Hypergeometric decomposition of Delsarte K3 pencils

TL;DR

This work studies five invertible quartic K3 pencils $X_{A,\psi}$ (Delsarte pencils) and develops a framework tying finite-field point counts to complex periods and to arithmetic $L$-functions via hypergeometric motives. Over $\mathbb{F}_q$, point counts are expressed through finite-field hypergeometric sums; over $\mathbb{C}$, holomorphic periods satisfy hypergeometric Picard-Fuchs equations with parameters extracted from dual weights, while the Adolphson–Sperber method yields the remaining hypergeometric periods via a gamma-triple construction. The authors prove a main decomposition: the $L$-function of each pencil factors into hypergeometric $L$-functions and Dedekind zeta factors, corresponding to a holomorphic hypergeometric piece plus additional hypergeometric and Dirichlet components aligned with Kloosterman-type cohomology decompositions. This provides explicit, computable realizations of hypergeometric motives for these K3 families and strengthens the bridge between hypergeometric functions, PF theory, and arithmetic geometry of Calabi–Yau-type varieties.

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.

Theorems & Definitions (64)

• Theorem 2.1: KS92
• Proposition 3.1
• Example 3.2
• Theorem 3.3: G11
• Remark 3.4
• Proposition 3.5
• Remark 3.6
• Theorem 3.7: AS23
• Remark 3.8
• Proposition 3.9