Period Matrices of Real Riemann Surfaces and Fundamental Domains

Abstract

For some positive integers g and n we consider a subgroup G_{g,n} of the 2g-dimensional modular group keeping invariant a certain locus 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 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 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.

Keywords

• real Riemann surfaces
• period matrices
• modular action
• fundamental domain
• reduction theory of positive definite quadratic forms