Related papers: On obstructions to the Euler system method for Ran…

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 this Euler system to prove a…

We construct a new Euler system for the Galois representation $V_{f,\chi}$ attached to a newform $f$ of weight $2r\geq 2$ twisted by an anticyclotomic Hecke character $\chi$. The Euler system is anticyclotomic in the sense of…

Let G be a connected reductive group defined over an algebraically closed field k of characteristic p > 0. The purpose of this paper is two-fold. First, when p is a good prime, we give a new proof of the ``order formula'' of D. Testerman…

We prove the existence of Euler systems for adjoint modular Galois representations using deformations of Galois representations coming from Hilbert modular forms and relate them to $p$-adic $L$-functions under a conjectural formula for the…

We construct a new Euler system (anticyclotomic, in the sense of Jetchev-Nekovar-Skinner) for the Galois representation $V_{f,\chi}$ attached to a newform $f$ of weight $k\geq 2$ twisted by an anticyclotomic Hecke character $\chi$ defined…

In this article, we study the pseudo-isomorphism class of the dual fine Selmer group $X$ attached to a $p$-adic Galois deformation whose deformation ring $\Lambda$ is isomorphic to the ring of formal power series. By using the "Kolyvagin…

Kings, Lei, Loeffler and Zerbes constructed a three-variable Euler system $\kappa({\bf g},{\bf h})$ of Beilinson-Flach elements associated to a pair of Hida families $({\bf g},{\bf h})$ and exploited it to obtain applications to the…

Consider the semisimple mod p reduction of the Galois representation associated to a Hilbert newform f by Carayol and Taylor. This paper discusses how, under certain conditions on f, the universal ring for deformations of this residual…

Following Ribet's seminal 1976 paper there have been many results employing congruences between stable cuspforms and lifted forms to construct non-split extensions of Galois representations. We show how this strategy can be extended to…

Let $p$ be an odd prime, $ f$ be a $ p $-ordinary newform of weight $ k $ and $ h $ be a normalized cuspidal $ p $-ordinary Hecke eigenform of weight $ l < k$. In this article, we study the $p$-adic $ L $-function and $ p^{\infty} $-Selmer…

Let $f$ and $g$ be two different newforms without complex multiplication having the same coefficient field. The main result of the present article proves that a congruence between the Galois representations attached to $f$ and to $g$ for a…

We consider questions in Galois cohomology which arise by considering mod $p$ Galois representations arising from automorphic forms. We consider a Galois cohomological analog for the standard heuristics about the distribution of Wieferich…

Given a weight two modular form f with associated p-adic Galois representation V_f, for certain quadratic imaginary fields K one can construct canonical classes in the Galois cohomology of V_f by taking the Kummer images of Heegner points…

Let k be a positive integer divisible by 4, l>k a prime, and f an elliptic cuspidal eigenform of weight k-1, level 4, and non-trivial character. Let \rho_f be the l-adic Galois representation attached to f. In this paper we provide evidence…

Let $\mathbb{F}_q$ be a finite field with $q$ elements, where $q$ is a prime power and let $A:= \mathbb{F}_{q}[T]$. By~\cite{PR09}, the adelic image of the Galois representation attached to a rank $2$ Drinfeld $A$-module $\varphi$ is open,…

We construct a mod $\ell$ congruence between a Klingen Eisenstein series (associated to a classical newform $\phi$ of weight $k$) and a Siegel cusp form $f$ with irreducible Galois representation. We use this congruence to show…

If a $p$-adic Galois representation $\rho_{f,\nu}:\Gamma_{\mathbb Q} \to \GL_2(E_{f,\nu})$ attached to some eigenform $f$ is residually reducible it will have 2 non-isomorphic reductions, which have the same semi-simplification. In this…

Let $k$ be an algebraically closed field of characteristic $p>0$ and let $\ell$ be another prime number. O. Gabber and F. Loeser proved that for any algebraic torus $T$ over $k$ and any perverse $\ell$-adic sheaf $\calF$ on $T$ the Euler…

Let $k > 3$ be an integer and $p$ a prime with $p > 2k-2$. Let $f$ be a newform of weight $2k-2$ and level 1 so that $f$ is ordinary at $p$ and $\bar{\rho}_{f}$ is irreducible. Under some additional hypotheses we prove that…

We revisit the construction of Castella and Do of an anticyclotomic Euler system for the $p$-adic Galois representation of a modular form, using diagonal classes. Combining this construction and some previous results of ours, we obtain new…