Tau Function Approach to Theta Functions

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.