HYPERGEOMETRIC FUNCTIONS OVER Fp AND RELATIONS TO ELLIPTIC CURVES AND MODULAR FORMS

For primes p equivalent 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 F-2(1)-hypergeometric function over F-p. We also give a formula for traces of Hecke operators 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.