Bigness of Canonical Quadratic Points on Curves of Genus 4

TL;DR

This paper introduces a new approach to constructing non-torsion rational points on elliptic curves via canonical quadratic points on genus 4 curves, using a notion of bigness and modular variation.

Contribution

It defines a bigness criterion for sections of abelian schemes and applies it to produce non-torsion points on elliptic curves from genus 4 curves.

Findings

Proves the bigness of the canonical quadratic point on certain loci.

Obtains non-torsion rational points on elliptic curves from genus 4 curves.

Establishes Northcott-type finiteness results for these points.

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.