We study congruences involving truncated hypergeometric series of the form
where
p is a prime and
m,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
e2f4" title="Click to view the MathML source">s=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=1 case and confirm some other supercongruence conjectures at special values of
λ.