherent sheaves appearing in the theory of nearly overconvergent padic modular forms (see [32]).
Recall from Corollary 5.1.4 that the composite of \(\delta _j^*\) with the moment map \(\rho _k\) is zero if \(0 \leqslant k < j\), which is somewhat undesirable. We can rectify this issue as follows. Recall that we have defined \(M_U(\mathscr {H}_0') = H^1_{{\acute{\mathrm{e}}{\mathrm{t}}}}(\overline{Y}, \mathcal {D}_{U, m}(\mathscr {H}_0')(1))\).
Proposition 5.2.5
Let U be an open disc contained in \(\mathcal {W}_0\), and \(\mathcal {F}\) a Coleman family defined over U. Suppose the following condition is satisfied: for any integer weight \(k \geqslant 0\) in U, the projection map \(M_k(\mathscr {H}'_0) \rightarrow M_k(\mathcal {F})^*\) factors through \(\rho _k\).
Proof
Note that \(\nabla \), regarded as a rigidanalytic function on \(\mathcal {W}\), takes the value k at an integer weight k. So the only points in \(\mathcal {W}_0\) at which \(\nabla (\nabla  1) \ldots (\nabla  j + 1)\) fails to be invertible are the positive integers \(\{0, \ldots , j1\}\), and it has simple zeroes at all of these points.
If k is one of these integers, then we have \(M_U(\mathscr {H}_0'){/}(\nabla  k) M_U(\mathscr {H}_0') = M_k(\mathscr {H}_0')\). Hence it suffices to show that \({{\mathrm{pr}}}_{\mathcal {F}} \circ \delta _j^*\) is zero on \(M_k(\mathscr {H}_0')\), but this is immediate since the specialisation of \({{\mathrm{pr}}}_{\mathcal {F}}\) at k factors through \(\rho _k\), and \(\rho _k \circ \delta _j^*\) is zero for \(0 \leqslant k < j\).
This shows that \({{\mathrm{pr}}}_\mathcal {F}\circ \delta _j^*\) lands in the stated submodule. Since \(M_U(\mathcal {F})^*\) is a free \(\varLambda _U[1/p]\)module (and \(\varLambda _U[1/p]\) is an integral domain), the map \({{\mathrm{pr}}}_\mathcal {F}^{[j]}\) is therefore well defined. \(\square \)
Remark 5.2.6
This proposition can be interpreted as follows: we can renormalise \(\delta _j^*\) to be an inverse to \(\beta _j^*\), as long as we avoid points on the eigencurve which are nonclassical but have classical weights.
5.3 Twoparameter families of Beilinson–Flach elements
Definition 5.3.1
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 \(\varLambda (\mathscr {H}_{\mathbf {Z}_p}\langle D' \rangle )\). Cf. Remark 3.3.6 above.
Proposition 5.3.3
Proof
This follows from the last statement of 5.1.4, since \({}_c \mathcal {BF}^{[k, k', j]}_{m, Np, a}\) is by definition the image of \({}_c \mathcal {BF}^{[j]}_{m, Np, a}\) under the map \(({{\mathrm{mom}}}^{kj} \cdot {{\mathrm{id}}}) \boxtimes ({{\mathrm{mom}}}^{k'j} \cdot {{\mathrm{id}}})\). \(\square \)
Now let us choose newforms f, g, of levels \(N_1, N_2\) and weights \(k_1 + 2, k_2 + 2 \geqslant 2\), and roots \(\alpha _1, \alpha _2\) of their Hecke polynomials, such that the pstabilisations \(f_{i, \alpha _i}\) both satisfy the hypotheses of Theorem 4.6.6. The theorem then gives us families of overconvergent eigenforms \(\mathcal {F}_1\), \(\mathcal {F}_2\) passing through the pstabilisations of f and g, defined over some discs \(U_1 \ni k_1, U_2 \ni k_2\).
Proposition 5.3.4
Proof
After shrinking the discs \(U_i\) if necessary so that all integerweight specialisations of \(\mathcal {F}\) and \(\mathcal {G}\) are classical, so that Proposition 5.2.5 applies, we can simply define \({}_c \mathcal {BF}^{[\mathcal {F}, \mathcal {G}, j]}_{m, a}\) as the image of \({}_c \mathcal {BF}^{[j]}_{m, a}\) under \({{\mathrm{pr}}}_{\mathcal {F}}^{[j]} \times {{\mathrm{pr}}}_{\mathcal {G}}^{[j]}\). \(\square \)
5.4 Interpolation in j
Now let \(\mathcal {F}, \mathcal {G}\) be Coleman families over open discs \(U_1, U_2\), satisfying the conditions of Proposition 5.3.4.
