English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Open

Presents the pdf analysis as premium feature

PDF Analysis

Insights

Presents the summarisation as premium feature

Summarisation

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

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   $V_{\ell }(X)$  .

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