Hypergeometric decomposition of symmetric K3 quartic pencils

TL;DR

This paper investigates hypergeometric functions related to deformations of K3 quartic hypersurfaces, deriving their differential equations, counting points over finite fields, and describing their motives explicitly in terms of hypergeometric motives.

Contribution

It provides a complete, explicit description of the motives for these K3 quartic pencils using hypergeometric motives, linking differential equations, point counts, and zeta functions.

Findings

Computed Picard--Fuchs differential equations for all deformations.

Expressed point counts using Gauss sums and finite field hypergeometric sums.

Matched differential equations to factors of the zeta function and related to global L-functions.

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.