Arithmetic geometry of algebraic curves and their moduli space

TL;DR

This paper reviews the fundamental aspects of algebraic curves, their moduli spaces, and uniformization theories, emphasizing their applications to arithmetic properties and automorphic forms on these moduli spaces.

Contribution

It synthesizes classical and arithmetic uniformization theories and explores their applications to the arithmetic of algebraic curves and automorphic forms.

Findings

Overview of algebraic curves and moduli space theory

Extension of Schottky uniformization to arithmetic uniformization

Applications to automorphic forms and arithmetic of moduli spaces

Abstract

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.