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

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 $n\le N$. We make use of an Arakelov inequality established by Lu and Zuo, as well as intersection theoretic methods, to prove our main results.