Les repr\'esentations ell-adiques associ\'ees aux courbes elliptiques sur \Q_p

Abstract

This paper is devoted to the study of the $\ell$-adic representations of the absolute Galois group $G$ of ${\mathbb Q}_p$, $p\geq 5$, associated to an elliptic curve over ${\mathbb Q}_p$, as $\ell$ runs through the set of all prime numbers (including $\ell =p$, in which case we use the theory of potentially semi-stable $p$-adic representations). For each prime $\ell$, we give the complete list of isomorphism classes of ${\mathbb Q}_{\ell}[G]$-modules coming from an elliptic curve over ${\mathbb Q}_p$, that is, those which are isomorphic to the Tate module of an elliptic curve over ${\mathbb Q}_p$. The $\ell =p$ case is the more delicate. It requires studying the liftings of a given elliptic curve over ${\mathbb F}_p$ to an elliptic scheme over the ring of integers of a totally ramified finite extension of ${\mathbb Q}_p$, and combining it with a descent theorem providing a Galois criterion for an elliptic curve having good reduction over a $p$-adic field to be defined over a closed subfield. This enables us to state necessary and sufficient conditions for an $\ell$-adic representation of $G$ to come from an elliptic curve over ${\mathbb Q}_p$, for each prime $\ell$.