Hypergeometric decomposition of symmetric K3 quartic pencils

Charles F. Doran, Tyler L. Kelly, Adriana Salerno, Steven Sperber, John Voight, Ursula Whitcher

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Abstract

We study the hypergeometric functions associated to five one-parameter deformations of Delsarte K3 quartic hypersurfaces in projective space. We compute all of their Picard–Fuchs differential equations; we count points using Gauss sums and rewrite this in terms of finite-field hypergeometric sums; then we match up each differential equation to a factor of the zeta function, and we write this in terms of global L-functions. This computation gives a complete, explicit description of the motives for these pencils in terms of hypergeometric motives.

Original languageEnglish (US)
Article number7
JournalResearch in Mathematical Sciences
Volume7
Issue number2
DOIs
StatePublished - Jun 1 2020

Bibliographical note

Funding Information:
The authors heartily thank Xenia de la Ossa for her input and many discussions about this project. They also thank Simon Judes for sharing his expertise, Frits Beukers, David Roberts, Fernando Rodríguez-Villegas, and Mark Watkins for numerous helpful discussions, Edgar Costa for sharing his code for computing zeta functions, and the anonymous referee for helpful corrections and comments. The authors would like to thank the American Institute of Mathematics (AIM) and its SQuaRE program, the Banff International Research Station, SageMath, and the MATRIX Institute for facilitating their work together. Doran acknowledges support from NSERC and the hospitality of the ICERM at Brown University and the CMSA at Harvard University. 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) and a Simons Collaboration Grant (550029).

Funding Information:
The authors heartily thank Xenia de la Ossa for her input and many discussions about this project. They also thank Simon Judes for sharing his expertise, Frits Beukers, David Roberts, Fernando Rodr?guez-Villegas, and Mark Watkins for numerous helpful discussions, Edgar Costa for sharing his code for computing zeta functions, and the anonymous referee for helpful corrections and comments. The authors would like to thank the American Institute of Mathematics (AIM) and its SQuaRE program, the Banff International Research Station, SageMath, and the MATRIX Institute for facilitating their work together. Doran acknowledges support from NSERC and the hospitality of the ICERM at Brown University and the CMSA at Harvard University. 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) and a Simons Collaboration Grant (550029).

Publisher Copyright:
© 2020, The Author(s).

Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

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