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This article is cited in 1 scientific paper (total in 1 paper)
Period Matrices of Real Riemann Surfaces and Fundamental Domains
Pietro Giavedoni Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
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.
Keywords:
real Riemann surfaces; period matrices; modular action; fundamental domain; reduction theory of positive definite quadratic forms.
Received: March 1, 2013; in final form October 14, 2013; Published online October 22, 2013
Citation:
Pietro Giavedoni, “Period Matrices of Real Riemann Surfaces and Fundamental Domains”, SIGMA, 9 (2013), 062, 25 pp.
Linking options:
https://www.mathnet.ru/eng/sigma845 https://www.mathnet.ru/eng/sigma/v9/p62
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Abstract page: | 341 | Full-text PDF : | 68 | References: | 41 |
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