Algebraic transformations of hypergeometric functions and automorphic forms on Shimura curves
HTML articles powered by AMS MathViewer
- by Fang-Ting Tu and Yifan Yang PDF
- Trans. Amer. Math. Soc. 365 (2013), 6697-6729 Request permission
Abstract:
In this paper, we will obtain new algebraic transformations of the ${}_2F_1$-hypergeometric functions. The main novelty in our approach is the interpretation of identities among ${}_2F_1$-hypergeometric functions as identities among automorphic forms on different Shimura curves.References
- P. Bayer and A. Travesa, Uniformizing functions for certain Shimura curves, in the case $D=6$, Acta Arith. 126 (2007), no. 4, 315–339. MR 2289964, DOI 10.4064/aa126-4-3
- Noam D. Elkies, Shimura curves for level-3 subgroups of the $(2,3,7)$ triangle group, and some other examples, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 4076, Springer, Berlin, 2006, pp. 302–316. MR 2282932, DOI 10.1007/11792086_{2}2
- Édouard Goursat, Sur l’équation différentielle linéaire, qui admet pour intégrale la série hypergéométrique, Ann. Sci. École Norm. Sup. (2) 10 (1881), 3–142 (French). MR 1508709
- Svetlana Katok, Fuchsian groups, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992. MR 1177168
- Goro Shimura, Construction of class fields and zeta functions of algebraic curves, Ann. of Math. (2) 85 (1967), 58–159. MR 204426, DOI 10.2307/1970526
- Goro Shimura, 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; Kanô Memorial Lectures, 1. MR 1291394
- Kisao Takeuchi, Arithmetic triangle groups, J. Math. Soc. Japan 29 (1977), no. 1, 91–106. MR 429744, DOI 10.2969/jmsj/02910091
- Kisao Takeuchi, Commensurability classes of arithmetic triangle groups, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), no. 1, 201–212. MR 463116
- Fang-Ting Tu, Schwarzian differential equations associated to Shimura curves of genus zero, preprint (2011).
- Raimundas Vidūnas, Transformations of some Gauss hypergeometric functions, J. Comput. Appl. Math. 178 (2005), no. 1-2, 473–487. MR 2127899, DOI 10.1016/j.cam.2004.09.053
- Marie-France Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, vol. 800, Springer, Berlin, 1980 (French). MR 580949
- Yifan Yang, Schwarzian differential equations and Hecke eigenforms on Shimura curves, Compositio Math. (to appear).
Additional Information
- Fang-Ting Tu
- Affiliation: Department of Applied Mathematics, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, Taiwan 30010, Republic of China
- Email: ft12.am95g@nctu.edu.tw
- Yifan Yang
- Affiliation: Department of Applied Mathematics, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, Taiwan 30010, Republic of China – and – National Center for Theoretical Sciences, Hsinchu 300, Taiwan, Republic of China
- MR Author ID: 633505
- Email: yfyang@math.nctu.edu.tw
- Received by editor(s): December 12, 2011
- Received by editor(s) in revised form: September 10, 2012
- Published electronically: July 26, 2013
- Additional Notes: The authors were partially supported by Grant 99-2115-M-009-011-MY3 of the National Science Council, Taiwan (R.O.C.).
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 6697-6729
- MSC (2010): Primary 11F12; Secondary 11G18, 33C05
- DOI: https://doi.org/10.1090/S0002-9947-2013-05960-0
- MathSciNet review: 3105767