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

Let X be an abelian variety over an algebraically closed field k and f a quasi-unipotent automorphism of X. When 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 l-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 Neron-Severi space N-1(X)(Q) of Q-divisors modulo numerical equivalence; and (3) the maximum size of Jordan blocks of (the Jordan canonical form of) f*|(N)1((X)Q) in terms of the action of f on the Tate space V-l(X).