Analyse micro-locale l-adique en caracteristique p>0: Le cas d'un trait

Sur la correspondance de Simpson p-adique. I : étude locale

We develop a new approach for the p-adic Simpson correspondence, closely related to the original approach of Faltings, but also inspired by the work of Ogus and Vologodsky on an analogue in characteristic p>0. This first article is devoted to the local aspects of the theory.

Uniformisation locale des schémas quasi-excellents de caractéristique nulle

We prove an embedded local uniformization theroem for a valuation centered on a point of a quasi-excellent scheme of characteristic zero. The proof reduces to valuations of rank 1 and consists in desingularizing the ideal formed by the elements of infinite value and monomializing the key polynomials. We then prove a monomialization theorem valid in all characteristic under certain conditions, including that of non-existence of limit key polynomials, a condition that is always satified in characteristic zero.

$p$-进局部 Langlands 对应与 Kisin 环

Correspondance de Langlands locale $p$-adique et anneaux de Kisin

本文借助 $\mathcal B$-进完备化与 $\mathrm{GL}_2(\mathbf Q_p)$ 的 $p$-进局部 Langlands 对应，从经典对应直接构造 Kisin 环及二维通用 Galois 表示，并统一证明超尖点情形的几何 Breuil–Mézard 猜想。