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
DOI:
https://doi.org/10.1090/S0002-9939-09-10068-0
Published electronically:
August 28, 2009
MathSciNet review:
2550175
Full-text PDF Free Access
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 $j$-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
MR Author ID:
882190
Email:
jenny.fuselier@usma.edu, jfuselie@highpoint.edu
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.
