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arXiv · 2509.17624

Hypergeometric Motives from Toric Hypersurfaces

Asem Abdelraouf
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Abstract

In this paper, we study two compactifications of general hypersurfaces defined by the vanishing of linear combinations of $d+2$ monomials in $d$-dimensional algebraic tori. We prove that the number of their $\mathbb{F}_q$-points is given by finite hypergeometric sums under certain general conditions. In the process, we introduce the notion of a gamma triple, which allows us to extend the classical definition of finite hypergeometric sums to prime powers corresponding to the cyclotomic field of definition of the associated monodromy representation. As a special case of our main results, we study the Dwork family and obtain a formula for the number of its $\mathbb{F}_q$-points. Our results generalise the work of Beukers, Cohen and Mellit for finite hypergeometric sums defined over $\mathbb{Q}$.

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Asem Abdelraouf. 2025-09-22. Hypergeometric Motives from Toric Hypersurfaces. https://arxiv.org/abs/2509.17624

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