Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-09T01:32:11.090Z Has data issue: false hasContentIssue false
  • English
  • Français

Schwarzian differential equations and Hecke eigenforms on Shimura curves

Part of: Discontinuous groups and automorphic forms Arithmetic algebraic geometry

Published online by Cambridge University Press:  31 October 2012

Yifan Yang
Yifan Yang*
Affiliation:
Department of Applied Mathematics, National Chiao Tung University and National Center for Theoretical Sciences, Hsinchu, Taiwan 300, Taiwan (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let X be a Shimura curve of genus zero. In this paper, we first characterize the spaces of automorphic forms on X in terms of Schwarzian differential equations. We then devise a method to compute Hecke operators on these spaces. An interesting by-product of our analysis is the evaluation and other similar identities.

Keywords

Hecke eigenformsHecke operatorsSchwarzian differential equationsShimura curves

MSC classification

Secondary: 11F12: Automorphic forms, one variable 11G18: Arithmetic aspects of modular and Shimura varieties 33C05: Classical hypergeometric functions, $_2F_1$
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 1 , January 2013 , pp. 1 - 31
Copyright
Copyright © The Author(s) 2012

References

[AB04]Alsina, M. and Bayer, P., Quaternion orders, quadratic forms, and Shimura curves, CRM Monograph Series, vol. 22 (American Mathematical Society, Providence, RI, 2004).Google Scholar
[BT07]Bayer, P. and Travesa, A., Uniformizing functions for certain Shimura curves, in the case D=6, Acta Arith. 126 (2007), 315–339.CrossRefGoogle Scholar
[Eic73]Eichler, M., The basis problem for modular forms and the traces of the Hecke operators, in Modular functions of one variable, I (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Mathematics, vol. 320 (Springer, Berlin, 1973), 75–151.Google Scholar
[Elk98]Elkies, N. D., Shimura curve computations, in Algorithmic number theory (Portland, OR, 1998), Lecture Notes in Computer Science, vol. 1423 (Springer, Berlin, 1998), 1–47.Google Scholar
[For72]Ford, L. R., Automorphic functions (Chelsea Publishing Company, New York, NY, 1972), Reprint of the second edition, 1952.Google Scholar
[Hil97]Hille, E., Ordinary differential equations in the complex domain (Dover Publications Inc, Mineola, NY, 1997), Reprint of the 1976 original.Google Scholar
[JL70]Jacquet, H. and Langlands, R. P., Automorphic forms on GL(2), Lecture Notes in Mathematics, vol. 114 (Springer, Berlin, 1970).CrossRefGoogle Scholar
[Shi72]Shimizu, H., Theta series and automorphic forms on GL2, J. Math. Soc. Japan 24 (1972), 638–683.CrossRefGoogle Scholar
[Shi67]Shimura, G., Construction of class fields and zeta functions of algebraic curves, Ann. of Math. (2) 85 (1967), 58–159.Google Scholar
[Shi94]Shimura, G., Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, vol. 11 (Princeton University Press, Princeton, NJ, 1994), Reprint of the 1971 original, Kano Memorial Lectures, 1.Google Scholar
[Sij12]Sijsling, J., Arithmetic (1;e)-curves and Belyĭ maps, Math. Comp. 81 (2012), 1823–1855.CrossRefGoogle Scholar
[Ste04]Stein, W., The modular forms database, (2004), http://modular.math.washington.edu/Tables.Google Scholar
[Sti84]Stiller, P., Special values of Dirichlet series, monodromy, and the periods of automorphic forms, Mem. Amer. Math. Soc. 49 (1984), iv+116.Google Scholar
[Tu11]Tu, F.-T., Schwarzian differential equations associated to Shimura curves of genus zero, Preprint (2011).Google Scholar
[Voi06]Voight, J., Computing CM points on Shimura curves arising from cocompact arithmetic triangle groups, in Algorithmic number theory (ANTS VII, Berlin, 2006), Lecture Notes in Computer Science, vol. 4076, eds Hess, F., Pauli, S. and Pohst, M. (Springer, Berlin, 2006), 406–420.CrossRefGoogle Scholar
[Voi09]Voight, J., Shimura curve computations, in Arithmetic geometry, Clay Mathematics Proceedings, vol. 8 (American Mathematical Society, Providence, RI, 2009), 103–113.Google Scholar
[Yam73]Yamauchi, M., On the traces of Hecke operators for a normalizer of Γ0(N), J. Math. Kyoto Univ. 13 (1973), 403–411.Google Scholar
[Yan04]Yang, Y., On differential equations satisfied by modular forms, Math. Z. 246 (2004), 1–19.CrossRefGoogle Scholar
[Yan11]Yang, Y., Modular forms of half-integral weights on , Preprint (2011), arXiv:1110.1810.Google Scholar