Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 062, 25 pages      arXiv:1202.3560      https://doi.org/10.3842/SIGMA.2013.062

Period Matrices of Real Riemann Surfaces and Fundamental Domains

Pietro Giavedoni
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

Received March 01, 2013, in final form October 14, 2013; Published online October 22, 2013

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.

Key words: real Riemann surfaces; period matrices; modular action; fundamental domain; reduction theory of positive definite quadratic forms.

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