Essential dimensions of polarized endomorphisms of abelian varieties

TL;DR

The paper investigates the essential dimension of polarized endomorphisms on abelian varieties, providing counterexamples to a conjecture and establishing conditions under which the essential dimension equals the variety's dimension.

Contribution

It offers counterexamples to Kollár and Zhuang's question and proves that under certain dynamical conditions, the essential dimension of some iterates equals the dimension.

Findings

Counterexamples to the equality of essential dimension and dimension

Conditions under which the essential dimension of an iterate equals the dimension

Affirmative answer for simple abelian surfaces with non-2-polarized endomorphisms

Abstract

Let $f$ be a polarized endomorphism of an abelian variety $A$. Koll\'ar and Zhuang asked whether the essential dimension $\mathrm{ed}(f)$ equals $\mathrm{dim}(A)$. 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 $\mathrm{ed}(f^s)=\mathrm{dim}(A)$ for some integer $s>0$. Our examples also show the necessity of both the hypothesis and iteration. We also give an affirmative answer to Koll\'ar and Zhuang's original question when $A$ is a simple abelian surface and $f$ is not $2$-polarized.