Gaussian hypergeometric evaluations of traces of Frobenius for elliptic curves
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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.References
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Additional Information
- Catherine Lennon
- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachu- setts Avenue, Cambridge, Massachusetts 02139
- Email: clennon@math.mit.edu
- 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
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 1931-1938
- MSC (2010): Primary 11T24, 11G20
- DOI: https://doi.org/10.1090/S0002-9939-2010-10609-3
- MathSciNet review: 2775369