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Gaussian hypergeometric evaluations of traces of Frobenius for elliptic curves

Author: Catherine Lennon
Journal: Proc. Amer. Math. Soc. 139 (2011), 1931-1938
MSC (2010): Primary 11T24, 11G20
Published electronically: November 3, 2010
MathSciNet review: 2775369
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Abstract: We present here a formula for expressing the trace of the Frobenius endomorphism of an elliptic curve $ E$ over $ \mathbb{F}_q$ satisfying $ j(E)\neq 0, 1728$ and $ q\equiv 1 \pmod{12}$ in terms of special values of Gaussian hypergeometric series. This paper uses methods introduced by Fuselier for one-parameter families of curves to express the trace of Frobenius of $ E$ as a function of its $ j$-invariant and discriminant instead of a parameter, which are more intrinsic characteristics of the curve.

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Catherine Lennon
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139

Received by editor(s): March 22, 2010
Received by editor(s) in revised form: May 28, 2010
Published electronically: November 3, 2010
Additional Notes: This work was supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program.
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.