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

Abstract

The aim of the present paper is to provide a new aspect of the $p$-adic Teichmüller theory established by S. Mochizuki. We study the symplectic geometry of the $p$-adic formal stacks ${\hat{\mathcal{M}}}_{g,{\mathbb{Z}}_p}$ (= the moduli classifying $p$-adic formal curves of fixed genus $g>1$) and ${\hat{\mathcal{S}}}_{g,{\mathbb{Z}}_p}$ (= the moduli classifying $p$-adic formal curves of genus $g$ equipped with an indigenous bundle). A major achievement in the (classical) $p$-adic Teichmüller theory is the construction of the locus ${\hat{\mathcal{N}}}_{g,{\mathbb{Z}}_p}^{\mathrm{ord}}$ in ${\hat{\mathcal{S}}}_{g,{\mathbb{Z}}_p}$ classifying $p$-adic canonical liftings of ordinary nilpotent indigenous bundles. The formal stack ${\hat{\mathcal{N}}}_{g,{\mathbb{Z}}_p}^{\mathrm{ord}}$ 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 $T_{{\mathbb{Z}}_p}^{\vee }{\hat{\mathcal{M}}}_{g,{\mathbb{Z}}_p}$ of ${\hat{\mathcal{M}}}_{g,{\mathbb{Z}}_p}$ is compared to Goldman's symplectic structure defined on ${\hat{\mathcal{S}}}_{g,{\mathbb{Z}}_p}$ after base-change by the projection ${\hat{\mathcal{N}}}_{g,{\mathbb{Z}}_p}^{\mathrm{ord}}\to {\hat{\mathcal{M}}}_{g,{\mathbb{Z}}_p}$. 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.

38 pages

References in corpus (2)

• The complex symplectic geometry of the deformation space of complex projective structures
• Motivic integration over wild Deligne-Mumford stacks