×

The Deligne-Mumford compactification of the real multiplication locus and Teichmüller curves in genus 3. (English) Zbl 1250.14014

The extensive paper under review investigates the real multiplication locus \(\mathcal{RM}\) in the moduli space \(\mathcal{M}_g\) of Riemann surfaces of genus \(g\). In particular, it is concerned with the boundary of \(\mathcal{RM}\) in the Deligne-Mumford compactification \(\bar{\mathcal{M}}_g\).
For \(g=3\), the authors give a complete description of the corresponding stable curves. They also identify those stable curves equipped with holomorphic 1-forms which are in the boundary of the eigenform locus in the bundle \[ \Omega\mathcal{M}_g\to\mathcal{M}_g \] of holomorphic 1-forms (the case \(g=2\) has been treated in [Geom. Topol. 11, 1887–2073 (2007; Zbl 1131.32007)]).
For genus \(g > 3\), the authors prove quite strong restrictions on the stable curves in the boundary of \(\mathcal{RM}\). Many questions about Riemann surfaces with real multiplication can thus be reduced to concrete problems in algebraic geometry and number theory by passing to the boundary of \(\mathcal{M}_g\).
The boundary classification is used to prove, for instance, finiteness results for Teichmüller curves in \(\mathcal{M}_3\) and the non-invariance of the eigenform locus under the action of GL\(^+_2(\mathbb{R})\) on \(\Omega\mathcal{M}_3\). Relations to Hilbert modular varieties are also discussed.

MSC:

14G35 Modular and Shimura varieties
14D22 Fine and coarse moduli spaces
14H10 Families, moduli of curves (algebraic)
14H15 Families, moduli of curves (analytic)
14H55 Riemann surfaces; Weierstrass points; gap sequences
30F30 Differentials on Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)

Citations:

