Dynamical canonical heights for Jordan blocks, arithmetic degrees of orbits, and nef canonical heights on abelian varieties
HTML articles powered by AMS MathViewer
- by Shu Kawaguchi and Joseph H. Silverman PDF
- Trans. Amer. Math. Soc. 368 (2016), 5009-5035 Request permission
Corrigendum: Trans. Amer. Math. Soc. 373 (2020), 2253-2253.
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 $P\in B_D(\bar {\mathbb {Q}})+A(\bar {\mathbb {Q}} )_{\mathrm {tors}}$.References
- Sheldon Axler, Linear algebra done right, 2nd ed., Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1997. MR 1482226, DOI 10.1007/b97662
- Arthur Baragar, Canonical vector heights on algebraic $K3$ surfaces with Picard number two, Canad. Math. Bull. 46 (2003), no. 4, 495–508. MR 2011389, DOI 10.4153/CMB-2003-048-x
- Garrett Birkhoff, Linear transformations with invariant cones, Amer. Math. Monthly 74 (1967), 274–276. MR 214605, DOI 10.2307/2316020
- Gregory S. Call and Joseph H. Silverman, Canonical heights on varieties with morphisms, Compositio Math. 89 (1993), no. 2, 163–205. MR 1255693
- Marc Hindry and Joseph H. Silverman, Diophantine geometry, Graduate Texts in Mathematics, vol. 201, Springer-Verlag, New York, 2000. An introduction. MR 1745599, DOI 10.1007/978-1-4612-1210-2
- S. Kawaguchi and J. H. Silverman, On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties, J. reine angew Math., to appear.
- Shu Kawaguchi and Joseph H. Silverman, Examples of dynamical degree equals arithmetic degree, Michigan Math. J. 63 (2014), no. 1, 41–63. MR 3189467, DOI 10.1307/mmj/1395234358
- Serge Lang, Fundamentals of Diophantine geometry, Springer-Verlag, New York, 1983. MR 715605, DOI 10.1007/978-1-4757-1810-2
- Terry A. Loring, Factorization of matrices of quaternions, Expo. Math. 30 (2012), no. 3, 250–267. MR 2990115, DOI 10.1016/j.exmath.2012.08.006
- David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1970. MR 0282985
- Joseph H. Silverman, Dynamical degree, arithmetic entropy, and canonical heights for dominant rational self-maps of projective space, Ergodic Theory Dynam. Systems 34 (2014), no. 2, 647–678. MR 3233709, DOI 10.1017/etds.2012.144
Additional Information
- Shu Kawaguchi
- Affiliation: Department of Mathematics, Faculty of Science, Kyoto University, Kyoto, 606-8502, Japan
- MR Author ID: 655244
- Email: kawaguch@math.kyoto-u.ac.jp
- Joseph H. Silverman
- Affiliation: Department of Mathematics, Box 1917, Brown University, Providence, Rhode Island 02912
- MR Author ID: 162205
- ORCID: 0000-0003-3887-3248
- Email: jhs@math.brown.edu
- 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. - © Copyright 2015 American Mathematical Society
- 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
- MathSciNet review: 3456169