Powers of abelian varieties over $\overline{\mathbb Q(t)}$ not isogenous to a Jacobian

TL;DR

This paper demonstrates the existence of abelian varieties over algebraic closures of rational function fields that are not isogenous to Jacobians, even when considering their powers, using advanced intersection theory and Arakelov inequalities.

Contribution

It establishes the existence of such abelian varieties with maximal monodromy and no power isogenous to a Jacobian, extending previous understanding in the field.

Findings

Existence of abelian varieties over (t) not isogenous to Jacobians

Construction of varieties with maximal monodromy and non-Jacobian powers

Application of Arakelov inequality and intersection theory

Abstract

We prove the existence of abelian varieties over $\overline{\mathbb Q(t)}$ with no power isogenous to a Jacobian. Moreover, given a positive integer $N$, we prove the existence of abelian varieties over $\overline{\mathbb Q(t)}$ with maximal monodromy such that the $n$th power is not isogenous to a Jacobian for $n \leq N$. We make use of an Arakelov inequality established by Lu and Zuo, as well as intersection theoretic methods, to prove our main results.