Period Matrices of Real Riemann Surfaces and Fundamental Domains

Abstract

For some positive integers $g$ and $n$ we consider a subgroup $\mathbb{G}_{g,n}$ of the $2g$-dimensional modular group keeping invariant a certain locus $\mathcal{W}_{g,n}$ in the Siegel upper half plane of degree $g$. We address the problem of describing a fundamental domain for the modular action of the subgroup on $\mathcal{W}_{g,n}$. Our motivation comes from geometry: $g$ and $n$ represent the genus and the number of ovals of a generic real Riemann surface of separated type; the locus $\mathcal{W}_{g,n}$ contains the corresponding period matrix computed with respect to some specific basis in the homology. In this paper we formulate a general procedure to solve the problem when $g$ is even and $n$ equals one. For $g$ equal to two or four the explicit calculations are worked out in full detail.