# Minicorso di Geometria Algebrica e Teoria dei Numeri: “Almost Cp-representations and vector bundles”

## 13 Ottobre - 9 Novembre 2017, ore 14:30 - J. M. Fontaine (Paris-Sud)

Minicorso di Geometria Algebrica e Teoria dei Numeri: “Almost $C_p$-representations and vector bundles”

Venerdì 13 Ottobre - Aula 2AB45
Giovedì 19 Ottobre - Aula 2BC30
Venerdì 20 Ottobre - Aula 2AB45
Giovedì 26 Ottobre - Aula 2BC30
Venerdì 27 Ottobre - Aula 2AB45
Giovedì 2 Novembre - Aula 2BC30
Venerdì 3 Novembre - Aula 2AB45
Giovedì 9 Novembre - Aula 2BC30

Abstract
Let $K$ be a finite extension of $\mathbb Q_p$ and $G_K$ the absolute Galois group. Then $G_K$ acts on the fundamental curve $X$ of $p$-adic Hodge theory and we may consider the abelian category $\mathcal{M}(G_K)$ of coherent $\mathcal{O}_X$-modules equipped with a continuous and semi-linear action of $G_K$.
An $\text{ almost$C_p$-representation of$G_K$}$ is a $p$-adic Banach space $V$ equipped with a linear and continuous action of $G_K$ such that there exists $d\in\mathbb N$, two $G_K$-stable finite dimensional sub-$\mathbb Q_p$-vector spaces $U$ of $V$, $U'$ of $C_p^d$, and a $G_K$-equivariant isomorphism
$$V/U\longrightarrow C_p^d/U'\ .$$
These representations form an abelian category $\mathcal{C}(G_K)$.
The main purpose of this series of lectures is to construct an equivalence of triangulated categories
$$D^b(\mathcal{M}(G_K))\longrightarrow D^b(\mathcal{C}(G_K))$$
and to describe possible generalisations.
Main topics:
- Construction and main properties of the curve X, generalisations,
- Classification of coherent $\mathcal{O}_X$-modules,
- The category $\mathcal{M}(G_K)$,
- Almost $C_p$-representations,
- Effective coherent $\mathcal{O}_X[G_K]$-modules and effective almost $C_p$-representations,
- The main theorem.
- The étale, the pro-étale and the $v$ sites of an adic space $S$.
- Possible generalisations.
- The case where $S$ is the adic space associated to $C_p$. Banach-Colmez spaces.