We prove that if two Calabi–Yau invertible pencils have the same dual weights, then they share a common factor in their zeta functions. By using Dwork cohomology, we demonstrate that this common factor is related to a hypergeometric Picard–Fuchs differential equation. The factor in the zeta function is defined over the rationals and has degree at least the order of the Picard–Fuchs equation. As an application, we relate several pencils of K3 surfaces to the Dwork pencil, obtaining new cases of arithmetic mirror symmetry.
Bibliographical noteFunding Information:
Acknowledgements. The authors heartily thank Marco Aldi, Amanda Francis, Xenia de la Ossa, Andrija Peruniˇcić, and Noriko Yui for many interesting discussions, as well as Alan Adolphson, Remke Kloosterman, Yang Liping, Fer-nando Rodriguez–Villegas, Duco van Straten, and the anonymous referee for helpful comments. They thank the American Institute of Mathematics and its SQuaRE program, the Banff International Research Station, the Clay Mathematics Institute, MATRIX in Australia, and SageMath for facilitating their work together. Doran was supported by NSERC and the Campobassi Professorship at the University of Maryland. Kelly acknowledges that this material is based upon work supported by the NSF under Award No. DMS-1401446 and the EPSRC under EP/N004922/1. Voight was supported by an NSF CAREER Award (DMS-1151047).
© 2018, The Hebrew University of Jerusalem.