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.

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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