Uniformization, Unipotent Flows and the Riemann Hypothesis

Schottky uniformization and Bloch-Wigner dilogarithm on higher genus curves

The question of convergence of iterated integrals on Riemann surfaces goes back to Bloch, Levin, and Zagier, who have proved this fact for various iterated integrals in the context of elliptic curves. In this work, I prove the convergence of the iterated integral defining Bloch-Wigner function on higher genus curves, utilizing some novel results of Hou in the theory of Schottky uniformization of Riemann surfaces, as well as some classical results of Bers and more recent results of Nayatani on Poincare metrics on domains of discontinuity of Kleinian groups.

代数黎曼曲面均匀化映射的实几何超越性

Real geometric transcendence for uniformization maps of algebraic Riemann surfaces

研究非单连通光滑连通复代数曲线\(X\)中包含其万有覆盖\(\tilde{X}\)中实代数曲线弧像的不可约实代数曲线集，给出该集合非空或无限时\(X\)的条件，还明确描述特定\(X\)情形下的该集合。

Lauricella超几何函数、穿孔Riemann球面的幂单基本群及其motivic余作用

Lauricella hypergeometric functions, unipotent fundamental groups of the punctured Riemann sphere, and their motivic coactions

本文提出一种与超几何积分相关的motives理论，证明散射振幅背景下Gauss超几何函数上的motivic Galois群作用猜想，并展示Lauricella超几何函数的局部与全局作用的兼容性。

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.

Equidistribution Rates, Closed String Amplitudes, and the Riemann Hypothesis

We study asymptotic relations connecting unipotent averages of $Sp(2g,\mathbb{Z})$ automorphic forms to their integrals over the moduli space of principally polarized abelian varieties. We obtain reformulations of the Riemann hypothesis as a class of problems concerning the computation of the equidistribution convergence rate in those asymptotic relations. We discuss applications of our results to closed string amplitudes. Remarkably, the Riemann hypothesis can be rephrased in terms of ultraviolet relations occurring in perturbative closed string theory.

关于酉 Shimura 曲线的 p-adic 一致化

On the p-adic uniformization of unitary Shimura curves

本文证明了关于酉相似群 Shimura 曲线的 p-adic 一致化，通过引入 Kottwitz 条件等定义积分模型，并利用收缩函子与 display 理论给出了沿特殊纤维形式完备化的一致化证明。