2012/10/16 by Jonas Kibelbek, Ling Long, Kibelbek, Jonas +7
Mathematics · #11G07 #11G15 #33C20 #44A20 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11G07 #msc:11G15 #msc:33C20 #msc:44A20
paper · pdf · doi:10.48550/arxiv.1210.4489
19 pages
arxiv created 2012/11/20 · arxiv updated 2012/11/21
We study congruences involving truncated hypergeometric series of the formrFr-1(1/2,...,1/2;1,...,1;λ)(mps-1)/2 = ∑k=0(mps-1)/2 ((1/2)k/k!)r λk 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 λ, 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 λ=1 case and confirm some other supercongruence conjectures at special values of λ.