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Dynamical canonical heights for Jordan blocks, arithmetic degrees of orbits, and nef canonical heights on abelian varieties


Authors: Shu Kawaguchi and Joseph H. Silverman
Journal: Trans. Amer. Math. Soc. 368 (2016), 5009-5035
MSC (2010): Primary 37P15; Secondary 11G10, 11G50, 37P30, 37P55
DOI: https://doi.org/10.1090/tran/6596
Published electronically: November 6, 2015
MathSciNet review: 3456169
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Abstract: Let $ f:X\to X$ be an endomorphism of a normal projective variety defined over a global field $ K$. We prove that for every $ x\in X(\bar {K})$, the arithmetic degree  $ \alpha _f(x)=\lim _{n\to \infty }h_X(f^n(x))^{1/n}$ of $ x$ exists, is an algebraic integer, and takes on only finitely many values as $ x$ varies over  $ X(\bar {K})$. Further, if $ X$ is an abelian variety defined over a number field, $ f$ is an isogeny, and $ x\in X(\bar {K})$ is a point whose $ f$-orbit is Zariski dense in $ X$, then  $ \alpha _f(x)$ is equal to the dynamical degree of $ f$. The proofs rely on two results of independent interest. First, if  $ D_0,D_1,\ldots \in \mathrm {Div}(X)\otimes \mathbb{C}$ form a Jordan block with eigenvalue $ \lambda $ for the action of $ f^*$ on  $ \mathrm {Pic}(X)\otimes \mathbb{C}$, then we construct associated canonical height functions  $ \hat {h}_{D_k}$ satisfying Jordan transformation formulas $ \hat {h}_{D_k}\circ f = \lambda \hat {h}_{D_k} + \hat {h}_{D_{k-1}}$. Second, if $ A/\bar {\mathbb{Q}}$ is an abelian variety and $ \hat {h}_D$ is the canonical height on $ A$ associated to a nonzero nef divisor $ D$, then there is a unique abelian subvariety $ B_D\subsetneq A$ such that $ \hat {h}_D(P)=0$ if and only if $ P\in B_D(\bar {\mathbb{Q}})+A(\bar {\mathbb{Q}} )_{\mathrm {tors}}$.


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Additional Information

Shu Kawaguchi
Affiliation: Department of Mathematics, Faculty of Science, Kyoto University, Kyoto, 606-8502, Japan
Email: kawaguch@math.kyoto-u.ac.jp

Joseph H. Silverman
Affiliation: Department of Mathematics, Box 1917, Brown University, Providence, Rhode Island 02912
Email: jhs@math.brown.edu

DOI: https://doi.org/10.1090/tran/6596
Keywords: Canonical height, arithmetic degree, nef divisor, abelian variety
Received by editor(s): February 20, 2014
Received by editor(s) in revised form: June 3, 2014
Published electronically: November 6, 2015
Additional Notes: The first author’s research was supported by KAKENHI 24740015.
The second author’s research was supported by NSF DMS-0854755 and Simons Collaboration Grant #241309.
Article copyright: © Copyright 2015 American Mathematical Society