### 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.

pdf (508 kb)   tex (37 kb)

References

1. Barnes E.S., Cohn M.J., On Minkowski reduction of positive quaternary quadratic forms, Mathematika 23 (1976), 156-158.
2. Beauville A., Le problème de Schottky et la conjecture de Novikov, Astérisque 1986/87 (1987), 101-112.
3. Belokolos E.D., Bobenko A.I., Enol'ski V.Z., Its A.R., Matveev V.B., Algebro-geometric approach to nonlinear integrable equations, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1994.
4. Bujalance E., Cirre F.J., Gamboa J.M., Gromadzki G., Symmetries of compact Riemann surfaces, Lecture Notes in Mathematics, Vol. 2007, Springer-Verlag, Berlin, 2010.
5. Conway J.H., Curtis R.T., Norton S.P., Parker R.A., Wilson R.A., Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups (with computational assistance from J.G. Thackray), Oxford University Press, Eynsham, 1985.
6. Costa A.F., Natanzon S.M., Poincaré's theorem for the modular group of real Riemann surfaces, Differential Geom. Appl. 27 (2009), 680-690, math.AG/0602413.
7. Debarre O., The Schottky problem: an update, in Current Topics in Complex Algebraic Geometry (Berkeley, CA, 1992/93), Math. Sci. Res. Inst. Publ., Vol. 28, Cambridge University Press, Cambridge, 1995, 57-64.
8. Dubrovin B.A., Flickinger R., Segur H., Three-phase solutions of the Kadomtsev-Petviashvili equation, Stud. Appl. Math. 99 (1997), 137-203.
9. Fay J.D., Theta functions on Riemann surfaces, Lecture Notes in Mathematics, Vol. 352, Springer-Verlag, Berlin, 1973.
10. Grushevsky S., The Schottky problem, in Current developments in algebraic geometry, Math. Sci. Res. Inst. Publ., Vol. 59, Cambridge University Press, Cambridge, 2012, 129-164, arXiv:1009.0369.
11. Lascurain Orive A., Molina Hernandez R., On fundamental domains for subgroups of isometries acting in $\mathbb{H}^{n}$, ISRN Geometry 2007 (2007), 685103, 27 pages.
12. Minkowski H., Diskontinuitätsbereich für arithmetische Äquivalenz, J. für Math. 129 (1905), 220-274.
13. Natanzon S.M., Moduli of Riemann surfaces, real algebraic curves, and their superanalogs, Translations of Mathematical Monographs, Vol. 225, American Mathematical Society, Providence, RI, 2004.
14. Newman M., Reiner I., Inclusion theorems for congruence subgroups, Trans. Amer. Math. Soc. 91 (1959), 369-379.
15. Newman M., Smart J.R., Symplectic modulary groups, Acta Arith. 9 (1964), 83-89.
16. Riera G., Automorphisms of abelian varieties associated with Klein surfaces, J. London Math. Soc. 51 (1995), 442-452.
17. Ryshkov S.S., The theory of Hermite-Minkowski reduction of positive definite quadratic forms, J. Soviet Math. 6 (1976), 651-671.
18. Siegel C.L., Topics in complex function theory. Vol. II. Automorphic functions and abelian integrals, Wiley Classics Library, John Wiley & Sons Inc., New York, 1988.
19. Siegel C.L., Topics in complex function theory. Vol. III. Abelian functions and modular functions of several variables, Wiley Classics Library, John Wiley & Sons Inc., New York, 1989.
20. Silhol R., Real algebraic surfaces, Lecture Notes in Mathematics, Vol. 1392, Springer-Verlag, Berlin, 1989.
21. Silhol R., Compactifications of moduli spaces in real algebraic geometry, Invent. Math. 107 (1992), 151-202.
22. Silhol R., Normal forms for period matrices of real curves of genus 2 and 3, J. Pure Appl. Algebra 87 (1993), 79-92.