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 2gdimensional 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.
 Publication:

SIGMA
 Pub Date:
 October 2013
 DOI:
 10.3842/SIGMA.2013.062
 arXiv:
 arXiv:1202.3560
 Bibcode:
 2013SIGMA...9..062G
 Keywords:

 real Riemann surfaces;
 period matrices;
 modular action;
 fundamental domain;
 reduction theory of positive definite quadratic forms;
 Mathematical Physics;
 Mathematics  Algebraic Geometry;
 Mathematics  Complex Variables
 EPrint:
 SIGMA 9 (2013), 062, 25 pages