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Essential dimensions of polarized endomorphisms of abelian varieties

Yujie Luo, Keiji Oguiso, De-Qi Zhang

TL;DR

The paper investigates essential dimension for polarized endomorphisms of abelian varieties, showing that ed(f) need not equal the dimension in general but that some iterate f^s achieves full dimension under a dynamical Manin–Mumford-type condition (every subtorus is f-preperiodic up to translation). It proves incompressibility results for broad classes, including abelian varieties isogenous to products of elliptic curves and abelian surfaces, and provides uniform bounds on kernel ranks and iteration indices. By tying essential dimension to dynamical Manin–Mumford phenomena, the work highlights both counterexamples and situations where ed is forced to reach the ambient dimension, and it poses several natural questions for simple varieties and positive characteristic.

Abstract

Let $f$ be a polarized endomorphism of an abelian variety $A$. Kollár 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 Kollár and Zhuang's original question when $A$ is a simple abelian surface and $f$ is not $2$-polarized.

Essential dimensions of polarized endomorphisms of abelian varieties

TL;DR

The paper investigates essential dimension for polarized endomorphisms of abelian varieties, showing that ed(f) need not equal the dimension in general but that some iterate f^s achieves full dimension under a dynamical Manin–Mumford-type condition (every subtorus is f-preperiodic up to translation). It proves incompressibility results for broad classes, including abelian varieties isogenous to products of elliptic curves and abelian surfaces, and provides uniform bounds on kernel ranks and iteration indices. By tying essential dimension to dynamical Manin–Mumford phenomena, the work highlights both counterexamples and situations where ed is forced to reach the ambient dimension, and it poses several natural questions for simple varieties and positive characteristic.

Abstract

Let be a polarized endomorphism of an abelian variety . Kollár and Zhuang asked whether the essential dimension equals . We provide counterexamples to this question. Instead, we prove that, under the hypothesis that every subtorus of is -preperiodic up to translation (a condition arising from the dynamical Manin--Mumford conjecture), we have for some integer . 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 is a simple abelian surface and is not -polarized.

Paper Structure

This paper contains 14 sections, 37 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Exponents and ranks of kernels of polarized isogenies: Proof of Theorem \ref{['thm: rank main']}
  4. Essential dimensions of polarized self-isogenies
  5. Proof of Theorem \ref{['thm: ed main']} and Proposition \ref{['thm: quasi-abelian and non-uniruled']}
  6. Isogenous to products of elliptic curves case: Proof of Theorem \ref{['thm: essential dimension isogenous to product elliptic curves intro']}
  7. Abelian surfaces
  8. Counterexamples
  9. Proof of Theorem \ref{['thm: rank main uniform']}
  10. Uniform incompressible index in dimension two: Proof of Theorem \ref{['thm: simple abelian surface uniform index']}
  11. Further questions
  12. Essential dimensions of simple abelian varieties
  13. Essential dimensions of polarized endomorphisms in positive characteristic
  14. Dynamical Manin--Mumford conjecture and essential dimension

Key Result

Theorem 1.4

Let $A$ be an abelian variety, and let $f$ be a polarized endomorphism of $A$. Suppose that every subtorus of $A$ is $f$-preperiodic up to translation. Then $f^s$ is incompressible for some positive integer $s$.

Theorems & Definitions (88)

  • Remark 1.2
  • Example 1.3
  • Theorem 1.4: Theorem \ref{['thm: main with all subtori f numerically periodic']}
  • Corollary 1.5
  • Theorem 1.6: Corollary \ref{['cor: essential dimension isogenous to product elliptic curves periodic']}
  • Theorem 1.7
  • Theorem 1.8: Theorem \ref{['thm: rank main 1']}
  • Theorem 1.9
  • Proposition 1.10
  • Definition 2.1: cf. BR97
  • ...and 78 more