Proposition 5.4.1
Proof
We now choose affinoid discs \(V_i\) contained in the \(U_i\) (so the \(M_{V_i}(\mathcal {F}_i)^*\) become Banach spaces).
Theorem 5.4.2
Proof
6 Phi–Gamma modules and triangulations
6.1 Phi–Gamma modules in families
As is well known, there is a functor \(\mathbf {D}^{\dag }_{\mathrm {rig}}\) mapping padic representations of \(G_{{\mathbf {Q}_p}}\) to \((\varphi , \varGamma )\)modules over \(\mathscr {R}\) (finitely generated free \(\mathscr {R}\)modules with commuting \(\mathscr {R}\)semilinear operators \(\varphi \) and \(\varGamma \)), and this is a fully faithful functor whose essential image is the subcategory of \((\varphi , \varGamma )\)modules of slope 0.
Remark 6.1.1
Strictly speaking, the definition of the functor \(\mathbf {D}^{\dag }_{\mathrm {rig}}\) depends on the auxiliary choice of a compatible system of ppower roots of unity \((\zeta _{p^n})_{n \geqslant 0}\) in \(\overline{\mathbf {Q}}_p\). We shall fix, once and for all, such a choice, and in applications to global problems we shall often assume that \(\zeta _{p^n}\) corresponds to \(e^{2\pi i{/}p^n} \in \mathbf {C}\).
Now let A be a reduced affinoid algebra over \({\mathbf {Q}_p}\), and write \(\mathscr {R}_A = \mathscr {R}\mathop {\hat{\otimes }}A\) and similarly for \(\mathscr {R}_A^+\). We define an Arepresentation of \(G_{{\mathbf {Q}_p}}\) to be a finitely generated locally free Amodule endowed with an Alinear action of \(G_{{\mathbf {Q}_p}}\) (continuous with respect to the canonical Banach topology of M).
Theorem 6.1.2
Definition 6.1.3
Corollary 6.1.4
Proof
Finally, if the base A is a finite field extension of \({\mathbf {Q}_p}\), then the functors \(\mathbf {D}_{\mathrm {cris}}()\) and \(\mathbf {D}_{\mathrm {dR}}()\) can be extended from Alinear representations of \(G_{{\mathbf {Q}_p}}\) to the larger category of \((\varphi , \varGamma )\)modules over \(\mathscr {R}_A\), and one has the following fact:
Theorem 6.1.5
6.2 PerrinRiou logarithms in families
Throughout this section, A denotes a reduced affinoid algebra, with supremum norm \(\Vert \cdot \Vert \), and \(\alpha \in A^\times \).
Definition 6.2.1
We write \(\mathscr {R}_A(\alpha ^{1})\) for the free rank 1 \((\varphi , \varGamma )\)module over \(\mathscr {R}_A\) with basis vector e such that \(\varphi (e) = \alpha ^{1} e\) and \(\gamma e = e\) for all \(\gamma \in \varGamma \). We write \(\mathscr {R}^+_A(\alpha ^{1})\) for the submodule \(\mathscr {R}^+_A \cdot e\) of \(\mathscr {R}_A(\alpha ^{1})\).
Lemma 6.2.2
6.3 Triangulations
Definition 6.3.1
Theorem 6.3.2
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
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
Proof of Theorem 6.4.1
Corollary 6.4.3
Proof
This follows by dualising \(\omega _\mathcal {F}\) using the Ohta pairing \(\{,\}\); the computations are exactly the same as in the ordinary case, for which see [19, Proposition 10.1.2]. \(\square \)
7 The explicit reciprocity law
7.1 Regulator maps for Rankin convolutions
Now let us choose two newforms f, g and pstabilisations \((\alpha _f, \alpha _g)\) satisfying the hypotheses of Theorem 4.6.4.
Notation 7.1.1
Theorem 7.1.2
If \(V_1\) and \(V_2\) are sufficiently small, then (for any m coprime to p) the image of \({}_c \mathcal {BF}^{[\mathcal {F}, \mathcal {G}]}_{m, a}\) under projection to the module \(H^1_{\mathrm {Iw}}(\mathbf {Q}(\mu _m) \otimes \mathbf {Q}_{p, \infty }, \mathscr {F}^{} D_{V_1 \times V_2}(\mathcal {F}\otimes \mathcal {G})^*)\) is zero.
