Polynomial Volume Growth of Quasi-Unipotent Automorphisms of Abelian Varieties (with an Appendix in Collaboration with Chen Jiang)

Abstract

Abstract Let $X$ be an abelian variety over an algebraically closed field $\textbf{k}$ and $f$ a quasi-unipotent automorphism of $X$. When $\textbf{k}$ 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 the following: (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 Néron–Severi space $\textsf{N}^{1}(X)_{\textbf{Q}}$ of $\textbf{Q}$-divisors modulo numerical equivalence; and (3) the maximum size of Jordan blocks of (the Jordan canonical form of) $f^{*}|_{\textsf{N}^{1}(X)_{\textbf{Q}}}$ in terms of the action of $f$ on the Tate space $V_{\ell }(X)$.