# p-adic Hodge theory

Thursday, November 15, 2018 - 11:00am - 12:00pm

Matthew Emerton (University of Chicago)

I will describe some of the history of, progress in, and future prospects for the p-adic Langlands program. This is an aspect of the Langlands program that grew out of the successful proof of Langlands reciprocity in various important cases (in particular, the modularity of elliptic curves over Q) twenty or so years ago. It relates the deformation theory of Galois representations to the representation-theoretic aspects of the theory of automorphic forms, for example via the investigation of representations of p-adic groups on p-adic vector spaces.

Wednesday, January 5, 2011 - 1:00pm - 2:00pm

Jean-Marc Fontaine (Université de Paris XI (Paris-Sud))

Let $F$ be a perfect field of characteristic $p>0$ equipped with a non trivial absolute value, $E$ a non archimedean locally compact field whose residue field is contained in $F$ and $pi$ a uniformizing parameter of $E$. We associate functorially to these datas a separated integral noetherian regular scheme $X=X_{F,E,pi}$ of dimension $1$ defined over $E$. There is an equivalence of categories between semi-stable vector bundles of slope $0$ over $X$ and continuous $E$-linear representations of the absolute Galois group $H_F$ of $F$.