Tau Function Approach to Theta Functions

TL;DR

This paper explores theta functions of Riemann surfaces through tau functions of soliton hierarchies, providing new series expansions, refining classical theorems, and establishing modular invariance of sigma functions.

Contribution

It introduces novel series expansions of theta functions, refines Riemann's singularity theorem, and determines modular-invariant normalization constants for higher genus sigma functions.

Findings

Series expansions of theta functions at divisor points and on Abel-Jacobi images.

Refinement of Riemann's singularity theorem.

Normalization constants for higher genus sigma functions are modular invariant.

Abstract

We study theta functions of a Riemann surface of genus g from the view point of tau function of a hierarchy of soliton equations. We study two kinds of series expansions. One is the Taylor expansion at any point of the theta divisor. We describe the initial term of the expansion by the Schur function corresponding to the partition determined by the gap sequence of a certain flat line bundle. The other is the expansion of the theta function and its certain derivatives in one of the variables on the Abel-Jacobi images of k points on a Riemann surface with k less than or equal to g. We determine the initial term of the expansion as certain derivatives of the theta function successively. As byproducts, firstly we obtain a refinement of Riemann's singularity theorem. Secondly we determine normalization constants of higher genus sigma functions of a Riemann surface, defined by Korotkin and Shramchenko, such that they become modular invariant.