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

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.