Proof
By taking the \(V_i\) sufficiently small, we may assume that \(\mathscr {F}^{} D_{V_1 \times V_2}(\mathcal {F}\otimes \mathcal {G})^*\) is actually isomorphic to \(\mathscr {R}_A(\alpha ^{1})\), where \(\alpha = \alpha _{\mathcal {F}} \alpha _{\mathcal {G}}\) and \(A = \mathcal {O}(V_1 \times V_2)\), and that \(\Vert \alpha ^{1}\Vert < p^{1 + h}\) and \(\alpha  1\) is not a zerodivisor. It suffices, therefore, to show that \(\mathcal {L}_{\mathscr {R}_A(\alpha ^{1})}\) maps the image of \({}_c \mathcal {BF}^{[\mathcal {F}, \mathcal {G}]}_{m, a}\) to zero.
However, for each pair of integers \((\ell , \ell ') \in V_1 \times V_2\) with \(\ell , \ell ' \geqslant 1 + 2h\) and such that \(\mathcal {F}_\ell \) and \(\mathcal {G}_{\ell '}\) are not twists of each other, we know that the image of \(\mathcal {L}_{\mathscr {R}_A(\alpha ^{1})}({}_c \mathcal {BF}^{[\mathcal {F}, \mathcal {G}]}_{m, a})\) vanishes when restricted to \((\ell , \ell ') \times \mathcal {W}\subseteq \mathrm{Max}(A) \times \mathcal {W}\), by Proposition 3.5.11. Since such pairs \((\ell , \ell ')\) are Zariskidense in \(\mathrm{Max}(A)\), the result follows. \(\square \)
Remark 7.1.3
Cf. [19, Lemma 8.1.5], which is an analogous (but rather stronger) statement in the ordinary case.
Theorem 7.1.4
Proof
The construction of the map \(\mathcal {L}\) is immediate from (6.2.1). The content of the theorem is that the map \(\mathcal {L}\) recovers the maps \(\exp ^*\) and \(\log \) for the specialisations of \(\mathcal {F}\) and \(\mathcal {G}\); this follows from Nakamura’s construction of \(\exp ^*\) and \(\log \) for \((\varphi , \varGamma )\)modules. \(\square \)
Theorem 7.1.5
Proof
The two sides of the desired formula agree at every \((k, k', j)\) with \(k \in V_1\), \(k' \in V_2\) and \(0 \leqslant j \leqslant \min (k, k')\), by [18, Theorem 6.5.9]. These points are manifestly Zariskidense, and the result follows. \(\square \)
Remark 7.1.6
The construction of \(\omega _{\mathcal {G}}\) and the proof of the explicit reciprocity law are also valid if \(\mathcal {G}\) is a Coleman family passing through a pstabilisation \(g_\alpha \) of a pregular 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 covering \(\tilde{V}_2\). In this setting, \(g_\alpha \) is automatically ordinary, so \(\mathcal {G}\) is in fact a Hida family, and one can use the construction of \(\omega _{\mathcal {G}}\) given in [19, 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 \(k \geqslant 0\), and we choose an integer j such that \(k' + 1 \leqslant j \leqslant k\). If \(j = \frac{k + k'}{2} + 1\), then we assume that \(\varepsilon _f \varepsilon _g\) is not trivial, where \(\varepsilon _f\) and \(\varepsilon _g\) are the characters of f and g.
As usual, we let E be a finite extension of \({\mathbf {Q}_p}\) with ring of integers \(\mathcal {O}\), containing the coefficients of f and g. Our goal will be to bound the Selmer group associated with the Galois representation \(M_\mathcal {O}(f \otimes g)(1 + j)\), in terms of the Lvalue \(L(f, g, 1 + j)\); our hypotheses on \((k, k', j)\) are precisely those required to ensure that this Lvalue is a critical value.

(pregularity) We have \(\alpha _f \ne \beta _f\) and \(\alpha _g \ne \beta _g\), where \(\alpha _f, \beta _f\) are the roots of the Hecke polynomial of f at p, and similarly for g.

