Hypergeometric Motives from Euler Integral Representations

TL;DR

This work builds a geometric realization of hypergeometric motives by studying affine cyclic covers arising from Euler integral representations and constructing a partial compactification that yields a motivic decomposition. The zeta functions of fibers factor explicitly into L-series attached to nondegenerate hypergeometric motives and twists of torus zeta factors by Hecke Grossencharacters, enabling computation of Hodge numbers via known motivic data. The authors develop a robust finite-field framework, relate point counts to period-normalized hypergeometric sums, and provide explicit L-series factorization and examples for curves and surfaces. Overall, the paper advances the arithmetic and geometric understanding of hypergeometric motives beyond nondegenerate parameter regimes and offers concrete tools for computing motivic invariants from Euler-integral-based constructions.

Abstract

We revisit certain one-parameter families of affine covers arising naturally from Euler's integral representation of hypergeometric functions. We introduce a partial compactification of this family. We show that the zeta function of the fibers in the family can be written as an explicit product of $L$-series attached to nondegenerate hypergeometric motives and zeta functions of tori, twisted by Hecke Grossencharacters. This permits a combinatorial algorithm for computing the Hodge numbers of the family.

Theorems & Definitions (68)

• Theorem 1.2.4
• Definition 2.1.1
• Lemma 2.1.4
• proof
• Remark 2.1.5
• Lemma 2.2.2
• proof
• Remark 2.2.4
• Lemma 3.1.4
• proof