Arithmetic rank bounds for abelian varieties over function fields

TL;DR

This paper refines bounds on the rank of abelian varieties over function fields using descent techniques, extending previous results from elliptic curves to higher-dimensional abelian varieties and providing explicit maps for Jacobians of hyperelliptic curves.

Contribution

It introduces an arithmetic refinement of rank bounds for abelian varieties over function fields, generalizes descent methods to higher dimensions, and constructs explicit 2-descent maps for Jacobians of hyperelliptic curves.

Findings

Derived a refined rank bound depending on genus and bad reduction data.

Extended descent techniques from elliptic curves to abelian varieties.

Constructed explicit 2-descent maps for Jacobians of hyperelliptic curves.

Abstract

It follows from the Grothendieck-Ogg-Shafarevich formula that the rank of an abelian variety (with trivial trace) defined over the function field of a curve is bounded by a quantity which depends on the genus of the base curve and on bad reduction data. Using a function field version of classical $\ell$-descent techniques, we derive an arithmetic refinement of this bound, extending previous work of the second and third authors from elliptic curves to abelian varieties, and improving on their result in the case of elliptic curves. When the abelian variety is the Jacobian of a hyperelliptic curve, we produce a more explicit $2$-descent map. Then we apply this machinery to studying points on the Jacobians of certain genus $2$ curves over $k(t)$, where $k$ is some perfect base field of characteristic not $2$.