(no local zero) None of the pairwise productsis 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 \(s = 1 + j\).$$\begin{aligned} \left\{ \alpha _f \alpha _g, \alpha _f \beta _g, \beta _f \alpha _g, \beta _f \beta _g\right\} \end{aligned}$$

(nobility of \(f_\alpha \)) If f is ordinary, then either \(\alpha _f\) is the unit root of the Hecke polynomial, or \(M_E(f) _{G_{{\mathbf {Q}_p}}}\) is not the direct sum of two characters (so the eigenform \(f_\alpha \) is noble in the sense of 4.6.3).

(nobility of \(g_\alpha \) and \(g_\beta \)) If \(k' \geqslant 0\), then \(M_E(g)_{G_{{\mathbf {Q}_p}}}\) does not split as a direct sum of characters, so both pstabilisations \(g_\alpha \) and \(g_\beta \) are noble.
Remark 8.1.1
 (1)
In our arguments we will use both pstabilisations \(g_\alpha \) and \(g_\beta \) of g, but only the one pstabilisation \(f_\alpha \) of f; in particular, we do not require that the other pstabilisation \(f_\beta \) be noble.
 (2)
Note that the “no local zero” hypothesis is automatic, for weight reasons, unless \(k + k'\) is even and \(j = \frac{k + k'}{2}\) or \(j = \frac{k + k'}{2} + 1\) (so the Lvalue \(L(f, g, 1 + j)\) is a “nearcentral” value).
Definition 8.1.2
Proposition 8.1.3

\(H^1({\mathbf {Q}_p}, W)\) is 4dimensional (as an Evector space), and \(H^0({\mathbf {Q}_p}, W) = H^2({\mathbf {Q}_p}, W) = 0\);

we haveand this space has dimension 2;$$\begin{aligned} H^1_\mathrm {e}({\mathbf {Q}_p}, W) = H^1_\mathrm {f}({\mathbf {Q}_p}, W) = H^1_\mathrm {g}({\mathbf {Q}_p}, W), \end{aligned}$$
Proof
This is an elementary exercise using local Tate duality, Tate’s local Euler characteristic formula, and the “no local zero” hypothesis. \(\square \)
Theorem 8.1.4
 (i)for every prime \(v \not \mid p\) of \(\mathbf {Q}(\mu _m)\), we have$$\begin{aligned} \mathrm{loc}_v\left( c^{\alpha _f \alpha _g}_m\right) \in H^1_\mathrm {f}\left( \mathbf {Q}(\mu _m)_v, M_E(f \otimes g)^*(j)\right) ; \end{aligned}$$
 (ii)there is a constant R (independent of m) such thatwhere \(M_{\mathcal {O}_E}(f \otimes g)^*\) is the lattice in \(M_E(f \otimes g)^*\) which is the image of the étale cohomology with \(\mathcal {O}_E\)coefficients;$$\begin{aligned} R c^{\alpha _f \alpha _g}_m, Rc^{\alpha _f \beta _g}_m \in H^1(\mathbf {Q}(\mu _m), M_{\mathcal {O}_E}(f \otimes g)^*(j)){/}\{\text {torsion}\}, \end{aligned}$$
 (iii)for \(\ell \not \mid m N_f N_g\), we havewhere \(P_\ell (X)\) is the local Euler factor of L(f, g, s) at \(\ell \), and similarly for \(c^{\alpha _f \beta _g}_{\ell m}\);$$\begin{aligned} {{\mathrm{norm}}}_{m}^{\ell m} \left( c^{\alpha _f \alpha _g}_{\ell m}\right) = P_\ell (\ell ^{1j} \sigma _\ell ^{1}) \cdot c^{\alpha _f \alpha _g}_{m}, \end{aligned}$$
 (iv)
 (v)
for \(m = 1\), the projections \({{\mathrm{pr}}}_{\alpha }\left( \exp ^* c^{\alpha _f \alpha _g}_1\right) \) and \({{\mathrm{pr}}}_{\alpha }\left( \exp ^* c^{\alpha _f \beta _g}_1\right) \) are nonzero (for some suitable choice of c) if and only if \(L(f \otimes g, 1 + j) \ne 0\).
Proof
These classes satisfy (i), by Proposition 2.4.4. They also satisfy (ii), by Proposition 2.4.7 (using the fact that f and g have differing weights, by hypothesis, so we have \(H^0(\mathbf {Q}^{\mathrm {ab}}, M_E(f_\alpha \otimes g_\alpha )^*) = 0\)).
