Traces of Hecke operators in level 1 and Gaussian hypergeometric functions

Author:
Jenny G. Fuselier

Journal:
Proc. Amer. Math. Soc. **141** (2013), 1871-1881

MSC (2010):
Primary 11F30; Secondary 11T24, 11G20, 33C99

DOI:
https://doi.org/10.1090/S0002-9939-2012-11540-0

Published electronically:
December 10, 2012

MathSciNet review:
3034414

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We provide formulas for traces of Hecke operators in level 1 in terms of values of finite field -hypergeometric functions, extending previous work of the author to all odd primes instead of only those . We first give a general level 1 trace formula in terms of the trace of Frobenius on a family of elliptic curves, and then we draw on recent work of Lennon to produce level 1 trace formulas in terms of hypergeometric functions for all primes .

**1.**S. Ahlgren,*The points of a certain fivefold over finite fields and the twelfth power of the eta function*, Finite Fields Appl.**8**(2002), no. 1, 18-33. MR**1872789 (2002h:11056)****2.**S. Ahlgren and K. Ono,*Modularity of a certain Calabi-Yau threefold*, Monatsh. Math.**129**(2000), no. 3, 177-190. MR**1746757 (2001b:11059)****3.**S. Ahlgren and K. Ono,*A Gaussian hypergeometric series evaluation and Apéry number congruences*, J. reine angew. Math.**518**(2000), 187-212. MR**1739404 (2001c:11057)****4.**B.J. Birch,*How the number of points of an elliptic curve over a fixed prime field varies*,

J. London Math. Soc.**43**(1968), 57-60. MR**0230682 (37:6242)****5.**S. Frechette, K. Ono, and M. Papanikolas,*Gaussian hypergeometric functions and traces of Hecke operators*, Int. Math. Res. Not. (2004), no. 60, 3233-3262. MR**2096220 (2006a:11055)****6.**J.G. Fuselier,*Hypergeometric functions over finite fields and relations to modular forms and elliptic curves*, Ph.D. Thesis, Texas A&M University, 2007. MR**2710790****7.**J.G. Fuselier,*Hypergeometric functions over and relations to elliptic curves and modular forms*. Proc. Amer. Math. Soc.**138**(2010), no. 1, 109-123. MR**2550175 (2011c:11068)****8.**J. Greene,*Hypergeometric functions over finite fields*, Trans. Amer. Math. Soc.**301**(1987), no. 1, 77-101. MR**879564 (88e:11122)****9.**H. Hijikata, A.K. Pizer, and T.R. Shemanske,*The basis problem for modular forms on*, Mem. Amer. Math. Soc.**82**(1989), no. 418, vi+159 pp. MR**960090 (90d:11056)****10.**Ihara, Y.,*Hecke Polynomials as congruence functions in elliptic modular case*, Ann. of Math. (2)**85**(1967), 267-295. MR**0207655 (34:7470)****11.**Lennon, C.,*Gaussian hypergeometric evaluations and traces of Frobenius for elliptic curves*, Proc. Amer. Math. Soc.**139**(2011), no. 6, 1931-1938. MR**2775369 (2012c:11260)****12.**Lennon, C.,*A Trace Formula for Certain Hecke Operators and Gaussian Hypergeometric Functions*,`http://arxiv.org/abs/1003.11578`.**13.**J. Riordan,*Combinatorial Identities*, John Wiley & Sons, New York, 1968. MR**0231725 (38:53)****14.**R. Schoof,*Nonsingular plane cubic curves over finite fields*, J. Combin. Theory, Ser. A**46**(1987), no. 2, 183-211. MR**914657 (88k:14013)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
11F30,
11T24,
11G20,
33C99

Retrieve articles in all journals with MSC (2010): 11F30, 11T24, 11G20, 33C99

Additional Information

**Jenny G. Fuselier**

Affiliation:
Department of Mathematics and Computer Science, High Point University, High Point, North Carolina 27262

Email:
jfuselie@highpoint.edu

DOI:
https://doi.org/10.1090/S0002-9939-2012-11540-0

Received by editor(s):
September 15, 2011

Published electronically:
December 10, 2012

Communicated by:
Ken Ono

Article copyright:
© Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.