March 4, 2024

Anabelian properties of Berkovich curves

Sylvain Gaulhiac, IMPAN Warsaw, Poland RIMS 110 Kyoto + Zoom · JP: 16:30 · FR: 08:30

The question that anabelian geometry asks is the following : what information can be recovered about a geometric object from its fundamental group?

As many of you know, some deep results due in particular to A. Tamagawa and S. Mochizuki were obtained for hyperbolic curves over some fields with interesting arithmetic flavour like number fields or $p$-adic number fields. Over algebraically closed fields the same techniques don't apply. However, S. Mochizuki and E. Lepage were able to prove some anabelian results for hyperbolic curves over algebraically closed non-archimedean fields using the tempered fundamental group of the analytifications of the curves, where the analytification lives in the realm of Berkovich analytic geometry. Starting from that, it is natural to ask whether anabelian phenomenons appear for Berkovich spaces, even those, purely analytic, that are not analytifications of algebraic varieties.

The tempered fundamental group of a Berkovich space encapsulates both algebraic (finite étale) and topological behaviours, and I will first explain why it is interesting to look at this group for such questions. Then I will bring to light some anabelian behaviours of analytic Berkovich curves, showing for instance that there is a big family of curves whose analytic skeleton is completely determined by the tempered group. The famous Drinfeld half-plane is an example of such a curve.

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