Hypergeometric functions over F_p and relations to elliptic curves and modular forms

2008/05/19 by Jenny G. Fuselier, Fuselier, Jenny G.
Mathematics · #11F30 (Primary) 11T24 #11G20 #33C99 (Secondary) #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.0805.2885

openalex publication_date 2008/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For primes p congruent to 1 mod 12, we present an explicit relation between the traces of Frobenius on a family of elliptic curves with j-invariant 1728/t and values of a particular 2F1-hypergeometric function over F_p. Additionally, we determine a formula for traces of Hecke operators T_k(p) 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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