Cohomologie des courbes analytiques $p$-adiques

The cohomology of affinoids does not behave well; often, this is remedied by making affinoids overconvergent. In this paper, we focus on dimension $1$ and compute, using analogs of pants decompositions of Riemann surfaces, various cohomologies of affinoids. To give a meaning to these decompositions we modify slightly the notion of $p$-adic formal scheme, which gives rise to the adoc (an interpolation between adic and ad hoc) geometry. It turns out that the cohomology of affinoids (in dimension $1$) is not that pathological.

From this we deduce a computation of cohomologies of curves without boundary (like the Drinfeld half-plane and its coverings). In particular, we obtain a description of their $p$-adic proétale cohomology in terms of de the Rham complex and the Hyodo–Kato cohomology, the later having properties similar to the ones of $\ell$-adic proétale cohomology, for $\ell \neq p$.

Keywords

analytic curves, adic spaces, Berkovich spaces, crystalline cohomology, de Rham cohomology, Hyodo–Kato cohomology, Proétale cohomology, syntomic cohomology, comparison theorem, Picard–Lefschetz formula, $p$-adic integration, Jacobian, Picard group, universal extension

2010 Mathematics Subject Classification

Primary 14Fxx, 14Hxx. Secondary 14F20, 14F30, 14G22, 14H99.

The full text of this article is unavailable through your IP address: 172.17.0.1

À la mémoire de Robert Coleman et Michel Raynaud.

Les trois auteurs sont membres du projet ANR-19-CE40-0015-02 COLOSS.

Received 9 February 2021

Published 22 July 2022