Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

Hypergeometric functions over ${\mathbb{F}_p}$ and relations to elliptic curves and modular forms

Author(s): Jenny G. Fuselier
Journal: Proc. Amer. Math. Soc. 138 (2010), 109-123.
MSC (2000): Primary 11F30; Secondary 11T24, 11G20, 33C99
Posted: August 28, 2009
MathSciNet review: 2550175
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: For primes $p\equiv 1 \pmod{12}$ , we present an explicit relation between the traces of Frobenius on a family of elliptic curves with -invariant $\frac{1728}{t}$ and values of a particular -hypergeometric function over ${\mathbb{F}_p}$ . We also give a formula for traces of Hecke operators on spaces of cusp forms of weight and level 1 in terms of the same traces of Frobenius. This leads to formulas for Ramanujan's $\tau$ -function in terms of hypergeometric functions.

References:

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.: F. Beukers, Algebraic values of G-functions, J. Reine Angew. Math. 434 (1993), 45-65. MR 1195690 (94d:11055)
5.: D.A. Cox, Primes of the form . In Fermat, Class Field Theory and Complex Multiplication, John Wiley & Sons, New York, 1989. MR 1028322 (90m:11016)
6.: 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)
7.: J.G. Fuselier, Hypergeometric functions over finite fields and relations to modular forms and elliptic curves, Ph.D. thesis, Texas A&M University, 2007.
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 $\Gamma_0(N)$ , Mem. Amer. Math. Soc. 82 (1989), no. 418. MR 960090 (90d:11056)
10.: K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, 2nd ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990. MR 1070716 (92e:11001)
11.: M. Koike, Hypergeometric series over finite fields and Apéry numbers, Hiroshima Math. J. 22 (1992), no. 3, 461-467. MR 1194045 (93i:11146)
12.: S. Lang, Cyclotomic Fields I and II, Graduate Texts in Mathematics, vol. 121, Springer-Verlag, New York, 1990. MR 1029028 (91c:11001)
13.: K. Ono, Values of Gaussian hypergeometric series, Trans. Amer. Math. Soc. 350 (1998), no. 3, 1205-1223. MR 1407498 (98e:11141)
14.: M. Papanikolas, A formula and a congruence for Ramanujan's $\tau$ -function, Proc. Amer. Math. Soc. 134 (2006), no. 2, 333-341. MR 2175999 (2007d:11046)
15.: J. Riordan, Combinatorial Identities, John Wiley & Sons, New York, 1968. MR 0231725 (38:53)
16.: R. Schoof, Nonsingular plane cubic curves over finite fields, J. Combin. Theory, Ser. A 46 (1987), no. 2, 183-211. MR 914657 (88k:14013)
17.: J.H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210 (87g:11070)
18.: P.F. Stiller, Classical automorphic forms and hypergeometric functions, J. Number Theory 28 (1988), no. 2, 219-232. MR 927661 (89b:11037)

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11F30, 11T24, 11G20, 33C99

Retrieve articles in all Journals with MSC (2000): 11F30, 11T24, 11G20, 33C99

Additional Information:

Jenny G. Fuselier
Affiliation: United States Military Academy, 646 Swift Road, West Point, New York 10996
Address at time of publication: Department of Mathematics & Computer Science, Drawer 31, High Point University, High Point, North Carolina 27262
Email: jenny.fuselier@usma.edu, jfuselie@highpoint.edu

DOI: 10.1090/S0002-9939-09-10068-0
PII: S 0002-9939(09)10068-0
Received by editor(s): June 3, 2009
Posted: August 28, 2009
Additional Notes: The author thanks her advisor, Matt Papanikolas, for his advice and support during the preparation of this paper. The author also thanks the Department of Mathematics at Texas A&M University, where the majority of this research was conducted.
Communicated by: Ken Ono
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|