Supercongruences and complex multiplication

Ling Long, Jonas Kibelbek, Kevin Moss, Benjamin Sheller, Hao Yuan

Abstract:

We study congruences involving truncated hypergeometric se-ries of the form
3F2( 1/2, 1/2, 1/2
1, 1 ; λ)(mps−1)/2 :=
(mps−1)/2
k=0
((1/2)k/k!)3λk
where pis a prime and m, sare positive integers. These trun-cated hypergeometric series are related to the arithmetic of a family of K3 surfaces. For special values of λ, with 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 =1case and confirm some other supercongruence conjectures at special values ofλ.