Related papers: The local Langlands correspondence for $\DeclareMa…

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…

Let $F$ be a non-archimedean local field of characteristic different from 2 and residual characteristic $p$. This paper concerns the $\ell$-modular representations of a connected reductive group $G$ distinguished by a Galois involution,…

Any finite-dimensional $p$-adic representation of the absolute Galois group of a $p$-adic local field with imperfect residue field is characterized by its arithmetic and geometric Sen operators defined by Sen and Brinon. We generalize their…

Let H be a connected reductive group defined over a non-archimedean local field F of characteristic p>0. Using Poincar\'e series, we globalize supercuspidal representations of H(F) in such a way that we have control over ramification at all…

Let $F$ be a non-Archimedean local field. Let $\mathcal{A}_n(F)$ be the set of equivalence classes of irreducible admissible representations of $\textrm{GL}_n(F)$, and $\mathcal{G}_n(F)$ be the set of equivalence classes of n-dimensional…

This article is the third and last part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. This part proves the following new result for the inverse…

The p-adic local Langlands correspondence for GL_2(Q_p) is given by an exact functor from unitary Banach representations of GL_2(Q_p) to representations of the absolute Galois group G_{Q_p} of Q_p. We prove, using characteristic 0 methods,…

We survey results related to our geometrization of a part of the $p$-adic local Langlands correspondence for ${\mathrm{GL}}_2({\mathbf Q}_p)$.

Let $F$ be a non-Archimedean locally compact field of residual characteristic $p$ with $p\neq 2$. Let $n$ be a power of $p$ and let $G$ be an inner form of the general linear group $\text{\rm GL}_n(F)$. We give a transparent parametrization…

This is an exposition of our joint work with Kakde, Silliman, and Wang, in which we prove a version of Ribet's Lemma for $\mathrm{GL}_2$ in the residually indistinguishable case. We suppose we are given a Galois representation taking values…

This paper deals with the geometric local theta correspondence at the Iwahori level for dual reductive pairs of type II over a non Archimedean field $F$ of characteristic $p\neq 2$ in the framework of the geometric Langlands program. First…

This paper proves local-global principles for Galois cohomology groups over function fields $F$ of curves that are defined over a complete discretely valued field. We show in particular that such principles hold for $H^n(F, Z/mZ(n-1))$, for…

Let $E/F$ be a finite and Galois extension of non-archimedean local fields. Let $G$ be a connected reductive group defined over $E$ and let $M: = \mathfrak{R}_{E/F}\, G$ be the reductive group over $F$ obtained by Weil restriction of…

This is an introduction to the article "Chtoucas pour les groupes r\'eductifs et param\'etrisation de Langlands globale", arXiv:1209.5352. We explain all the ideas of the proof. For any reductive group G over a global function field, we use…

Let G be the group of points of a quasi-split reductive algebraic group over a local field F. It follows from the local Langlands conjectures that to every non-trivial additive character of F and every representation of the Langlands dual…

We construct the local Galois representations over the complex field whose Swan conductors are one by using etale cohomology of Artin-Schreier sheaves on affine lines over finite fields. Then, we study the Galois representations, and give…

In previous works, we described algorithms to compute the number field cut out by the mod ell representation attached to a modular form of level N=1. In this article, we explain how these algorithms can be generalised to forms of higher…

In this paper, we prove new instances of the inverse Galois problem over global function fields for finite groups of Lie type. This is done by constructing compatible systems of $\ell$-adic Galois representations valued in a semisimple…

In \cite{lafforgue2012chtoucas}, Vicent Lafforgue attaches a semisimple Langlands parameter (or, what amounts to the same thing, a $\hat{G}$-pseudocharacter) to every cuspidal automorphic representation of a reductive group $G$ over the…

We can associate local constant to every continuous finite dimensional complex representation of the absolute Galois group $G_F$ of a non-archimedean local field $F/\mathbb{Q}_p$ by Deligne and Langlands. To give explicit formula of local…