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...
Abstract. We define a family of Coleman maps for positive crystalline p-adic representations of the absolute Galois group of Qp using the theory of Wach modules. Let f = anq n be a normalized new eigenform and p an odd prime at which f is either good ordinary or supersingular. By applying our theory to the p-adic representation associated to f , we define Coleman maps Col i for i = 1, 2 with values in Q p ⊗ Zp Λ, where Λ is the Iwasawa algebra of Z × p . Applying these maps to the Kato zeta elements gives a decomposition of the (generally unbounded) p-adic L-functions of f into linear combinations of two power series of bounded coefficients, generalizing works of Pollack (in the case ap = 0) and Sprung (when f corresponds to a supersingular elliptic curve). Using ideas of Kobayashi for elliptic curves which are supersingular at p, we associate to each of these power series a Λ-cotorsion Selmer group. This allows us to formulate a "main conjecture". Under some technical conditions, we prove one inclusion of the "main conjecture" and show that the reverse inclusion is equivalent to Kato's main conjecture.
We generalise works of Kobayashi to give a formulation of the Iwasawa main conjecture for modular forms at supersingular primes. In particular, we give analogous definitions of the plus and minus Coleman maps for normalised new forms of arbitrary weights and relate Pollack's p-adic L-functions to the plus and minus Selmer groups. In addition, by generalising works of Pollack and Rubin on CM elliptic curves, we prove the 'main conjecture' for CM modular forms.
We construct an Euler system -a compatible family of global cohomology classes -for the Galois representations appearing in the geometry of Hilbert modular surfaces. If a conjecture of Bloch and Kato on injectivity of regulator maps holds, this Euler system is non-trivial, and we deduce bounds towards the Iwasawa main conjecture for these Galois representations.
In this paper, we study the Iwasawa theory of a motive whose Hodge-Tate weights are 0 or 1 (thence in practice, of a motive associated to an abelian variety) at a non-ordinary prime, over the cyclotomic tower of a number field that is either totally real or CM. In particular, under certain technical assumptions, we construct Sprung-type Coleman maps on the local Iwasawa cohomology groups and use them to define (one unconditional and other conjectural) integral p-adic L-functions and cotorsion Selmer groups. This allows us to reformulate Perrin-Riou's main conjecture in terms of these objects, in the same fashion as Kobayashi's ±-Iwasawa theory for supersingular elliptic curves. By the aid of the theory of Coleman-adapted Kolyvagin systems we develop here, we deduce parts of Perrin-Riou's main conjecture from an explicit reciprocity conjecture.
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