$p$-Adic hypergeometric functions and certain weight three newforms

Abstract

For an odd prime $p$ and a positive integer $n$, let ${}_n{G}_n[\cdots {]}_p$ 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 ${}_3{G}_3[\cdots {]}_p$. 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.

17 pages