The classes \(z_m^{\alpha _f \alpha _g}\) do not satisfy (iii); instead, they satisfy the a slightly more complicated normcompatibility relation \({{\mathrm{norm}}}_{m}^{\ell m} \left( c^{\alpha _f \alpha _g}_{\ell m}\right) = Q_\ell (\ell ^{1j} \sigma _\ell ^{1}) c^{\alpha _f \alpha _g}_{m}\) where \(Q_\ell (X) \in X^{1} \mathcal {O}_L[X]\) is a polynomial congruent to \(X^{1} P_\ell (X)\) modulo \(\ell  1\). However, the “correct” Euler system relation can be obtained by modifying each class \(z^{\alpha _f \alpha _g}_m\) by an appropriate element of \(\mathcal {O}_L[(\mathbf {Z}{/}m\mathbf {Z})^\times ]\), as in [20, §7.3]. This gives classes \(c_m^{\alpha _f \alpha _g}\) satisfying (i)–(iv).
The horizontal arrows in the diagram are induced by the morphism of \((\varphi , \varGamma )\)modules \(D \rightarrow \mathscr {F}^{} D\). We know that the image of \({}_c \mathcal {BF}^{[f_\alpha , g_\alpha ]}_{m, 1}\) in \(H^1_{\mathrm {Iw}}(K_\infty , \mathscr {F}^{} D )\) is zero, by Theorem 7.1.2, so its image in the bottom righthand corner is zero. However, the projection \(\mathbf {D}_\mathrm {cris}(V) \rightarrow \mathbf {D}_\mathrm {cris}(\mathscr {F}^{} D)\) factors through projection to the eigenspace \(\mathbf {D}_\mathrm {cris}(V)^{\alpha _f \alpha _g}\) and is an isomorphism on this eigenspace, so we recover the statement that \(\exp ^*(c^{\alpha _f \alpha _g}_m)\) projects to zero in \(\mathbf {D}_\mathrm {cris}(V)^{\alpha _f \alpha _g}\), as required.
This completes the construction of classes \(c^{\alpha _f \alpha _g}_m\) with the required properties. The construction of \(c^{\alpha _f \beta _g}_m\) is identical, using the pstabilisation \(g_\beta \) in place of \(g_\alpha \). \(\square \)
8.2 Bounding the Bloch–Kato Selmer group
Theorem 8.2.1

(big image) There exists an element \(\tau \in {{\mathrm{Gal}}}(\overline{\mathbf {Q}}{/}\mathbf {Q}(\mu _{p^\infty }))\) such that \(V{/}(\tau  1) V\) is 1dimensional, where \(V = M_E(f \otimes g)(1 + j)\).
Remark 8.2.2
It is shown in [22] that, under fairly mild hypotheses on f and g, the “big image” hypothesis is satisfied for all but finitely many primes \(\mathfrak {P}\) of the coefficient field.
Proof
We have constructed two classes in \(H^1_{\mathrm {relaxed}}({\mathbf {Q}_p}, V^*(1))\), namely \(c_1^{\alpha _f \alpha _g}\) and \(c_1^{\alpha _f \beta _g}\), whose images in \(\frac{H^1({\mathbf {Q}_p}, V^*(1))}{H^1_{\mathrm {f}}({\mathbf {Q}_p}, V^*(1))}\) are linearly independent (since their images under \(\exp ^*\) span distinct eigenspaces). So \({{\mathrm{loc}}}_p^{\mathrm {s}}\) is surjective, and consequently \({{\mathrm{loc}}}^{\mathrm {f}}_p\) is the zero map. As we have already shown that \(H^1_{\mathrm {strict}}({\mathbf {Q}_p}, V) = 0\), this shows that \(H^1_{\mathrm {f}}({\mathbf {Q}_p}, V)\) is zero. \(\square \)
Remark 8.2.3
The above argument is an adaptation of the ideas of [13, §6.2], in which Poitou–Tate duality is used to bound the image of the map \({{\mathrm{loc}}}^\mathrm {f}_p\) for a Galois representation arising from the product of three cusp forms. In our setting, since we have a full Euler system rather than just the two classes \(c_1^{\alpha _f \alpha _g}\) and \(c_1^{\alpha _f \beta _g}\), we can also bound the kernel of this map.
8.3 Corollaries
From Theorem 8.2.1 one obtains a rather precise description of the global cohomology groups. We continue to write \(V = M_E(f \otimes g)(1 + j)\).
