Vector bundles and p-adic Galois representations

Wednesday, January 5, 2011 - 1:00pm - 2:00pm
Keller 3-180
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 $F$.

When $F$ is algebraically closed, the closed points of $F$ can be described in terms of the Lubin-Tate formal group of $E$ corresponding to $pi$.

If $C$ is the $p$-adic completion of $overline Q_p$, one can associate to $C$ an algebraically closed field $F=F(C)$ as above and ${rm Gal)(overlineQ_p/Q_p)$ acts on the curve $X=X_{F(C),Q_p,p}$. The two main results of $p$-adic Hodge theory can be recovered from the classification of vector bundles over $X$.

(joint work with Laurent Fargues)

MSC Code: