Essential dimensions of polarized endomorphisms of abelian varieties

Let $f$ be a polarized endomorphism of an abelian variety $A$. Kollár and Zhuang asked whether the essential dimension $\mathrm{e}\mathrm{d}(f)$ equals $\mathrm{d}\mathrm{i}\mathrm{m}(A)$. We provide counterexamples to this question. Instead, we prove that, under the hypothesis that every subtorus of $A$ is $f$-preperiodic up to translation (a condition arising from the dynamical Manin–Mumford conjecture), we have $\mathrm{e}\mathrm{d}(f^s)=\mathrm{d}\mathrm{i}\mathrm{m}(A)$ for some integer $s>0$. Our examples also show the necessity of both the hypothesis and iteration. We also give an affirmative answer to Kollár and Zhuang’s original question when $A$ is a simple abelian surface and $f$ is not $2$-polarized.