Remark 8.3.1
Since \({{\mathrm{Gal}}}(\mathbf {Q}^S\!/\mathbf {Q})\) is the étale fundamental group of \(\mathbf {Z}[1/S]\), we may interpret any continuous \({\mathbf {Q}_p}\)linear representation of \({{\mathrm{Gal}}}(\mathbf {Q}^S\!/\mathbf {Q})\) as a padic étale sheaf on \(\mathrm{Spec} \mathbf {Z}[1/S]\), and the continuous cohomology groups \(H^i(\mathbf {Q}^S\!/\mathbf {Q}, )\) coincide with the étale cohomology groups \(H^i_{{\acute{\mathrm{e}}{\mathrm{t}}}}(\mathbf {Z}[1/S], )\). The latter language is used in [19] for instance, but in the present work we have found it easier to use the language of group cohomology, since this makes the arguments of Sect. 2 easier to state.
Corollary 8.3.2
 (1)The localisation mapsare isomorphisms.$$\begin{aligned} H^2(\mathbf {Q}^S\!/\mathbf {Q}, V)\rightarrow & {} \bigoplus _{\ell \in S} H^2(\mathbf {Q}_\ell , V)\quad \text {and}\\ H^2(\mathbf {Q}^S\!/\mathbf {Q}, V^*(1))\rightarrow & {} \bigoplus _{\ell \in S} H^2(\mathbf {Q}_\ell , V^*(1)) \end{aligned}$$
 (2)
The space \(H^1_{\mathrm {f}}(\mathbf {Q}, V^*(1))\) is zero.
 (3)
The space \(H^1_{\mathrm {relaxed}}(\mathbf {Q}, V^*(1))\) is 2dimensional, and \(c_1^{\alpha _f \alpha _g}\) and \(c_1^{\alpha _f \beta _g}\) are a basis.
Proof
Now let \(c_\ell = \dim H^2(\mathbf {Q}_\ell , V^*(1))\). Using Tate’s local Euler characteristic formula, for any \(\ell \in S {\setminus } \{p\}\) we have \(\dim H^1(\mathbf {Q}_\ell , V^*(1)) = c_\ell \); while for \(\ell = p\) we have \(c_p = 0\) and \(\dim H^1_{\mathrm {s}}({\mathbf {Q}_p}, V^*(1)) = 2\). Thus \(\dim \bigoplus _{\ell \in S} H^1_{\mathrm {s}}(\mathbf {Q}_\ell , V^*(1)) = 2 + \sum c_\ell = 2 + \dim H^2(\mathbf {Q}^S\!/\mathbf {Q}, V^*(1))\). However, Tate’s global Euler characteristic formula gives \(\dim H^1(\mathbf {Q}^S\!/\mathbf {Q}, V^*(1)) = 2 +\dim H^2(\mathbf {Q}^S\!/\mathbf {Q}, V^*(1))\).
Thus the map is a surjection between finitedimensional vector spaces of the same dimension, so it is injective and we conclude that \(H^1_{\mathrm {f}}(\mathbf {Q}, V^*(1)) = 0\). Repeating the duality argument with \(V^*(1)\) in place of V we now deduce that the localisation map for \(H^2(\mathbf {Q}^S\!/\mathbf {Q}, V)\) is an isomorphism.
Finally, since \(H^1_{\mathrm {f}}(\mathbf {Q}, V^*(1)) = 0\), we deduce that \(H^1_{\mathrm {relaxed}}(\mathbf {Q}, V^*(1))\) maps isomorphically to its image in \(H^1_{\mathrm {s}}({\mathbf {Q}_p}, V^*(1))\), but the images of \(c_1^{\alpha _f \alpha _g}\) and \(c_1^{\alpha _f \beta _g}\) are a basis of \(H^1_{\mathrm {s}}({\mathbf {Q}_p}, V^*(1))\), so these two classes must be a basis of \(H^1_{\mathrm {relaxed}}(\mathbf {Q}, V^*(1))\). \(\square \)
Corollary 8.3.3
Let \(L_S(f, g, s) = \prod _{\ell \notin S} P_\ell (\ell ^{s})^{1}\) be the Lfunction without its local factors at places in S. If the hypotheses of Theorem 8.2.1 are satisfied and \(L_S(f, g, 1 + j) \ne 0\), then \(H^2(\mathbf {Q}^S\!/\mathbf {Q}, M_E(f \otimes g)^*(j)) = 0\).
