Polynomial volume growth of quasi-unipotent automorphisms of abelian varieties (with an appendix in collaboration with Chen Jiang)

TL;DR

This paper provides an algebraic proof for the polynomial volume growth formula of quasi-unipotent automorphisms on abelian varieties, extending previous complex-analytic results to arbitrary characteristic and exploring their actions on Tate spaces.

Contribution

It offers a new algebraic proof of the polynomial volume growth formula, applicable in any characteristic, and introduces new insights into the action of endomorphisms on Tate spaces and Néron–Severi groups.

Findings

Algebraic proof of volume growth formula in arbitrary characteristic.

New description of endomorphism actions on Tate spaces.

Bounds on Jordan block sizes for automorphisms.

Abstract

Let $X$ be an abelian variety over an algebraically closed field $\mathbf{k}$ and $f$ a quasi-unipotent automorphism of $X$. When $\mathbf{k}$ is the field of complex numbers, Lin, Oguiso, and D.-Q. Zhang provide an explicit formula for the polynomial volume growth of (or equivalently, for the Gelfand--Kirillov dimension of the twisted homogeneous coordinate ring associated with) the pair $(X, f)$, by an analytic argument. We give an algebraic proof of this formula that works in arbitrary characteristic. In the course of the proof, we obtain: (1) a new description of the action of endomorphisms on the $\ell$-adic Tate spaces, in comparison with recent results of Zarhin and Poonen--Rybakov; (2) a partial converse to a result of Reichstein, Rogalski, and J.J. Zhang on quasi-unipotency of endomorphisms and their pullback action on the rational N\'eron--Severi space $\mathsf{N}^1(X)_{\mathbf{Q}}$ of $\mathbf{Q}$-divisors modulo numerical equivalence; (3) the maximum size of Jordan blocks of (the Jordan canonical form of) $f^*|_{\mathsf{N}^1(X)_{\mathbf{Q}}}$ in terms of the action of $f$ on the Tate space $V_\ell(X)$.