Period Matrices of Real Riemann Surfaces and Fundamental Domains

Details Period Matrices of Real Riemann Surfaces and Fundamental Domains Period Matrices of Real Riemann Surfaces and Fundamental Domains

2012-02-16

arxiv.org/abs/1202.3560v2

Physics

Mathematical Physics

29 pages; minor corrections, added affiliation and acknowledgements

Scientific paper

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.

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