Traces of Hecke operators in level 1 and Gaussian hypergeometric functions
Author:
Jenny G. Fuselier
Journal:
Proc. Amer. Math. Soc. 141 (2013), 18711881
MSC (2010):
Primary 11F30; Secondary 11T24, 11G20, 33C99
Published electronically:
December 10, 2012
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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, 1833. MR 1872789 (2002h:11056)
 2.
S. Ahlgren and K. Ono, Modularity of a certain CalabiYau threefold, Monatsh. Math. 129 (2000), no. 3, 177190. 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), 187212. 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), 5760. 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, 32333262. 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, 109123. MR 2550175 (2011c:11068)
 8.
J. Greene, Hypergeometric functions over finite fields, Trans. Amer. Math. Soc. 301 (1987), no. 1, 77101. 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), 267295. 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, 19311938. 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, 183211. MR 914657 (88k:14013)
 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, 1833. MR 1872789 (2002h:11056)
 2.
 S. Ahlgren and K. Ono, Modularity of a certain CalabiYau threefold, Monatsh. Math. 129 (2000), no. 3, 177190. 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), 187212. 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), 5760. 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, 32333262. 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, 109123. MR 2550175 (2011c:11068)
 8.
 J. Greene, Hypergeometric functions over finite fields, Trans. Amer. Math. Soc. 301 (1987), no. 1, 77101. 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), 267295. 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, 19311938. 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, 183211. MR 914657 (88k:14013)
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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:
http://dx.doi.org/10.1090/S000299392012115400
PII:
S 00029939(2012)115400
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.
