We provide a new aspect of the -adic Teichmüller theory established by Mochizuki. The formal stack classifying -adic canonical liftings of ordinary nilpotent indigenous bundles embodies a -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 -adic analogue of comparison theorems in the theory of projective structures on Riemann surfaces proved by Kawai and other mathematicians.
"Symplectic geometry of -adic Teichmüller uniformization for ordinary nilpotent indigenous bundles." Tunisian J. Math. 4 (2) 203 - 247, 2022. https://doi.org/10.2140/tunis.2022.4.203