Abelian varieties are not quotients of low-dimension Jacobians

TL;DR

The paper demonstrates that for certain dimensions, many abelian varieties over algebraic closures of rationals cannot be obtained as quotients of Jacobians, highlighting limitations in the relationship between these classes of varieties.

Contribution

It establishes the existence of abelian varieties not arising as quotients of Jacobians in specific dimension ranges, using height-based counting methods.

Findings

Most abelian varieties are not quotients of Jacobians in certain dimensions.

Existence of abelian varieties not related to Jacobians for g ≥ 4.

Method applies height counting to prove generic properties.

Abstract

We prove that for any two integers $g\geq 4$ and $g'\leq 2g-1$, there exist abelian varieties over $\overline{\mathbb{Q}}$ which are not quotients of a Jacobian of dimension $g'$. Our method in fact proves that most Abelian varieties satisfy this property, when counting by height relative to a fixed finite map to projective space.