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

TL;DR

This paper establishes explicit relations between hypergeometric functions over finite fields, elliptic curve Frobenius traces, and modular forms, providing new formulas connecting these areas and applications to Ramanujan's tau-function.

Contribution

It introduces explicit formulas linking hypergeometric functions over F_p with elliptic curve traces and Hecke operator traces, advancing understanding of their interrelations.

Findings

Explicit relation between Frobenius traces and hypergeometric functions for p ≡ 1 mod 12

Formulas for Hecke operator traces in terms of Frobenius traces

New expressions for Ramanujan's tau-function using hypergeometric functions

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.