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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Author: Jenny G. Fuselier
Journal: Proc. Amer. Math. Soc. 138 (2010), 109-123
MSC (2000): Primary 11F30; Secondary 11T24, 11G20, 33C99
Published electronically: August 28, 2009
MathSciNet review: 2550175
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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 $ j$-invariant $ \frac{1728}{t}$ and values of a particular $ _2F_1$-hypergeometric function over $ {\mathbb{F}_p}$. We also give a formula for traces of Hecke operators on spaces of cusp forms of weight $ k$ 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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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: http://dx.doi.org/10.1090/S0002-9939-09-10068-0
PII: S 0002-9939(09)10068-0
Received by editor(s): June 3, 2009
Published electronically: 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
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.