相关论文: Adjoint Selmer groups of automorphic Galois repres…

Let F be a number field with adele ring A_F, and \pi an isobaric, algebraic automorphic representation of GL_4(A_F) of a fixed archimedean weight, which is quasi-regular, meaning that at every archimedean place v of F, the 4-dimensional…

Let $K$ be a totally real field and $\pi$ be a regular algebraic polarized cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb A_K)$. Let $\{\rho_{\pi,\lambda}:\mathrm{Gal}_K\to\mathrm{GL}_n(\overline E_\lambda)\}_\lambda$ be the…

We prove automorphy lifting results for geometric representations $\rho:G_F \rightarrow GL_2(\mathcal{O})$, with $F$ a totally real field, and $\mathcal{O}$ the ring of integers of a finite extension of $\mathbb{Q}_p$ with $p$ an odd prime,…

In this paper, we consider Galois representations of the absolute Galois group $\text{Gal}(\overline {\mathbb Q}/\mathbb Q)$ attached to modular forms for noncongruence subgroups of $\text{SL}_2(\mathbb Z)$. When the underlying modular…

We present methods to compute Selmer groups associated to mod p Galois representations rho over a number field K, with a particular focus on comparing their ranks with periods coming from cohomology classes associated to rho by Serre's…

Let $N$ be a positive integer and $K$ be a number field. Suppose that $f_1,f_2 \in S_k(\Gamma_0(N))$ are two newforms such that their residual Galois representations at $2$ are isomorphic. Let $\omega_2: G_{\mathbb Q} \rightarrow {\mathbb…

We study the arithmetic of curves and Jacobians endowed with the action of a finite group $G$. This includes a study of the basic properties, as $G$-modules, of their $\ell$-adic representations, Selmer groups, rational points and…

We prove the Sato-Tate conjecture for Hilbert modular forms. More precisely, we prove the natural generalisation of the Sato-Tate conjecture for regular algebraic cuspidal automorphic representations of $\GL_2(\A_F)$, $F$ a totally real…

We construct an Euler system for the adjoint Galois representation of a modular form, using motivic cohomology classes arising from Hilbert modular surfaces. We use this Euler system to give an upper bound for the Selmer group of the…

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…

Let $p\geq 5$ be a prime number. In this paper, we construct Galois representations associated with modular forms for which the dimension of the $p$-torsion in the Bloch-Kato Selmer group can be made arbitrarily large. Our result extends…

We consider the Galois representation associated with a finite slope $p$-adic family of modular forms. We prove that the Lie algebra of its image contains a congruence Lie subalgebra of a non-trivial level. We describe the largest such…

We prove a companion forms theorem for ordinary n-dimensional automorphic Galois representations, by use of automorphy lifting theorems developed by the second author, and a technique for deducing companion forms theorems due to the first…

We introduce the adjoint homological Selmer module for an SL$_2$-representation of a knot group, which may be seen as a knot theoretic analogue of the dual adjoint Selmer module for a Galois representation. We then show finitely generated…

In this paper, we study the fine Selmer groups of two congruent Galois representations over an admissible $p$-adic Lie extension. We show that under appropriate congruence conditions, if the dual fine Selmer group of one is pseudo-null, so…

For a totally real field $F$, a finite extension $\mathbf{F}$ of $\mathbf{F}_p$ and a Galois character $\chi: G_F \to \mathbf{F}^{\times}$ unramified away from a finite set of places $\Sigma \supset \{\mathfrak{p} \mid p\}$ consider the…

Let $X$ be a smooth, separated, geometrically connected scheme defined over a number field $K$ and $\{\rho_\lambda\}_\lambda$ a system of n-dimensional semisimple $\lambda$-adic representations of the \'etale fundamental group of $X$ such…

This paper is devoted to the study of the $\ell$-adic representations of the absolute Galois group $G$ of ${\mathbb Q}_p$, $p\geq 5$, associated to an elliptic curve over ${\mathbb Q}_p$, as $\ell$ runs through the set of all prime numbers…

We prove the indecomposability of Galois representation restricted to the p-decomposition group attached to a non CM nearly p-ordinary weight two Hilbert modular form under mild conditions.

Let $K$ be a finitely generated extension of $\mathbb{Q}$. We consider the family of $\ell$-adic representations ($\ell$ varies through the set of all prime numbers) of the absolute Galois group of $K$, attached to $\ell$-adic cohomology of…