论文标题
Abelian品种的准独裁者自动形态的多项式体积增长（与Chen Jiang合作的附录）
Polynomial volume growth of quasi-unipotent automorphisms of abelian varieties (with an appendix in collaboration with Chen Jiang)
论文作者
论文摘要
在代数封闭的字段$ \ mathbf {k} $和$ f $ a quasi-unipotent自动形态的$ x $上，让$ x $为ABELIAN品种。当$ \ mathbf {k} $是复数的字段时，lin，oguiso和d.-q。张提供了一个明确的公式，用于通过分析论点（x，x，f）$相关的扭曲均质坐标环的gelfand-kirillov尺寸的多项式体积生长。我们给出了以任意特征起作用的该公式的代数证明。 在证明过程中，我们获得：（1）与Zarhin和Poonen-rybakov的最新结果相比，对内态对$ \ ell $ ad-adic tate空间的作用的新描述； （2）与Reichstein，Rogalski和J.J.的结果进行部分交谈。 Zhang关于内态的准独裁者及其对理性的néron-severi space $ \ mathsf {n}^1（x）_ {\ Mathbf {q}} $ $ \ 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 $V_\ell(X)$.
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 Néron--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 $V_\ell(X)$.
