L-Adic representations associated with elliptic curves over Qp

This paper is devoted to the study of the l-adic representations of the absolute Galois group G of Q(p), p greater than or equal to 5, associated to an elliptic curve over Q(p), as l runs through the set of all prime numbers (including l = p, in which case we use the theory of potentially semi-stable p-adic representations). For each prime l, we give the complete list of isomorphism classes of Q(l)[G]-modules coming from an elliptic curve over Q(p), that is, those which are isomorphic to the Tate module of an elliptic curve over Q(p). The l = p case is the more delicate. It requires studying the liftings of a given elliptic curve over F-p to an elliptic scheme over the ring of integers of a totally ramified finite extension of 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 l-adic representation of G to come from an elliptic curve over Q(p), for each prime l.