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Some Differential Equations for the Riemann $θ$-Function on Jacobians

Robert Wilms

Abstract

We prove some differential equations for the Riemann theta function associated to the Jacobian of a Riemann surface. The proof is based on some variants of a formula by Fay for the theta function, which are motivated by their analogues in Arakelov theory of Riemann surfaces.

Some Differential Equations for the Riemann $θ$-Function on Jacobians

Abstract

We prove some differential equations for the Riemann theta function associated to the Jacobian of a Riemann surface. The proof is based on some variants of a formula by Fay for the theta function, which are motivated by their analogues in Arakelov theory of Riemann surfaces.

Paper Structure

This paper contains 3 sections, 5 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Variations of Fay's formula
  3. Analogous results in Arakelov theory

Key Result

Theorem 1.1

Let $X$ be a compact and connected Riemann surface of genus $g\ge 1$ and $p_1,\dots,p_g$, $q\in X$ arbitrary points on $X$ in general position. We denote the degree $(g-1)$ divisor $D=\sum_{j=1}^g p_j-q$ and the effective degree $(g-1)$ divisors $D_k=\sum_{j=1}^g p_j-p_k$ for $1\le k\le g$. Then the

Theorems & Definitions (5)

  • Theorem 1.1
  • Proposition 2.1
  • Proposition 2.2
  • Corollary 2.5
  • Proposition 3.1