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The aim of the present paper is to provide a new aspect of the $p$-adicTeichm\"{u}ller theory established by S. Mochizuki. We study the symplecticgeometry 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). Amajor achievement in the (classical) $p$-adic Teichm\"{u}ller theory is theconstruction 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 indigenousbundles. The formal stack $\widehat{\mathcal{N}}_{g,\mathbb{Z}_p}^{\mathrm{ord}}$ embodies a $p$-adic analogue of uniformization ofhyperbolic Riemann surfaces, as well as a hyperbolic analogue of Serre-Tatetheory of ordinary abelian varieties. In the present paper, the canonicalsymplectic 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 onRiemann surfaces proved by S. Kawai and other mathematicians.

Symplectic geometry of $p$-adic Teichm\"{u}ller uniformization for ordinary nilpotent indigenous bundles