Schottky uniformization and Bloch-Wigner dilogarithm on higher genus curves

Berkovich curves and Schottky uniformization

This text is an exposition of non-Archimedean curves and Schottky uniformization from the point of view of Berkovich geometry. It consists of two parts, the first one of an introductory nature, and the second one more advanced. The first part is meant to be an introduction to the theory of Berkovich spaces focused on the case of the affine line. We define the Berkovich affine line and present its main properties, with many details: classification of points, path-connectedness, metric structure, variation of rational functions, etc. Contrary to many other introductory texts, we do not assume th…

Arithmetic geometry of algebraic curves and their moduli space

We review the following subjects: 1. Basic theory on algebraic curves and their moduli space, 2. Schottky uniformization theory of Riemann surfaces, and its extension called arithmetic uniformization theory, 3. Application to these theories to the arithmetic of the moduli space of algebraic curves, especially to automorphic forms on this space.

On Abelian Automorphism Groups of Mumford Curves

We use rigid analytic uniformization by Schottky groups to give a bound for the order of the abelian subgroups of the automorphism group of a Mumford curve in terms of its genus.

超椭圆Mumford曲线的算法：$p$-adic 均匀化、$p$-adic 积分和 $p$-adic 高度

Algorithms for hyperelliptic Mumford Curves $p$-adic Uniformization, $p$-adic integrals and $p$-adic heights

针对超椭圆Mumford曲线，提出计算$p$-adic Schottky群的算法，并利用该均匀化计算$p$-adic Abelian积分和Schneider高度，通过SageMath数值示例验证。

Schottky spaces and universal Mumford curves over $\mathbb{Z}$

For every integer $g \geq 1$ we define a universal Mumford curve of genus $g$ in the framework of Berkovich spaces over $\mathbb{Z}$. This is achieved in two steps: first, we build an analytic space $\mathcal{S}_g$ that parametrizes marked Schottky groups over all valued fields. We show that $\mathcal{S}_g$ is an open, connected analytic space over $\mathbb{Z}$. Then, we prove that the Schottky uniformization of a given curve behaves well with respect to the topology of $\mathcal{S}_g$, both locally and globally. As a result, we can define the universal Mumford curve $\mathcal{C}_g$ as a relat…

高亏格多重zeta值

Higher-genus multiple zeta values

该研究将多重zeta值的形式体系从经典及椭圆情形规范扩展到高亏格黎曼曲面，提出基于肖特基单值化的正则化方案，探究几何退化和多重对数关系衍生的高亏格多重zeta值关系，并通过数值验证发现了新的关系类别。