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- Title
Polynomial Volume Growth of Quasi-Unipotent Automorphisms of Abelian Varieties (with an Appendix in Collaboration with Chen Jiang).
- Authors
Hu, Fei
- 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)$|.
- Subjects
JORDAN; POLYNOMIALS; AUTOMORPHISMS; ABELIAN varieties; ENDOMORPHISMS; COMPLEX numbers
- Publication
IMRN: International Mathematics Research Notices, 2024, Vol 2024, Issue 8, p6374
- ISSN
1073-7928
- Publication type
Article
- DOI
10.1093/imrn/rnad170