1202.3560 (Pietro Giavedoni)

Period Matrices of Real Riemann Surfaces and Fundamental Domains    [PDF]

Pietro Giavedoni

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 seek 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 details.

View original: http://arxiv.org/abs/1202.3560