The aim of the present paper is to provide a new aspect of the $p$-adic Teichm\"{ u}ller theory established by S. Mochizuki. We study the symplectic geometry of the $p$-adic formal stacks $\widehat{ \mathcal{ M}}_{ g, \mathbb{ Z}_p}$ (= the moduli classifying $p$-adic formal curves of fixed genus $g>1$) and $\widehat{ \mathcal{ S}}_{ g, \mathbb{ Z}_p}$ (= the moduli classifying $p$-adic formal curves of genus $g$ equipped with an indigenous bundle). A major achievement in the (classical) $p$-adic Teichm\"{ u}ller theory is the construction of the locus $\widehat{ \mathcal{ N}}_{ g, \mathbb{ Z}_p}^{ \mathrm{ ord}}$ in $\widehat{ \mathcal{ S}}_{ g, \mathbb{ Z}_p}$ classifying $p$-adic canonical liftings of ordinary nilpotent indigenous bundles. The formal stack $\widehat{ \mathcal{ N}}_{ g, \mathbb{ Z}_p}^{ \mathrm{ ord}}$ 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. In the present paper, the canonical symplectic structure on the cotangent bundle $T^\vee_{ \mathbb{ Z}_p} \widehat{ \mathcal{ M}}_{ g, \mathbb{ Z}_p}$ of $\widehat{ \mathcal{ M}}_{ g, \mathbb{ Z}_p}$ is compared to Goldman's symplectic structure defined on $\widehat{ \mathcal{ S}}_{ g, \mathbb{ Z}_p}$ after base-change by the projection $\widehat{ \mathcal{ N}}_{ g, \mathbb{ Z}_p}^{ \mathrm{ ord}} \rightarrow \widehat{ \mathcal{ M}}_{ g, \mathbb{ Z}_p}$. We can think of this comparison as a $p$-adic analogue of certain results in the theory of projective structures on Riemann surfaces proved by S. Kawai and other mathematicians.
Categories: math.AG math.NT math.SG