Supercongruences and Complex Multiplication

Abstract

We study congruences involving truncated hypergeometric series of the form_rF_{r-1}(1/2,...,1/2;1,...,1;\lambda)_{(mp^s-1)/2} = \sum_{k=0}^{(mp^s-1)/2} ((1/2)_k/k!)^r \lambda^k where p is a prime and m, s, r are positive integers. These truncated hypergeometric series are related to the arithmetic of a family of algebraic varieties and exhibit Atkin and Swinnerton-Dyer type congruences. In particular, when r=3, they are related to K3 surfaces. For special values of \lambda, with s=1 and r=3, our congruences are stronger than what can be predicted by the theory of formal groups because of the presence of elliptic curves with complex multiplications. They generalize a conjecture made by Rodriguez-Villegas for the \lambda=1 case and confirm some other supercongruence conjectures at special values of \lambda.