Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T15:31:50.906Z Has data issue: false hasContentIssue false
  • English
  • Français

Modular embeddings of Teichmüller curves

Part of: Arithmetic problems. Diophantine geometry Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  21 September 2016

Martin Möller  and
Don Zagier
Martin Möller
Affiliation:
Institut für Mathematik, Goethe-Universität Frankfurt, Robert-Mayer-Str. 6–8, 60325 Frankfurt am Main, Germany email [email protected]
Don Zagier
Affiliation:
Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Fuchsian groups with a modular embedding have the richest arithmetic properties among non-arithmetic Fuchsian groups. But they are very rare, all known examples being related either to triangle groups or to Teichmüller curves. In Part I of this paper we study the arithmetic properties of the modular embedding and develop from scratch a theory of twisted modular forms for Fuchsian groups with a modular embedding, proving dimension formulas, coefficient growth estimates and differential equations. In Part II we provide a modular proof for an Apéry-like integrality statement for solutions of Picard–Fuchs equations. We illustrate the theory on a worked example, giving explicit Fourier expansions of twisted modular forms and the equation of a Teichmüller curve in a Hilbert modular surface. In Part III we show that genus two Teichmüller curves are cut out in Hilbert modular surfaces by a product of theta derivatives. We rederive most of the known properties of those Teichmüller curves from this viewpoint, without using the theory of flat surfaces. As a consequence we give the modular embeddings for all genus two Teichmüller curves and prove that the Fourier developments of their twisted modular forms are algebraic up to one transcendental scaling constant. Moreover, we prove that Bainbridge’s compactification of Hilbert modular surfaces is toroidal. The strategy to compactify can be expressed using continued fractions and resembles Hirzebruch’s in form, but every detail is different.

