Supercongruences and complex multiplication

We study congruences involving truncated hypergeometric series of the formF23(1/2,1/2,1/21,1;λ)(mps−1)/2:=∑k=0(mps−1)/2((1/2)k/k!)3λk where p   is a prime and m,sm,s are positive integers. These truncated hypergeometric series are related to the arithmetic of a family of K3 surfaces. For special values of λ  , with s=1s=1, our congruences are stronger than those predicted by the theory of formal groups, because of the presence of elliptic curves with complex multiplications. They generalize a conjecture made by Stienstra and Beukers for the λ=1λ=1 case and confirm some other supercongruence conjectures at special values of λ.