Kummers, spinors, and heights
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908442188316672 |
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| author | Laga, Jef Thorne, Jack A. |
| author_facet | Laga, Jef Thorne, Jack A. |
| contents | Let $f(x) = x^{2g+1} + c_1 x^{2g} + \dots + c_{2g+1} \in k[x]$ be a polynomial of nonzero discriminant, and let $J$ denote the Jacobian of the odd hyperelliptic curve $C : y^2 = f(x)$. We show that the morphism $J \to \mathbb{P}^{2^g-1}$ associated to the linear system $|2 Θ|$ may be described explicitly, for any $g \geq 1$, using the theory of pure spinors.
We apply this theory to study the heights of rational points in $J(k)$, when $k$ is a number field. As a particular consequence, we show that $100\%$ of monic, degree $2g+1$ polynomials $f(x) \in \mathbb{Z}[x]$ of nonzero discriminant $Δ(f)$ have the property that, for any non-trivial point $P \in J(\mathbb{Q})$, the canonical height of $P$ satisfies $ \widehat{h}_Θ(P) \geq \left(\frac{3g-1}{4g(2g+1)} - ε\right) \log | Δ(f) |$. This is a `density 1' form of the Lang--Silverman conjecture. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_06865 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Kummers, spinors, and heights Laga, Jef Thorne, Jack A. Number Theory Algebraic Geometry Let $f(x) = x^{2g+1} + c_1 x^{2g} + \dots + c_{2g+1} \in k[x]$ be a polynomial of nonzero discriminant, and let $J$ denote the Jacobian of the odd hyperelliptic curve $C : y^2 = f(x)$. We show that the morphism $J \to \mathbb{P}^{2^g-1}$ associated to the linear system $|2 Θ|$ may be described explicitly, for any $g \geq 1$, using the theory of pure spinors. We apply this theory to study the heights of rational points in $J(k)$, when $k$ is a number field. As a particular consequence, we show that $100\%$ of monic, degree $2g+1$ polynomials $f(x) \in \mathbb{Z}[x]$ of nonzero discriminant $Δ(f)$ have the property that, for any non-trivial point $P \in J(\mathbb{Q})$, the canonical height of $P$ satisfies $ \widehat{h}_Θ(P) \geq \left(\frac{3g-1}{4g(2g+1)} - ε\right) \log | Δ(f) |$. This is a `density 1' form of the Lang--Silverman conjecture. |
| title | Kummers, spinors, and heights |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2507.06865 |