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arXiv · 2605.11888

Bigness of Canonical Quadratic Points on Curves of Genus 4

Jiahui Gao
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Abstract

A central problem in arithmetic geometry is to construct non-torsion rational points on elliptic curves. We study a canonical quadratic point $\xi_C \in {\rm Jac}(C)$ attached to a smooth non-hyperelliptic curve of genus 4 and use it to produce such points on elliptic curves arising from families of genus $4$ curves. We introduce a notion of bigness for sections of abelian schemes and establish a criterion in terms of modular variation of abelian quotients, using adelic line bundles and Betti maps. As applications, we prove that $\xi_C$ is big on the triple-involution locus and on certain CM families, obtaining in particular non-torsion rational points on the associated elliptic curves and Northcott-type finiteness results.

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Jiahui Gao. 2026-05-12. Bigness of Canonical Quadratic Points on Curves of Genus 4. https://arxiv.org/abs/2605.11888

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