A complete proof of a supercongruence conjecture of Rodriguez-Villegas

For an odd prime p and a positive integer n, let ₙGₙₚ 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 modular forms of weight three in terms of special values of ₃G₃ₚ. As a consequence, we prove a conjecture of Rodriguez-Villegas which was earlier proved by Mortenson for certain particular cases.