Zbl 1131.32007
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Abikoff, W., Degenerating families of Riemann surfaces. Ann. of Math., 105 (1977), 29–44. · Zbl 0347.32010
[2] Accola, R., Differentials and extremal length on Riemann surfaces. Proc. Natl. Acad. Sci. USA, 46 (1960), 540–543. · Zbl 0093.07801
[3] Ahlfors, L., Lectures on Quasiconformal Mappings. Van Nostrand Mathematical Studies, 10. Van Nostrand, Toronto–New York–London, 1966.
[4] Bainbridge, M., Euler characteristics of Teichmüller curves in genus two. Geom. Topol., 11 (2007), 1887–2073. · Zbl 1131.32007
[5] – Billiards in L-shaped tables with barriers. Geom. Funct. Anal., 20 (2010), 299–356. · Zbl 1213.37062
[6] Bass, H., Torsion free and projective modules. Trans. Amer. Math. Soc., 102 (1962), 319–327. · Zbl 0103.02304
[7] Belabas, K., A fast algorithm to compute cubic fields. Math. Comp., 66 (1997), 1213–1237. · Zbl 0882.11070
[8] Bers, L., Spaces of degenerating Riemann surfaces, in Discontinuous Groups and Riemann Surfaces (University of Maryland, College Park, MD, 1973), Ann. of Math. Studies, 79, pp. 43–55. Princeton Univ. Press, Princeton, NJ, 1974.
[9] – Finite-dimensional Teichmüller spaces and generalizations. Bull. Amer. Math. Soc., 5 (1981), 131–172. · Zbl 0485.30002
[10] Bombieri, E., Masser, D. & Zannier, U., Intersecting a curve with algebraic subgroups of multiplicative groups. Int. Math. Res. Not., 1999 (1999), 1119–1140. · Zbl 0938.11031
[11] Borel, A. & Ji, L., Compactifications of Symmetric and Locally Symmetric Spaces. Mathematics: Theory & Applications. Birkhäuser, Boston, MA, 2006. · Zbl 1100.22001
[12] Borevich, A. & Shafarevich, I., Number Theory. Pure and Applied Mathematics, 20. Academic Press, New York, 1966. · Zbl 0145.04902
[13] Bouw, I. & Möller, M., Teichmüller curves, triangle groups, and Lyapunov exponents. Ann. of Math., 172 (2010), 139–185. · Zbl 1203.37049
[14] Calta, K., Veech surfaces and complete periodicity in genus two. J. Amer. Math. Soc., 17 (2004), 871–908. · Zbl 1073.37032
[15] Cohen, H., Advanced Topics in Computational Number Theory. Graduate Texts in Mathematics, 193. Springer, New York, 2000. · Zbl 0977.11056
[16] Douady, A. & Hubbard, J., A proof of Thurston’s topological characterization of rational functions. Acta Math., 171 (1993), 263–297. · Zbl 0806.30027
[17] Eskin, A., Masur, H. & Zorich, A., Moduli spaces of abelian differentials: the principal boundary, counting problems, and the Siegel–Veech constants. Publ. Math. Inst. Hautes Études Sci., 97 (2003), 61–179. · Zbl 1037.32013
[18] Freitag, E., Hilbert Modular Forms. Springer, Berlin-Heidelberg, 1990. · Zbl 0702.11029
[19] van der Geer, G., Hilbert Modular Surfaces. Ergebnisse der Mathematik und ihrer Gren-zgebiete, 16. Springer, Berlin-Heidelberg, 1988. · Zbl 0634.14022
[20] Goren, E., Lectures on Hilbert Modular Varieties and Modular Forms. CRM Monograph Series, 14. Amer. Math. Soc., Providence, RI, 2002.
[21] Griffiths, P. & Harris, J., Principles of Algebraic Geometry. Pure and Applied Mathematics. Wiley, New York, 1978. · Zbl 0408.14001
[22] Habegger, P., Intersecting subvarieties of $$ \(\backslash\)mathrm{G}_m\^n $$ with algebraic subgroups. Math. Ann., 342 (2008), 449–466. · Zbl 1168.14019
[23] Harris, J. & Morrison, I., Moduli of Curves. Graduate Texts in Mathematics, 187. Springer, New York, 1998. · Zbl 0913.14005
[24] Hindry, M. & Silverman, J. H., Diophantine Geometry. Graduate Texts in Mathematics, 201. Springer, New York, 2000.
[25] van Hoeij, M., An algorithm for computing the Weierstrass normal form of hyperelliptic curves. Preprint, 2002. arXiv:0203130 [math.AG].
[26] Hubert, P. & Lanneau, E., Veech groups without parabolic elements. Duke Math. J., 133 (2006), 335–346. · Zbl 1101.30044
[27] Imayoshi, Y. & Taniguchi, M., An Introduction to Teichmüller Spaces. Springer, Tokyo, 1992. · Zbl 0754.30001
[28] Kenyon, R. & Smillie, J., Billiards on rational-angled triangles. Comment. Math. Helv., 75 (2000), 65–108. · Zbl 0967.37019
[29] Laurent, M., Équations diophantiennes exponentielles. Invent. Math., 78 (1984), 299–327. · Zbl 0554.10009
[30] Masur, H., On a class of geodesics in Teichmüller space. Ann. of Math., 102 (1975), 205–221. · Zbl 0322.32010
[31] – Extension of the Weil-Petersson metric to the boundary of Teichmüller space. Duke Math. J., 43 (1976), 623–635. · Zbl 0358.32017
[32] Mcmullen, C. T., Billiards and Teichmüller curves on Hilbert modular surfaces. J. Amer. Math. Soc., 16 (2003), 857–885. · Zbl 1030.32012
[33] – Teichmüller curves in genus two: discriminant and spin. Math. Ann., 333 (2005), 87–130. · Zbl 1086.14024
[34] – Prym varieties and Teichmüller curves. Duke Math. J., 133 (2006), 569–590. · Zbl 1099.14018
[35] – Teichmüller curves in genus two: torsion divisors and ratios of sines. Invent. Math., 165 (2006), 651–672. · Zbl 1103.14014
[36] – Dynamics of SL2(R) over moduli space in genus two. Ann. of Math., 165 (2007), 397–456. · Zbl 1131.14027
[37] Möller, M., Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve. Invent. Math., 165 (2006), 633–649. · Zbl 1111.14019
[38] – Variations of Hodge structures of a Teichmüller curve. J. Amer. Math. Soc., 19 (2006), 327–344. · Zbl 1090.32004
[39] – Finiteness results for Teichmüller curves. Ann. Inst. Fourier (Grenoble), 58 (2008), 63–83. · Zbl 1140.14010
[40] Philippon, P., Sur des hauteurs alternatives. III. J. Math. Pures Appl., 74 (1995), 345–365. · Zbl 0878.11025
[41] Roman, S., Advanced Linear Algebra. Graduate Texts in Mathematics, 135. Springer, New York, 1992. · Zbl 0754.15002
[42] Veech, W., Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards. Invent. Math., 97 (1989), 553–583. · Zbl 0676.32006
[43] Voisin, C., Hodge Theory and Complex Algebraic Geometry. I. Cambridge Studies in Advanced Mathematics, 76. Cambridge University Press, Cambridge, 2007. · Zbl 1129.14019
[44] Ward, C., Calculation of Fuchsian groups associated to billiards in a rational triangle. Ergodic Theory Dynam. Systems, 18 (1998), 1019–1042. · Zbl 0915.58059
[45] Wolpert, S., Riemann surfaces, moduli and hyperbolic geometry, in Lectures on Riemann Surfaces (Trieste, 1987), pp. 48–98. World Scientific, Teaneck, NJ, 1989. · Zbl 0800.32005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.