Period Matrices of Real Riemann Surfaces and Fundamental Domains

TL;DR

This paper studies the structure of fundamental domains for certain subgroups of the modular group acting on loci of period matrices of real Riemann surfaces, providing explicit calculations for specific cases.

Contribution

It introduces a general method to describe fundamental domains for the action of subgroups on loci of period matrices of real Riemann surfaces, with detailed examples for genus two and four.

Findings

Explicit fundamental domain descriptions for genus two and four.

A general procedure for even genus and one oval case.

Connections between modular group actions and real Riemann surface geometry.

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.