Counting quadratic points on Fano varieties
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arXiv
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| Hauptverfasser: | , , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908398534000640 |
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| author | Balestrieri, Francesca Destagnol, Kevin Lyczak, Julian Park, Jennifer Rome, Nick |
| author_facet | Balestrieri, Francesca Destagnol, Kevin Lyczak, Julian Park, Jennifer Rome, Nick |
| contents | This paper initiates the systematic study of the number of points of bounded height on symmetric squares of weak Fano varieties. We provide a general framework for establishing the point count on $\text{Sym}^2 X$. In the specific case of surfaces, we relate this to the Manin--Peyre conjecture for $\text{Hilb}^2 X$, and prove the conjecture for an infinite family of non-split quadric surfaces. In order to achieve the predicted asymptotic, we show that a type II thin set of a new flavour must be removed.
To establish our counting result for the specific family of surfaces, we generalise existing lattice point counting techniques to lattices defined over rings of integers. This reduces the dimension of the problem and yields improved error terms. Another key tool we develop is a collection of results for summing Euler products over quadratic extensions. We use this to show moments of $L$-functions at $s=1$ are constant on average in quadratic twist families. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_17940 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counting quadratic points on Fano varieties Balestrieri, Francesca Destagnol, Kevin Lyczak, Julian Park, Jennifer Rome, Nick Number Theory Algebraic Geometry This paper initiates the systematic study of the number of points of bounded height on symmetric squares of weak Fano varieties. We provide a general framework for establishing the point count on $\text{Sym}^2 X$. In the specific case of surfaces, we relate this to the Manin--Peyre conjecture for $\text{Hilb}^2 X$, and prove the conjecture for an infinite family of non-split quadric surfaces. In order to achieve the predicted asymptotic, we show that a type II thin set of a new flavour must be removed. To establish our counting result for the specific family of surfaces, we generalise existing lattice point counting techniques to lattices defined over rings of integers. This reduces the dimension of the problem and yields improved error terms. Another key tool we develop is a collection of results for summing Euler products over quadratic extensions. We use this to show moments of $L$-functions at $s=1$ are constant on average in quadratic twist families. |
| title | Counting quadratic points on Fano varieties |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2505.17940 |