Proof
From the definition of \(P_\ell (X)\), the fact that \(P_\ell (\ell ^{1j}) \ne 0\) implies that \(H^0(\mathbf {Q}_\ell , M_E(f \otimes g)(1+j)) = 0\) for all \(\ell \in S\). Thus \(H^2(\mathbf {Q}_\ell , M_E(f \otimes g)^*(j)) = 0\) for all \(\ell \in S\), and since the global \(H^2\) injects into the direct product of these groups, it must also vanish. \(\square \)
Remark 8.3.4
8.4 Application to elliptic curves
Theorem 8.2.1 above allows us to strengthen one of the results of [19] to cover elliptic curves which are not necessarily ordinary at p:
Theorem 8.4.1
 (i)
The conductors \(N_E\) and \(N_\rho \) are coprime;
 (ii)
\(p \geqslant 5\);
 (iii)
\(p \not \mid N_E N_\rho \);
 (vi)
the map \(G_\mathbf {Q}\rightarrow \mathrm{Aut}_{\mathbf {Z}_p}(T_p E)\) is surjective;
 (v)
\(\rho (\mathrm{Frob}_p)\) has distinct eigenvalues.
Proof
This is exactly Theorem 8.2.1 applied with \(f = f_E\), the weight 2 form attached to E, and \(g = g_{\rho }\), the weight 1 form attached to \(\rho \). Compare Theorem 11.7.4 of [19], which is exactly the same theorem under the additional hypotheses that E is ordinary at p and \(\rho (\mathrm{Frob}_p)\) has distinct eigenvalues modulo a prime of L above p. \(\square \)
9 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 [32]: the construction of a 3variable padic Rankin–Selberg Lfunction as a rigidanalytic function on \(V_1 \times V_2 \times \mathcal {W}\), where \(V_i\) are small discs in the Coleman–Mazur eigencurve surrounding classical pstabilised eigenforms, and \(\mathcal {W}\) is weight space.
Unfortunately, since the present paper was submitted, it has emerged that there are some unresolved technical issues in the paper [32], so the existence of this padic Lfunction 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 threevariable geometric padic Lfunction
We place ourselves in the situation of Sect. 7.1, so \(f_\alpha , g_\alpha \) are noble eigenforms, obtained as pstabilisations 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 \(\mathcal {F}, \mathcal {G}\) passing through \(f_\alpha \), \(g_\alpha \). We also allow the possibility that \(k'_0 = 1\), g is a pregular weight 1 newform, and g does not have real multiplication by a field in which p splits. (The exceptional realmultiplication 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 \(\varepsilon _\mathcal {F}\varepsilon _{\mathcal {G}}\) is nontrivial and is not of ppower order. Thus there is a \(c > 1\) coprime to \(6p N_f N_g\) for which the factor \(c^2  c^{2\mathbf {j}  \mathbf {k}\mathbf {k}'} \varepsilon _\mathcal {F}(c)^{1} \varepsilon _{\mathcal {G}}(c)^{1}\) is a unit in \(\mathcal {O}(V_1 \times V_2 \times \mathcal {W})\), and we may define \(\mathcal {BF}^{[\mathcal {F}, \mathcal {G}]}_{1, 1}\) (without c) by dividing out by this factor.
We shall begin by turning Theorem C on its head and defining a padic Lfunction to be the output of this theorem:
Definition 9.1.1
Our goal is now to show that this geometrically defined padic Lfunction is related to critical values of complex Lfunctions.
9.2 Values in the geometric range
By construction, for integer points of \(V_1 \times V_2 \times \mathcal {W}\) in the “geometric range”—that is, the points \((k, k', j)\) with \(0 \leqslant j \leqslant \min (k, k')\)—the geometric padic Lfunction interpolates the syntomic regulators of the Rankin–Eisenstein classes. From the computations of [18], we have the following explicit formula for these syntomic regulators.
Let \(f_{k, \alpha }\) be the pstabilised eigenform that is the specialisation of \(\mathcal {F}\) in weight \(k+2\), and let \(\lambda _{f_{k, \alpha }}\) be the unique linear functional on the space \(S_{k + 2}^\mathrm {oc}(N_f, E)\) of overconvergent cusp forms that factors through projection to the \(f_{k, \alpha }\)isotypical subspace and sends \(f_{k, \alpha }\) to 1. We view \(\lambda _{f_{k, \alpha }}\) as a linear functional on \(S_{k + 2}^\mathrm {oc}(N, E)\), where \(N = \mathrm{lcm}(N_f, N_g)\), by composing with the trace map from level N to level \(N_f\).
