普通幂零固有丛的$p$ -adic Teichmüller一致化的辛几何
Symplectic geometry of $p$-adic Teichmüller uniformization for ordinary nilpotent indigenous bundles
摘要：
本文的目的是为S.望月研究了$p$ -adic形式栈$\widehat{\mathcal{M}}_{g, \mathbb{Z}_p}$（=固定亏格$g>1$的$p$ -adic形式曲线的模分类）和$\widehat{\mathcal{S}}_{g, \mathbb{Z}_p}$（=具有固有丛的亏格$g$的$p$ -adic形式曲线的模分类）的辛几何.（经典的）$p$ -adic Teichmüller理论的一个主要成就是构造了$\widehat{\mathcal{S}}_{g, \mathbb{Z}_p}$中的轨迹$\widehat{\mathcal{N}}_{g, \mathbb{Z}_p}^{\mathrm{ord}}$，分类了普通幂零固有丛的$p$ -adic典范提升.正式的堆栈$\widehat{\mathcal{N}}_{g, \mathbb{Z}_p}^{\mathrm{ord}}$体现了$p$ -adic模拟的双曲黎曼曲面的一致化，以及双曲模拟的Serre-Tate理论的普通阿贝尔品种。本文将$\widehat{\mathcal{M}}_{g, \mathbb{Z}_p}$的余切丛$T^\vee_{\mathbb{Z}_p} \widehat{\mathcal{M}}_{g, \mathbb{Z}_p}$上的正则辛结构与投影$\widehat{\mathcal{N}}_{g, \mathbb{Z}_p}^{\mathrm{ord}} \rightarrow \widehat{\mathcal{M}}_{g, \mathbb{Z}_p}$基变换后定义在$\widehat{\mathcal{S}}_{g, \mathbb{Z}_p}$上的Goldman辛结构进行了比较。我们可以把这种比较看作是S. S.证明的黎曼曲面上射影结构理论中某些结果的$p$ -adic模拟。Kawai和其他数学家。
英文摘要：
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 $\widehat{\mathcal{M}}_{g, \mathbb{Z}_p}$ (= 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 Teichmüller 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.
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