Period Matrices of Real Riemann Surfaces and Fundamental Domains

Pietro Giavedoni

Symmetry, integrability and geometry: methods and applications, Tome 9 (2013)

Résumé

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.

Détail

Keywords: real Riemann surfaces; period matrices; modular action; fundamental domain; reduction theory of positive definite quadratic forms.

Bibliographie

[1] Barnes E. S., Cohn M. J., “On {M}inkowski reduction of positive quaternary quadratic forms”, Mathematika, 23 (1976), 156–158 | DOI | MR | Zbl

[2] Beauville A., “Le problème de {S}chottky et la conjecture de {N}ovikov”, Astérisque, 1987, 101–112 | MR | Zbl

[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 | Zbl

[4] Bujalance E., Cirre F. J., Gamboa J. M., Gromadzki G., Symmetries of compact {R}iemann surfaces, Lecture Notes in Mathematics, 2007, Springer-Verlag, Berlin, 2010 | DOI | MR | Zbl

[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, Oxford University Press, Eynsham, 1985 (with computational assistance from J. G. Thackray) | MR | Zbl

[6] Costa A. F., Natanzon S. M., “Poincaré's theorem for the modular group of real {R}iemann surfaces”, Differential Geom. Appl., 27 (2009), 680–690, arXiv: math.AG/0602413 | DOI | MR | Zbl

[7] Debarre O., “The {S}chottky problem: an update”, Current Topics in Complex Algebraic Geometry ({B}erkeley, CA, 1992/93), Math. Sci. Res. Inst. Publ., 28, Cambridge University Press, Cambridge, 1995, 57–64 | MR | Zbl

[8] Dubrovin B. A., Flickinger R., Segur H., “Three-phase solutions of the {K}adomtsev–{P}etviashvili equation”, Stud. Appl. Math., 99 (1997), 137–203 | DOI | MR | Zbl

[9] Fay J. D., Theta functions on {R}iemann surfaces, Lecture Notes in Mathematics, 352, Springer-Verlag, Berlin, 1973 | MR | Zbl

[10] Grushevsky S., “The {S}chottky problem”, Current developments in algebraic geometry, Math. Sci. Res. Inst. Publ., 59, Cambridge University Press, Cambridge, 2012, 129–164, arXiv: 1009.0369 | MR | Zbl

[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 pp. | DOI

[12] Minkowski H., “Diskontinuitätsbereich für arithmetische Äquivalenz”, J. für Math., 129 (1905), 220–274

[13] Natanzon S. M., Moduli of {R}iemann surfaces, real algebraic curves, and their superanalogs, Translations of Mathematical Monographs, 225, American Mathematical Society, Providence, RI, 2004 | MR | Zbl

[14] Newman M., Reiner I., “Inclusion theorems for congruence subgroups”, Trans. Amer. Math. Soc., 91 (1959), 369–379 | DOI | MR | Zbl

[15] Newman M., Smart J. R., “Symplectic modulary groups”, Acta Arith., 9 (1964), 83–89 | MR | Zbl

[16] Riera G., J. London Math. Soc., 51 (1995), Automorphisms of abelian varieties associated with {K}lein surfaces | DOI | MR

[17] Ryshkov S. S., “The theory of Hermite–Minkowski reduction of positive definite quadratic forms”, J. Soviet Math., 6 (1976), 651–671 | DOI | Zbl

[18] Siegel C. L., Topics in complex function theory, v. II, Wiley Classics Library, Automorphic functions and abelian integrals, John Wiley Sons Inc., New York, 1988 | MR

[19] Siegel C. L., Topics in complex function theory, v. III, Wiley Classics Library, Abelian functions and modular functions of several variables, John Wiley Sons Inc., New York, 1989 | MR | Zbl

[20] Silhol R., Real algebraic surfaces, Lecture Notes in Mathematics, 1392, Springer-Verlag, Berlin, 1989 | MR | Zbl

[21] Silhol R., “Compactifications of moduli spaces in real algebraic geometry”, Invent. Math., 107 (1992), 151–202 | DOI | MR | Zbl

[22] Silhol R., “Normal forms for period matrices of real curves of genus {$2$} and {$3$}”, J. Pure Appl. Algebra, 87 (1993), 79–92 | DOI | MR | Zbl