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

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 $V_\ell(X)$.