Galois Representations and Automorphic Forms Seminar

Let $\overline K$ be an algebraic closure of a $p$-adic field $K$. We construct a separated noetherian regular scheme $X$ (nonalgebraic) equipped with an action of $G_K=\mathrm{Gal}(\overline{K}/K)$. We have $H^0(X, O_X) = Q_p$ and $H_1(X, O_X) = 0$. For each rational number $\lambda$, there is exactly one isomorphism class of stable vector bundles of slope $\lambda$. The two main theorems of $p$-adic Hodge theory can be deduced from the classification of vector bundles over $X$ (joint work with Laurent Fargues).