Hypergeometric functions over 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

DOI:
https://doi.org/10.1090/S0002-9939-09-10068-0

Published electronically:
August 28, 2009

MathSciNet review:
2550175

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Abstract: For primes , we present an explicit relation between the traces of Frobenius on a family of elliptic curves with -invariant and values of a particular -hypergeometric function over . 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 -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:
https://doi.org/10.1090/S0002-9939-09-10068-0

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.