Bigness of Canonical Quadratic Points on Curves of Genus 4

A central problem in arithmetic geometry is to construct non-torsion rational points on elliptic curves. We study a canonical quadratic point ${\xi }_C\in \mathrm{J}\mathrm{a}\mathrm{c}(C)$ 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.