Gaussian hypergeometric evaluations of traces of Frobenius for elliptic curves

By C. Lennon. 2010

60 citations recorded by the source corpus.

We present here a formula for expressing the trace of the Frobenius endomorphism of an elliptic curve E over F q satisfying j(E) ≠ 0,1728 and q ≡ 1 (mod 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.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e752c4dbf91078fb6285d4fc76873a14629e9f6b

How researchers see this paper

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In Computational Mechanics

1. Holly Swisher
2. Fang-Ting Tu
3. Dermot McCarthy
4. Alyson Deines
5. Ling Long
6. Jenny G. Fuselier

In Computational Theory and Mathematics

1. Holly Swisher
2. Fang-Ting Tu
3. Dermot McCarthy
4. Alyson Deines
5. Ling Long
6. Jenny G. Fuselier

In Geometry and Topology

1. Dermot McCarthy
2. Ling Long
3. Holly Swisher
4. Fang-Ting Tu
5. Alyson Deines
6. Jenny G. Fuselier
7. C. Lennon

Citations in the corpus

• Hypergeometric Series, Truncated Hypergeometric Series, and Gaussian Hypergeometric Functions (2015, methodology): “Gaussian hypergeometric functions have been used to count points on different types of varieties over Fq and they are related to coefficients of various modular forms [22, 31, 15, 16, 23, 34, 6, 1].”
• The trace of Frobenius of elliptic curves and the p-adic gamma function (2012, methodology): “This allows the main result in [12], which is stated in terms of j(E) and ∆(E), to be expressed independently of the Weierstrass model of the elliptic curve.”