On p-Adic Geometric Representations of G_Q

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

2000 Mathematics Subject Classification: : 11R32, 11R39

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