Hypergeometric decomposition of Delsarte K3 pencils

TL;DR

This paper analyzes five families of Delsarte K3 surfaces, providing explicit formulas for point counts over finite fields, matching periods with hypergeometric functions, and decomposing their L-functions in terms of hypergeometric series.

Contribution

It offers a detailed hypergeometric decomposition of Delsarte K3 pencils, linking geometric motives with hypergeometric sums and differential operators.

Findings

Explicit point count formulas over finite fields.

Matching of periods with hypergeometric differential operators.

Decomposition of L-functions into hypergeometric and Dedekind zeta components.

Abstract

We study five pencils of projective quartic Delsarte K3 surfaces. Over finite fields, we give explicit formulas for the point counts of each family, written in terms of hypergeometric sums. Over the complex numbers, we match the periods of the corresponding family with hypergeometric differential operators and series. We also obtain a decomposition of the $L$-function of each pencil in terms of hypergeometric $L$-series and Dedekind zeta functions. This gives an explicit description of the hypergeometric motives geometrically realised by each pencil.