Najmuddin Fakhruddin - 14h - Lifting global Galois representations.

In this talk I will discuss the question of when a representation of the absolute Galois group of a number field $F$ into the group of points of a reductive group $G$ over a finite field $k$ of characteristic p lifts to a representation of the Galois group into $G(\mathcal{O})$, where $\mathcal{O}$ is a discrete valuation ring finite over $\mathbb{Z}_p$. One would like to control the local behaviour of the lift at the places of $F$ above $p$, e.g., ask that it be de Rham or trianguline, and I will explain how this is related to the behaviour of the original representation at the real places of $F$.