Related papers: Conjecture de type de Serre et formes compagnons p…

We prove the semisimplicity conjecture for A-motives over finitely generated fields K. This conjecture states that the rational Tate modules V_p(M) of a semisimple A-motive M are semisimple as representations of the absolute Galois group of…

We show that if two continuous semi-simple \(\ell \)-adic Galois representations are locally potentially equivalent at a sufficiently large set of places then they are globaly potentially equivalent. We also prove an analogous result for…

We consider the p-adic Galois representation associated to a Hilbert modular form. We show the compatibility with the local Langlands correspondence at a place divising p under a certain assumption. We also prove the monodromy-weight…

We show that, under suitable assumptions, the systems of Hecke eigenvalues arising from (mod p) modular forms of PEL-type associated to an algebraic group G of type A or C coincide with the Hecke eigensystems arising from (mod p) algebraic…

Let $E$ be an elliptic curve without complex multiplication defined over the rationals. The purpose of this article is to define a positive integer $A(E)$, that we call the {\it Serre's constant associated to $E$}, that gives necessary…

Let $f$ and $f'$ be genus $2$ cuspidal Siegel paramodular newforms. We prove that if their Hecke eigenvalues $a_p$ and $a_p'$ satisfy a non-trivial polynomial relation $P(a_p, a_p') = 0$ for a set of primes $p$ of positive density, then $f$…

The Breuil-M\'{e}zard Conjecture predicts the existence of hypothetical "Breuil-Mezard cycles" in the moduli space of mod $p$ Galois representations of $\mathrm{Gal}(\overline{\mathbb{Q}}_q/\mathbb{Q}_q)$ that should govern congruences…

Let $p$ be an odd prime. Let $F$ be a non-archimedean local field of residue characteristic $p$, and let $\mathbb{F}_q$ be its residue field. Let $\mathcal{H}^{(1)}_{\mathbb{F}_q}$ be the pro-$p$-Iwahori-Hecke algebra of the $p$-adic group…

We extend the computations in our prior work to find the cohomology in degree five of a congruence subgroup Gamma of SL_4(Z) with coefficients in Sym^g(K^4), twisted by a nebentype character eta, along with the action of the Hecke algebra.…

Let $K/\mathbb{Q}_p$ be a finite extension with residue field $k$. By a work of Emerton--Gee, irreducible components inside the reduced special fiber of the moduli stack of rank $n$ \'etale $(\varphi,\Gamma)$-modules are labeled by Serre…

We prove an integral R = T theorem for odd two dimensional p-adic representations of the absolute Galois group which are unramified at p, extending results of [CG] to the non-minimal case. We prove, for any p, the existence of Katz modular…

Let $G$ be a finite classical group of Lie type of rank $\ell$, defined over a field of characteristic $p>2$. In this work, we classify the irreducible representations of $G$ whose dimensions are bounded by a constant proportional to…

Let $F$ be a totally real number field. We prove that a character of the spherical Hecke algebra appearing in the completed cohomology of Hilbert modular varieties is modular if the associated Galois representation is absolutely…

It is known that any Galois representation $\rho : G_{\mathbb{Q}} \rightarrow \mathrm{GL}(2,\mathbb{F}_p)$ with determinant equal to the mod-$p$ cyclotomic character, arises from the $p$-torsion of an elliptic curve over $\mathbb{Q}$, if…

In this article we report on extensive calculations concerning the Gorenstein defect for Hecke algebras of spaces of modular forms of prime weight p at maximal ideals of residue characteristic p such that the attached mod p Galois…

Let $p$ be prime, and $n,m \in \mathbb{N}$. When $K/F$ is a cyclic extension of degree $p^n$, we determine the $\mathbb{Z}/p^m\mathbb{Z}[\text{Gal}(K/F)]$-module structure of $K^\times/K^{\times p^m}$. With at most one exception, each…

Let $F$ be a totally real field in which $p$ is unramified and $B$ be a quaternion algebra which splits at at most one infinite place. Let $\overline{r}:\mathrm{Gal}(\overline{F}/F)\to \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ be a modular…

We prove a non-minimal modularity lifting theorem for ordinary Galois representations over imaginary quadratic fields, conditional on a local-global compatibility conjecture for ordinary torsion classes.

Let $p$ be an odd prime, and $\mathbf{Q}_{p^f}$ the unramified extension of $\mathbf{Q}_p$ of degree $f$. In this paper, we reduce the problem of constructing strongly divisible modules for $2$-dimensional semi-stable non-crystalline…

We compute the reductions of irreducible crystalline two-dimensional representations of $G_{\mathbf{Q}_p}$ of slope 1, for primes $p \geq 5$, and all weights. We describe the semisimplification of the reductions completely. In particular,…