Theorem 9.2.1
9.3 Twovariable analytic Lfunctions
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 [32], which concern interpolation of families where the degree of nearoverconvergence is unbounded.
 (i)If \(k \geqslant \max (t, 2t1)\) and \(k' \geqslant kt\), then the “geometric” interpolating property above applies, showing that for these values of \((k, k')\) we haveSince such \((k, k')\) are manifestly Zariskidense in \(V_1 \times V_2\), this relation must in fact hold for all points \((\kappa , \kappa ') \in V_1 \times V_2\).$$\begin{aligned} L_p^{(t)}(\mathcal {F}, \mathcal {G})(k, k') = L_p^{\mathrm {geom}}(\mathcal {F}, \mathcal {G})(k, k', kt). \end{aligned}$$
 (ii)
If \(k' \geqslant 0\) and \(k  k' \geqslant 2t + 1\), then both \(g_{k', \alpha }\) and \(E^{[p]}_{kk'2t}\) are classical modular forms (since, after possibly shrinking \(V_2\), we may arrange that the specialisations of the family \(\mathcal {G}\) at classical weights are classical). Thus the product \(g_{k', \alpha } \cdot \theta ^{t}\left( E^{[p]}_{kk'2t}\right) \) is a classical nearly holomorphic form, and on such forms Urban’s overconvergent projector coincides with the holomorphic projector \(\Pi ^{\mathrm {hol}}\). This shows that the values of \(L_p^{(t)}(\mathcal {F}, \mathcal {G})(k, k')\) for \((k,k')\) in this range are algebraic, and they compute the values of the Rankin–Selberg Lfunction in the usual way. This also holds for \(k' = 1\), as long as we assume that the weight 1 specialisation \(g_{k', \alpha }\) is classical (which is no longer automatic).
Theorem 9.3.2
This suffices to prove Theorem C of the introduction when \(j \geqslant \frac{k + k' + 1}{2}\). The remaining cases of Theorem C, when \(k' + 1 \leqslant j < \frac{k + k' + 1}{2}\), are easily reduced to these cases using the functional equation.
Remark 9.3.3
It is important to be clear about what this argument does not prove: we obtain no information at all about the values of the geometric padic Lfunction at points of the form \((k, k', j + \chi )\) for a nontrivial finiteorder character \(\chi \). In particular, we cannot determine by this method whether the specialisation of our 3variable geometric Lfunction to \(\{k_0\} \times \{k_0'\} \times \mathcal {W}\) coincides with other existing constructions of a singlevariable padic Rankin–Selberg Lfunction (cf. [30]).
Acknowledgements
During the preparation of this paper, we benefitted from conversations with a number of people, notably Fabrizio Andreatta, Pierre Colmez, Hansheng Diao, Henri Darmon, Adrian Iovita, Guido Kings, Ruochuan Liu, Jay Pottharst, Karl Rubin, and Chris Skinner. We would also like to thank the anonymous referee for several helpful comments and corrections.
Large parts of the paper were written while the authors were visiting the Mathematical Sciences Research Institute in Berkeley, California, for the programme “New Geometric Methods in Automorphic Forms”, and it is again a pleasure to thank MSRI for their support and the organisers of the programme for inviting us to participate.
The authors’ research was supported by the following grants: Royal Society University Research Fellowship (Loeffler); Leverhulme Trust Research Fellowship “Euler systems and Iwasawa theory” (Zerbes). Parts of this paper were written, while the authors were guests at MSRI in Berkeley, California, supported by NSF Grant 0932078 000.
This notation is somewhat misleading; it would be better to describe this as the \({weak\hbox {}star\, topology}\) and to reserve the term \({weak\, topology}\) for the topology on \(D_\lambda (\mathbf {Z}_p, {\mathbf {Q}_p})\) induced by its own continuous dual (for the strong topology), in line with the usual terminology in classical functional analysis. However, the above abuse of notation has become standard in the nonArchimedean theory, perhaps because the continuous duals of spaces such as \(D_\lambda (\mathbf {Z}_p, {\mathbf {Q}_p})\) are too pathological to be of much interest.
The rings \(\mathscr {R}\) and \(\mathscr {R}^+\) are often also denoted by \(\mathbf {B}^{\dag }_{\mathrm {rig}, {\mathbf {Q}_p}}\) and \(\mathbf {B}^+_{\mathrm {rig}, {\mathbf {Q}_p}}\), respectively; this notation is used in several earlier works of the present authors.
Declarations
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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