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

We investigate level $p$ Eisenstein congruences for GSp$_4$, generalisations of level $1$ congruences predicted by Harder. By studying the associated Galois and automorphic representations we see conditions that guarantee the existence of a…

We introduce non-abelian cohomology sets of Hopf algebras with coefficients in Hopf modules. We prove that these sets generalize Serre's non-abelian group cohomology theory. Using descent techniques, we establish that our construction…

We extend the computations in [AGM1, AGM2, AGM3] to find the cohomology in degree five of a congruence subgroup Gamma of SL(4,Z) with coefficients in a field K, twisted by a nebentype character eta, along with the action of the Hecke…

Let $p$ be a prime number, $F$ a totally real number field unramified at places above $p$ and $D$ a quaternion algebra of center $F$ split at places above $p$ and at no more than one infinite place. Let $v$ be a fixed place of $F$ above $p$…

Let $p$ be a prime number, $n>2$ an integer, and $F$ a CM field in which $p$ splits completely. Assume that a continuous automorphic Galois representation…

Let $F$ be a totally real field unramified at all places above $p$ and $D$ be a quaternion algebra which splits at either none, or exactly one, of the infinite places. Let $\bar{r}:\mathrm{Gal}(\bar{F}/F)\to…

Let $p$ be a prime number and $F$ a totally real number field. For each prime $\mathfrak{p}$ of $F$ above $p$ we construct a Hecke operator $T_\mathfrak{p}$ acting on $(\mathrm{mod}\, p^m)$ Katz Hilbert modular classes which agrees with the…

These are the lecture notes from a five-hour mini-course given at the Winter School on Galois Theory held at the University of Luxembourg in February 2012. Their aim is to give an overview of Serre's modularity conjecture and of its proof…

Let $N/K$ be a finite Galois extension of $p$-adic number fields and let $\rho^\mathrm{nr} : G_K \to \mathrm{Gl}_r(\mathbb Z_p)$ be an $r$-dimensional unramified representation of the absolute Galois group $G_K$ which is the restriction of…

Let $p>2$ be a prime. Let $K$ be a tamely ramified finite extension over $\mathbb Q_p$ with ramification index $e$, and let $G_K$ be the Galois group. We study Kisin modules attached to crystalline representations of $G_K$ whose labeled…

In this paper we explicitly compute mod-l Galois representations associated to modular forms. To be precise, we look at cases with l<=23 and the modular forms considered will be cusp forms of level 1 and weight up to 22. We present the…

We prove several dimension formulas for spaces of scalar-valued Siegel modular forms of degree $2$ with respect to certain congruence subgroups of level $4$. In case of cusp forms, all modular forms considered originate from cuspidal…

We construct infinite families of irreducible supersingular mod $p$ representations of $\mathrm{GL}_2(F)$ with $\mathrm{GL}_2(\mathcal{O}_F)$-socle compatible with Serre's modularity conjecture, where $F / \mathbb{Q}_p$ is any finite…

We prove modularity of certain residually reducible ordinary 2-dimensional $p$-adic Galois representations with determinant a finite order odd character $\chi$. For certain non-quadratic $\chi$ we prove an $R=T$ result for $T$ the weight 1…

Mazur's principle gives a criterion under which an irreducible mod $\ell$ Galois representation arising from a modular form of level $Np$ (with $p$ prime to $N$) can also arise from a modular form of level $N.$ We prove an analogous result…

Let V be a p-adic representation of the absolute Galois group G of Q_p that becomes crystalline over a finite tame extension, and assume p odd. We provide necessary and sufficient conditions for V to be isomorphic to the Tate module V_p(A)…

In a previous article we introduced various moduli stacks of two-dimensional tamely potentially Barsotti-Tate representations of the absolute Galois group of a p-adic local field, as well as related moduli stacks of Breuil-Kisin modules…

Let $F$ be a totally real field in which $p$ is unramified. Let $\overline{r}: G_F \rightarrow \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ be a modular Galois representation which satisfies the Taylor--Wiles hypotheses and is tamely ramified…

In this article we consider mod p modular Galois representations which are unramified at p such that the Frobenius element at p acts through a scalar matrix. The principal result states that the multiplicity of any such representation is…

This article gives a generalization of the work of Y.Ding in the context of $\mathrm{GSp}_4(\mathbb{Q}_p)$, where $p$ is an odd prime number. Let $\rho$ be a 4-dimensional generic non-critical crystalline representations of the absolute…