Hypergeometric Motives from Toric Hypersurfaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908552331788288 |
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| author | Abdelraouf, Asem |
| author_facet | Abdelraouf, Asem |
| contents | 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 $\mathbb{Q}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_17624 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hypergeometric Motives from Toric Hypersurfaces Abdelraouf, Asem Number Theory Algebraic Geometry 11G25 (Primary) 11S40, 14G10, 11T24, 14M25, 14C15 (Secondary) 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 $\mathbb{Q}$. |
| title | Hypergeometric Motives from Toric Hypersurfaces |
| topic | Number Theory Algebraic Geometry 11G25 (Primary) 11S40, 14G10, 11T24, 14M25, 14C15 (Secondary) |
| url | https://arxiv.org/abs/2509.17624 |