Vol. 4, No. 2, 2022

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Symplectic geometry of $p$-adic Teichmüller uniformization for ordinary nilpotent indigenous bundles

Yasuhiro Wakabayashi

Vol. 4 (2022), No. 2, 203–247
Abstract

We provide a new aspect of the p-adic Teichmüller theory established by Mochizuki. The formal stack classifying p-adic canonical liftings of ordinary nilpotent indigenous bundles embodies a p-adic analogue of uniformization of hyperbolic Riemann surfaces, as well as a hyperbolic analogue of Serre–Tate theory of ordinary abelian varieties. We prove a comparison theorem for the canonical symplectic structure on the cotangent bundle of this formal stack and Goldman’s symplectic structure. This result may be thought of as a p-adic analogue of comparison theorems in the theory of projective structures on Riemann surfaces proved by Kawai and other mathematicians.

Keywords
hyperbolic curve, indigenous bundle, symplectic structure, canonical lifting, $p$-adic Teichmüller theory, uniformization, crystal
Mathematical Subject Classification
Primary: 14H10
Secondary: 53D30
Milestones
Received: 19 May 2020
Revised: 26 September 2021
Accepted: 11 October 2021
Published: 24 August 2022
Authors
Yasuhiro Wakabayashi
Department of Mathematics
Tokyo Institute of Technology
Ookayama, Meguro-ku
Tokyo
Japan