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arXiv · 2508.15049

Hypergeometric decomposition of Delsarte K3 pencils

Rachel Davis·Jessamyn Dukes·Thais Gomes Ribeiro·Eli Orvis·Adriana Salerno·Leah Sturman·Ursula Whitcher
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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.

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Rachel Davis, Jessamyn Dukes, Thais Gomes Ribeiro, Eli Orvis, Adriana Salerno, Leah Sturman, Ursula Whitcher. 2025-08-20. Hypergeometric decomposition of Delsarte K3 pencils. https://arxiv.org/abs/2508.15049

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