A RIMS - Kyoto University &“Arithmetic and Homotopic Galois Theory” lecture

B ERKOVICH METHODS FOR ANABELIAN RECONSTRUCTIONSAND THE R ESOLUTION OF N ONSINGULARITIES
E. L EPAGE - April. 08, 10, & 12, 2024

Berkovich spaces are an approach to analytic geometry over non-archimedean fields. Compared to otherapproaches, such as rigid spaces or Huber’s adic spaces, they tend to have nicer topological properties. The
topology of Berkovich spaces is strongly related to the combinatorics of models over the ring of integers.
The recovery of such combinatorics from fundamental groups often plays an important role in anabeliangeometry. In these lectures, I will try to explain how Berkovich spaces and analytic methods can be helpful
in anabelian geometry.
Keywords: p-adic analytic geometry, Berkovich spaces, anabelian geometry

References[Ber90][MT23]

V. G. Berkovich. Spectral Theory and Analytic Geometry over Non-Archimedean Fields. Vol. 33. Mathematical Surveys and Monographs. Providence: American Mathematical Society, 1990.
S. Mochizuki and S. Tsujimura. Resolution of Nonsingularities, Point-theoreticity, and Metric-admissibility
for p-adic Hyperbolic Curves. RIMS preprint 1974. 2023. url: https://www.kurims.kyoto-u.ac.jp/preprint/file/RIMS1974.pdf.

G EOMETRY OF B ERKOVICH SPACESTALK 1In this talk, I will explain the definition of Berkovich spaces and explain its basic properties. I will statesome general applications of the theory. I will also discuss their relation to other approaches of analytic
geometry over non-archimedean fields and give a more precise description of curves.
Keywords: Berkovich spaces, p-adic analytic geometry, geometry of curves.

References[Ber90][Ber93][Duc][Duc07]V. G. Berkovich. Spectral Theory and Analytic Geometry over Non-Archimedean Fields. Vol. 33. Mathematical Surveys and Monographs. Providence: American Mathematical Society, 1990.
V. G. Berkovich. “Étale cohomology for non-Archimedean analytic spaces”. In: Publication mathématiquesde l’Institut des hautes études scientifiques 78 (1993), pp. 5–161.
A. Ducros. La structure des courbes analytique. url: https://webusers.imj-prg.fr/~antoine.ducros/
trirss.pdf.
A. Ducros. “Espaces analytiques p-adiques au sens de Berkovich”. fr. In: Séminaire Bourbaki : volume2005/2006, exposés 952-966. Astérisque 311. talk:958. Société mathématique de France, 2007, pp. 137–176.
url: http://www.numdam.org/item/SB_2005-2006__48__137_0/.

R ESOLUTION OF NON - SINGULARITIES AND LOG - DIFFERENTIALSTALK 2This talk will focus on Mochizuki and Tsujimura’s proof of the absolute anabelian conjecture: every
isomorphism between the étale fundamental groups of hyperbolic curves over finite extensions of Qp isgeometric. The new input of their work is the proof of resolution of non-singularities: given a hyperbolic
curve X over a finite extensions of Qp is geometric, every divisorial valuations on K(X) comes fromsome irreducible component of the special fiber of the stable model after replacing X by some finite étale
cover. If Mochizuki and Tsujimura’s proof is written in a purely scheme-theoretic framework, some ofits intuition comes from previous work using analytic methods: resolution of non-singularities can bereduced to the study of the vanishing of differentials appearing in the image of the Hodge-Tate mapH 1 (XCp , Zp (1)) → H 0 (XCp , Ω1 ). I will reformulate their proof using analytic geometry.
1/2

RIMS - Kyoto University & AHGT Project

Spring 2024

Keywords: p-adic anabelian geometry, absolute anabelian conjecture, resolution of non-singularities, vanishing of log-differentials.

References[Lep23][Lep13][MT23]

E. Lepage. Resolution of non-singularities and the absolute anabelian conjecture. 2023. url: https://arxiv.org/abs/2306.07058.
E. Lepage. “Resolution of nonsingularities for Mumford curves”. In: Publ. Res. Inst. Math. Sci. 49.4
(2013), pp. 861–891.
S. Mochizuki and S. Tsujimura. Resolution of Nonsingularities, Point-theoreticity, and Metric-admissibility
for p-adic Hyperbolic Curves. RIMS preprint 1974. 2023. url: https://www.kurims.kyoto-u.ac.jp/
preprint/file/RIMS1974.pdf.

A NABELIAN RECONSTRUCTION OF METRIC ON B ERKOVICH CURVESTALK 3In this talk, I will keep on studying the differentials in the image of the Hodge-Tate map introduced in
the previous talk. I will explain how the tropicalization of these differentials and a natural metric on theBerkovic space can be recovered from the geometric tempered fundamental group of a hyperbolic curve.
Keywords: Tempered fundamental group, log-differentials, metric on Berkovich curves.

References[Lep]

E. Lepage. Anabelian tempered metric-admissibility. in preparation.

Schedule Location: RIMS Kyoto University, room 110.
• Talk 1: April 8th (Mon.) 13:00-15:00• Talk 2: April 10th (Wed.) 10:00-12:00• Talk 3: April 12th (Fri.) 13:00-15:00

Emmanuel Lepage
Institut de Mathématiques de Jussieu
Paris Rive Gauche, France
https://webusers.imj-prg.fr/~emmanuel.lepage/emmanuel.lepage@imj-prg.fr
2/2