  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.
