Euler systems for Rankin-Selberg convolutions of modular forms

Abstract

We construct an Euler system in the cohomology of the tensor product of the Galois representations attached to two modular forms, using elements in the higher Chow groups of products of modular curves. We use these elements to prove a finiteness theorem for the strict Selmer group of the Galois representation when the associated p-adic Rankin-Selberg L-function is non-vanishing at s = 1. Dedicated to Kazuya KatoContents 1. Outline 1 2. Generalized Beilinson-Flach elements 2 3. Norm relations for generalized Beilinson-Flach elements 14 4. Relation to complex L-values 37 5. Relation to p-adic L-values 43 6. Families of cohomology classes 49 7. Bounding strict Selmer groups 62 8. Conjectures on higher-rank Euler systems 71 Appendix A. Ancillary results 74 References 77 1. OutlineIn [BDR12], Bertolini, Darmon and Rotger have studied certain canonical global cohomology classes (the "Beilinson-Flach elements", obtained from the constructions of [Beȋ84] and [Fla92]) in the cohomology of the tensor products of the p-adic Galois representations of pairs of weight 2 modular forms, and related their image under the Bloch-Kato logarithm maps to the values of p-adic Rankin-Selberg L-functions. These Beilinson-Flach elements are constructed as the image of elements of the higher Chow group of a product of modular curves.In this paper, we construct a form of Euler system -a compatible system of cohomology classes over cyclotomic fields -of which the Beilinson-Flach elements are the bottom layer. We first define elements of higher Chow groups of the product of two (affine) modular curves over a cyclotomic field,for integers m ≥ 1, N ≥ 5, and j ∈ Z/mZ (cf. Definition 2.7.3). These are obtained by considering the images of various maps from higher level modular curves to the surface Y 1 (N ) 2 , together with modular units (Siegel units) on these curves. For m = 1 our elements reduce to those considered in [BDR12], and as in op.cit., we show that after tensoring with Q we can construct preimages of our elements in the higher Chow group of the self-product of the projective modular curve X 1 (N ); however, in this paper (as in [Kat04]) we shall take the affine versions as the principal objects of study. We next turn to the relation between our elements and L-values. Theorem 4.3.7 shows, following an argument due to Beilinson, that the images of the elements c Ξ m,N,j under the Beilinson regulator map into complex de Rham cohomology are related to the derivatives at s = 1 of Rankin-Selberg L-functions of weight 2 modular forms. Theorem 5.6.4 is a p-adic analogue of this result, generalizing a theorem of Bertolini-Darmon-Rotger [BDR12]; it gives a formula for the image of our element for m = 1 under the p-adic syntomic regulator, for a prime p ∤ N , in terms of Hida's p-adic Rankin-Selberg L-functions.Next we consider the images of our elements inétale cohomology. Applying Huber's "continuousétale realization" functor and the Hochschild-Serre exact sequence, and projecting into the isotypical component corresponding to a pair of eige...

Rankin–Eisenstein classes in Coleman families

“…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 finite-order distributions. …”

“…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 distribution-valued cohomology class.…”

“…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].…”

Regulators for Rankin–Selberg products of modular forms

“…These elements were later exploited by Flach [11] to prove the finiteness of the Selmer group associated to the symmetric square of an elliptic curve. More recently, Bertolini, Darmon and Rotger [4] established a p-adic analogue of Beilinson's result, while Lei, Loeffler and Zerbes [18] constructed a cyclotomic Euler system whose bottom layer are the Beilinson-Flach elements. These results have many important arithmetic applications ( [5], [18]).…”

Derivative of symmetric square p-adic L-functions via pull-back formula

“…Kato's Euler system has been used in this way in [Ben12] to prove many instances of Conjecture 1.1 for modular forms. It might be possible that the construction of Lei-Loeffler-Zerbes [LLZ12] of an Euler system for the Rankin product of two modular forms could produce such Iwasawa classes for other Galois representations; in particular, for V = Sym 2 (V f )(1), where V f is the Galois representation associated to a weight two modular form (see also [PR98] and the upcoming work of Dasgupta on Greenberg's conjecture for the symmetric square p-adic L-functions of ordinary forms). We also refer the reader to [BDR14].…”