P-Adic hypergeometric functions and certain weight three newforms

Abstract

For an odd prime p and a positive integer n, let Gnn[⋯]p denote McCarthy's p-adic hypergeometric function. In this article, we prove p-adic analogue of certain classical hypergeometric identities and using these identities we express the p-th Fourier coefficient of certain weight three newforms in terms of special values of G33[⋯]p. Rodriguez-Villegas conjectured certain supercongruences between values of truncated hypergeometric series and the p-th Fourier coefficients of these newforms. As a consequence of our main results, we obtain another proof of these supercongruences which were earlier proved by Mortenson and Sun.