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.