Arithmetic and Dynamical Degrees on Abelian Varieties

Joseph H. Silverman

doi:10.5802/jtnb.973

Joseph H. Silverman ¹

¹ Mathematics Department, Box 1917 Brown University, Providence, RI 02912, USA

Journal de théorie des nombres de Bordeaux, Tome 29 (2017) no. 1, pp. 151-167.

Abstract (VO)
Résumé

Let $φ : X ⤏ X$ be a dominant rational map of a smooth variety and let $x \in X$ , all defined over $\bar{ℚ}$ . The dynamical degree $δ (φ)$ measures the geometric complexity of the iterates of $φ$ , and the arithmetic degree $α (φ, x)$ measures the arithmetic complexity of the forward $φ$ -orbit of $x$ . It is known that $α (φ, x) \leq δ (φ)$ , and it is conjectured that if the $φ$ -orbit of $x$ is Zariski dense in $X$ , then $α (φ, x) = δ (φ)$ , i.e. arithmetic complexity equals geometric complexity. In this note we prove this conjecture in the case that $X$ is an abelian variety, extending earlier work in which the conjecture was proven for isogenies.

Soit $φ : X ⤏ X$ une application rationnelle dominante d’une variété lisse et soit $x \in X$ , tous deux définis sur $\bar{ℚ}$ . Le degré dynamique $δ (φ)$ mesure la complexité géométrique des itérations de $φ$ , tandis que le degré arithmétique $α (φ, x)$ mesure la complexité arithmétique de la $φ$ -orbite de $x$ . Il est connu que $α (φ, x) \leq δ (φ)$ , et il est conjecturé que si la $φ$ -orbite de $x$ est Zariski dense dans $X$ , alors $α (φ, x) = δ (φ)$ . Dans cette note, nous prouvons cette conjecture dans le cas où $X$ est une variété abélienne, étendant des travaux antérieurs où la conjecture a été prouvée pour les isogénies.

Reçu le : 2015-03-01
Révisé le : 2015-06-30
Accepté le : 2015-07-06
Publié le : 2017-01-26

DOI : 10.5802/jtnb.973

Classification : 37P30, 11G10, 11G50, 37P15
Keywords: dynamical degree, arithmetic degree, abelian variety

Affiliations des auteurs :

Joseph H. Silverman ¹

¹ Mathematics Department, Box 1917 Brown University, Providence, RI 02912, USA

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{JTNB_2017__29_1_151_0,
     author = {Joseph H. Silverman},
     title = {Arithmetic and {Dynamical} {Degrees} on {Abelian} {Varieties}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {151--167},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {29},
     number = {1},
     year = {2017},
     doi = {10.5802/jtnb.973},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.973/}
}

TY  - JOUR
AU  - Joseph H. Silverman
TI  - Arithmetic and Dynamical Degrees on Abelian Varieties
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2017
SP  - 151
EP  - 167
VL  - 29
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.973/
DO  - 10.5802/jtnb.973
LA  - en
ID  - JTNB_2017__29_1_151_0
ER  -

%0 Journal Article
%A Joseph H. Silverman
%T Arithmetic and Dynamical Degrees on Abelian Varieties
%J Journal de théorie des nombres de Bordeaux
%D 2017
%P 151-167
%V 29
%N 1
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.973/
%R 10.5802/jtnb.973
%G en
%F JTNB_2017__29_1_151_0

Joseph H. Silverman. Arithmetic and Dynamical Degrees on Abelian Varieties. Journal de théorie des nombres de Bordeaux, Tome 29 (2017) no. 1, pp. 151-167. doi : 10.5802/jtnb.973. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.973/

Bibliographie
Cité par

[1] Marc P. Bellon; Claude Michel Viallet Algebraic entropy, Comm. Math. Phys., Volume 204 (199) no. 2, pp. 425-437 | DOI

[2] Tien-Cuong Dinh; Viêt-Anh Nguyên Comparison of dynamical degrees for semi-conjugate meromorphic maps, Comment. Math. Helv., Volume 86 (2011) no. 4, pp. 817-840 | DOI

[3] Tien-Cuong Dinh; Viêt-Anh Nguyên; Tuyen Trung Truong On the dynamical degrees of meromorphic maps preserving a fibration, Commun. Contemp. Math., Volume 14 (2012) no. 6, 1250042, 18 pages | DOI

[4] Dragos Ghioca; Thomas Scanlon Density of orbits of endomorphisms of abelian varieties (2014) (http://arxiv.org/abs/1412.2029)

[5] Vincent Guedj Ergodic properties of rational mappings with large topological degree, Ann. Math., Volume 161 (2055) no. 3, pp. 1589-1607 | DOI

[6] Robin Hartshorne Algebraic Geometry, Graduate Texts in Mathematics, 52, Springer-Verlag, New York, 1977, xvi+496 pages

[7] Marc Hindry; Joseph H. Silverman Diophantine Geometry: An Introduction, Graduate Texts in Mathematics, 201, Springer-Verlag, New York, 2000, xiii+558 pages

[8] Shu Kawaguchi; Joseph H. Silverman Examples of dynamical degree equals arithmetic degree, Michigan Math. J., Volume 63 (2014) no. 1, pp. 41-63 | DOI

[9] Shu Kawaguchi; Joseph H. Silverman Dynamical canonical heights for Jordan blocks, arithmetic degrees of orbits, and nef canonical heights on abelian varieties, Trans. Amer. Math. Soc., Volume 368 (2016) no. 7, pp. 5009-5035 | DOI

[10] Shu Kawaguchi; Joseph H. Silverman On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties, J. Reine Angew. Math., Volume 713 (2016), pp. 21-48

[11] Serge Lang Fundamentals of Diophantine Geometry, Springer-Verlag, New York, 1983, xviii+370 pages

[12] David Mumford Abelian Varieties, Tata Institute of Fundamental Research Studies in Mathematics, 5, London: Oxford University Press, 1970, viii+242 pages

[13] Joseph H. Silverman Dynamical degree, arithmetic entropy, and canonical heights for dominant rational self-maps of projective space, Ergodic Theory Dynam. Systems, Volume 34 (2014) no. 2, pp. 647-678 | DOI

Cité par Sources :