Proceedings of the American Mathematical Society

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
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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.

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

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.

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