Emmanuel Lepage (IMJ Paris)

Abstract: In various anabelian settings over p-adic fields, one can reconstruct from the fundamental group of a hyperbolic curve the dual graph of the stable reduction of the curve, and one can get more anabelian information on the curve by applying it to various finite étale covers. For each such finite étale cover, this graph defines a retract of the analytic space associated to the curve (in the adic or Berkovich sense), and resolution of non-singularities predicts that the Berkovich space is homeomorphic to the inverse limit of all these retracts. This was proven in 2023 by Mochizuki and Tsujimura over finite extensions of Q_p. I will try to give a sketch of their proof and explain how to deduce a characterization of geometric Galois sections.