On $p$-adic geometric representations of $G_{\mathbb{Q}}$.

Abstract

A conjecture of Fontaine and Mazur states that a geometric odd irreducible $p$-adic representation $\rho$ of the Galois group of $\Bbb Q$ comes from a modular form ([FM95]). Dieulefait proved that, under certain hypotheses, $\rho$ is a member of a compatible system of $\ell$-adic representations, as predicted by the conjecture ([Dieu]). Thanks to recent results of Kisin (Mark), we are able to apply the method of Dieulefait under weaker hypotheses. This is useful in the proof of Serre's conjecture (Serre) given in KW1, K,KW2,KW3.