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 $\mathbb{Q}$.