The periods of Calabi-Yau varieties realized as hypersurfaces in Gorenstein Fano toric varieties satisfy GKZ A-hypergeometric equations. We show that the same A-hypergeometric data yields point counts on Calabi-Yau varieties over finite fields, using a notion of finite field A-hypergeometric sums defined by Adolphson. This talk describes joint work with Thais Gomes Ribeiro, Tyler Kelly, Adriana Salerno, and John Voight.