Asem Abdelraouf | Searcharxiv

arXiv2026

Geometric realisation of hypergeometric local systems

We prove a realisation theorem for irreducible hypergeometric local systems defined over the rational numbers in terms of families of affine varieties in algebraic tori. The families we consider have been studied extensively in the literature and appear in mirror symmetry. Our result holds unconditionally for families with one-dimensional or even-dimensional fibres. It holds under a monodromy assumption for families with fibres of odd dimension greater than one.

math.AG↗

arXiv2025

Hypergeometric Motives from Toric Hypersurfaces

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 $\mathbb{Q}$.

math.NT↗