Hypergeometric Motives from Euler Integral Representations

TL;DR

This paper explores hypergeometric motives derived from Euler integral representations, providing explicit formulas for zeta functions and an algorithm to compute Hodge numbers of related families.

Contribution

It introduces a partial compactification of affine covers from Euler integrals and expresses fiber zeta functions via hypergeometric motives and L-series.

Findings

Zeta functions expressed as products of L-series and zeta functions of tori

Algorithm for computing Hodge numbers of the family

Connection between Euler integrals and hypergeometric motives

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.