Keywords

Teichmüller curvesHilbert modular surfacestheta functionsmodular forms

MSC classification

Primary: 11F27: Theta series; Weil representation; theta correspondences
Secondary: 14G35: Modular and Shimura varieties 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 11 , November 2016 , pp. 2269 - 2349
Copyright
© The Authors 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ash, A., Mumford, D., Rapoport, M. and Tai, Y., Smooth compactification of locally symmetric varieties , in Lie groups: history, frontiers and applications, Vol. IV (Mathematical Science Press, Brookline, NY, 1975).Google Scholar
Bainbridge, M., Euler characteristics of Teichmüllercurves in genus two , Geom. Topol. 11 (2007), 1887–2073.Google Scholar
Bainbridge, M. and Möller, M., The Deligne–Mumford compactification of the real multiplication locus and Teichmüllercurves in genus 3 , Acta Math. 208 (2012), 1–92.Google Scholar
Beukers, F., Irrationality proofs using modular forms , Astérisque 147–148 (1987), 271–283.Google Scholar
Bouw, I. and Möller, M., Differential equations associated with non-arithmetic Fuchsian groups , J. Lond. Math. Soc. (2) 81 (2010), 65–90.Google Scholar
Bouw, I. and Möller, M., Triangle groups, Teichmüllercurves and Lyapunov exponents , Ann. of Math. (2) 172 (2010), 139–185.Google Scholar
Bruinier, J., van der Geer, G. and Zagier, D., The 1–2–3 of modular forms (Springer, Berlin, 2009).Google Scholar
Calta, K., Veech surfaces and complete periodicity in genus two , J. Amer. Math. Soc. 17 (2004), 871–908.CrossRefGoogle Scholar
Carathéodory, C., Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen , Math. Ann. 64 (1907), 95–115.Google Scholar
Choie, Y. J. and Zagier, D., Rational period functions for PSL(2, ℤ), Contemporary Mathematics, vol. 143 (American Mathematical Society, Providence, RI, 1993), 89–107.Google Scholar
Cohen, P. and Wolfart, J., Modular embeddings for some non-arithmetic Fuchsian groups , Acta Arith. 56 (1990), 93–110.Google Scholar
Farkas, H. and Kra, I., Riemann surfaces (Springer, Berlin, 1980).Google Scholar
Franke, H.-G., Kurven in Hilbertschen Modulflächen und Humbertsche Flächen im Siegel-Raum. Dissertation, Universität Bonn, 1977.Google Scholar
Fulton, W., Introduction to toric varieties, Annals of Mathematics Studies, vol. 131 (Princeton University Press, Princeton, NJ, 1993).Google Scholar
Hammond, W., The modular groups of Hilbert and Siegel , Amer. J. Math. 88 (1966), 497–516.CrossRefGoogle Scholar
Hermann, C. F., Symmetrische Hilbertsche Modulformen und Modulfunktionen zu ℚ(√17) , Math. Ann. 256 (1981), 191–197.CrossRefGoogle Scholar
Hirzebruch, F., Hilbert modular surfaces , Enseign. Math. (2) 19 (1973), 183–281.Google Scholar
Hirzebruch, F. and Zagier, D., Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus , Invent. Math. 36 (1976), 57–113.Google Scholar
Hirzebruch, F. and Zagier, D., Classification of Hilbert modular surfaces , in Complex analysis and algebraic geometry (Iwanami Shoten & Cambridge University Press, Tokyo, 1977), 43–77.Google Scholar
Igusa, J.-I., Theta functions, Grundlehren der Mathematischen Wissenschaften, vol. 194 (Springer, Heidelberg, 1972).Google Scholar
Iwaniec, H., Introduction to the spectral theory of automorphic forms, Biblioteca of Revista Matemática Iberoamericana (Real Sociedad Matemática Española, Madrid, 1995).Google Scholar
Kumar, A. and Mukamel, R., Algebraic models and arithmetic geometry of Teichmüller curves in genus two, Preprint (2014), arXiv:1406.7057.Google Scholar
Lee, M. H., Mixed automorphic forms, torus bundles, and Jacobi forms, Lecture Notes in Mathematics, vol. 1845 (Springer, Berlin, 2004); MR 2067222, doi:10.1007/b97773.Google Scholar
McMullen, C., Billiards and Teichmüller curves on Hilbert modular surfaces , J. Amer. Math. Soc. 16 (2003), 857–885.Google Scholar
McMullen, C., Teichmüller curves in genus two: discriminant and spin , Math. Ann. 333 (2005), 87–130.Google Scholar
McMullen, C., Teichmüller curves in genus two: torsion divisors and ratios of sines , Invent. Math. 165 (2006), 651–672.Google Scholar
McMullen, C., Prym varieties and Teichmüller curves , Duke Math. J. 133 (2006), 569–590.Google Scholar
McMullen, C., Foliations of Hilbert modular surfaces , Amer. J. Math. 129 (2007), 365–425.Google Scholar
McMullen, C., Moduli spaces of isoperiodic forms on Riemann surfaces , Duke Math. J. 163 (2012), 2271–2323.Google Scholar
Möller, M., Variations of Hodge structures of Teichmüllercurves , J. Amer. Math. Soc. 19 (2006), 327–344.Google Scholar
Möller, M., Periodic points on Veech surfaces and the Mordell–Weil group over a Teichmüllercurve , Invent. Math. 165 (2006), 633–649.Google Scholar
Möller, M. and Viehweg, E., Kobayashi geodesics in A g , J. Differential Geom. 86 (2010), 355–379.Google Scholar
Möller, M., Prym covers, theta functions and Kobayashi geodesics in Hilbert modular surfaces , Amer. J. Math. 136 (2014), 995–1021.Google Scholar
Mukamel, R., Fundamental domains and generators for lattice Veech groups, Preprint (2012).Google Scholar
Mukamel, R., Orbifold points on Teichmüller curves and Jacobians with complex multiplication , Geom. Topol. 18 (2014), 779–829.Google Scholar
Ricker, S., Symmetric Fuchsian quadrilateral groups and modular embeddings , Quart. J. Math. 53 (2002), 75–86.Google Scholar
Schmutz-Schaller, P. and Wolfart, J., Semi-arithmetic Fuchsian groups and modular embeddings , J. Lond. Math. Soc. (2) 61 (2000), 13–24.CrossRefGoogle Scholar
Stömbergsson, A., On the uniform equidistribution of long closed horocycles , Duke Math. J. 123 (2004), 507–547.Google Scholar
van der Geer, G., Hilbert modular sufaces (Springer, Berlin, 1987).Google Scholar
Weiss, C., Twisted Teichmüller curves, Lecture Notes in Mathematics, vol. 2041 (Springer, Berlin, 2014), 166.Google Scholar
Wolfart, J., Eine arithmetische Eigenschaft automorpher Formen zu gewissen nicht-arithmetischen Gruppen , Math. Ann. 262 (1983), 1–21.Google Scholar
Wolfart, J., Der Überlagerungsradius gewisser algebraischer Kurven und die Werte der Betafunktion an rationalen Stellen , Math. Ann. 273 (1985), 1–15.Google Scholar
Zagier, D., Modular forms associated to real quadratic fields , Invent. Math. 30 (1975), 1–46.Google Scholar
Zagier, D., On the values at negative integers of the zeta-function of a real quadratic field , Enseign. Math. (2) 22 (1976), 55–95.Google Scholar
Zagier, D., Zetafuntionen und Quadratische Körper, Hochschultext (Springer, Berlin, 1981).Google Scholar
Zagier, D., Integral solutions of Apéry-like recurrence equations , in Groups and symmetries: from the Neolithic Scots to John McKay, CRM Proceedings and Lecture Notes, vol. 47 (American Mathematical Society, Providence, RI, 2009), 349–366.Google Scholar
Zorich, A., Flat surfaces , in Frontiers in number theory, physics and geometry. Volume 1: on random matrices, zeta functions and dynamical systems (Springer, Berlin, 2006), 439–586.Google Scholar