Let $\pi$ be a regular, algebraic, essentially self-dual cuspidal automorphic representation of ${\mathrm{GL}}_n\left({𝔸}_F\right)$, where $F$ is a totally real field and $n$ is at most $5$. We show that for all primes $l$, the $l$-adic Galois representations associated to $\pi$ are irreducible, and for all but finitely many primes $l$, the mod $l$ Galois representations associated to $\pi$ are also irreducible. We also show that the Lie algebras of the Zariski closures of the $l$-adic representations are independent of $l$.

Let $f$ be a cuspidal newform with complex multiplication (CM) and let $p$ be an odd prime at which $f$ is non-ordinary. We construct admissible $p$-adic $L$-functions for the symmetric powers of $f$, thus verifying conjectures of Dabrowski and Panchishkin in this special case. We combine this with recent work of Benois to prove the trivial zero conjecture in this setting. We also construct “mixed” plus and minus $p$-adic $L$-functions and prove an analogue of Pollack’s decomposition of the admissible $p$-adic $L$-functions....

La conjecture de Birch et Swinnerton-Dyer prédit que l’ordre $r_{\infty }$ du zéro en $s=1$ de la fonction $L$ d’une courbe elliptique $E$ définie sur $𝐐$ est égal au rang $r$ du groupe de ses points rationnels. On sait démontrer cette conjecture si $r_{\infty }=0$ ou $1$, mais on n’a aucun résultat reliant $r_{\infty }$ et $r$ si $r_{\infty }\ge 2$. Nous expliquerons comment Kato démontre que la fonction $L$$p$-adique attachée à $E$ a, en $s=1$, un...

La conjecture dit qu’une représentation continue irréductible impaire du groupe de Galois de $Q$ dans un espace vectoriel de dimension $2$ sur un corps fini $F$ de caractéristique $p$ provient d’une forme modulaire. C. Khare vient de la prouver pour les représentations qui sont non ramifiées hors de $p$.

In this lecture we introduce the reader to the proof of the p-adic monodromy theorem linking the p-adic differential equations theory and the local Galois p-adic representations theory.

We prove the indecomposability of the Galois representation restricted to the $p$-decomposition group attached to a non CM nearly $p$-ordinary weight two Hilbert modular form over a totally real field $F$ under the assumption that either the degree of $F$ over $\mathbb{Q}$ is odd or the automorphic representation attached to the Hilbert modular form is square integrable at some finite place of $F$.

Let $p$ be a rational prime and $K$ a complete discrete valuation field with residue field $k$ of positive characteristic $p$. When $k$ is finite, generalizing the theory of Deligne [1], we construct in [10] and [11] a theory of local ${\epsilon }_0$-constants for representations, over a complete local ring with an algebraically closed residue field of characteristic $\ne p$, of the Weil group $W_K$ of $K$. In this paper, we generalize the results in [10] and [11] to the case where $k$ is an arbitrary perfect field.

We prove the compatibility of the local and global Langlands correspondences at places dividing $l$ for the $l$-adic Galois representations associated to regular algebraic conjugate self-dual cuspidal automorphic representations of ${GL}_n$ over an imaginary CM field, under the assumption that the automorphic representations have Iwahori-fixed vectors at places dividing $l$ and have Shin-regular weight.

The $p$-adic local Langlands correspondence for ${\mathrm{GL}}_2\left({\mathbb{Q}}_p\right)$ attaches to any $2$-dimensional irreducible $p$-adic representation $V$ of $G_{{\mathbb{Q}}_p}$ an admissible unitary representation $\Pi \left(V\right)$ of ${\mathrm{GL}}_2\left({\mathbb{Q}}_p\right)$. The unitary principal series of ${\mathrm{GL}}_2\left({\mathbb{Q}}_p\right)$ are those $\Pi \left(V\right)$ corresponding to trianguline representations. In this article, for $p\>2$, using the machinery of Colmez, we determine the space of locally analytic vectors $\Pi {\left(V\right)}_{\mathrm{an}}$ for all non-exceptional unitary principal series $\Pi \left(V\right)$ of ${\mathrm{GL}}_2\left({\mathbb{Q}}_p\right)$ by proving a conjecture of Emerton.

In this paper, we prove that the representation $\rho$ from $G_{\mathbb{Q}}$ in GL${}_2\left(ℂ\right)$ with image $A_5$ in PGL${}_2\left(A_5\right)$ corresponding to the example $16$ in [B-K] is modular. This representation has conductor $5203$ and determinant ${\chi }_{-43}$; its modularity was not yet proved, since this representation does not satisfy the hypothesis of the theorems of [B-D-SB-T] and [Tay2].

This paper is essentially the text of the author’s lecture at the 2001 Journées Arithmétiques. It addresses the problem of identifying in Galois-theoretic terms those two-dimensional, $p$-adic Galois representations associated to holomorphic Hilbert modular newforms.

In this short note we give a new approach to proving modularity of $p$-adic Galois representations using a method of $p$-adic approximations. This recovers some of the well-known results of Wiles and Taylor in many, but not all, cases. A feature of the new approach is that it works directly with the $p$-adic Galois representation whose modularity is sought to be established. The three main ingredients are a Galois cohomology technique of Ramakrishna, a level raising result due to Ribet, Diamond, Taylor,...