Arithmetic and Dynamical Degrees on Abelian Varieties

Silverman, Joseph H.

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

Soit $\phi :X⤏X$ une application rationnelle dominante d’une variété lisse et soit $x\in X$, tous deux définis sur $\overline{\mathbb{Q}}$. Le degré dynamique $\delta \left(\phi \right)$ mesure la complexité géométrique des itérations de $\phi$, tandis que le degré arithmétique $\alpha (\phi ,x)$ mesure la complexité arithmétique de la $\phi$-orbite de $x$. Il est connu que $\alpha (\phi ,x)\le \delta \left(\phi \right)$, et il est conjecturé que si la $\phi$-orbite de $x$ est Zariski dense dans $X$, alors $\alpha (\phi ,x)=\delta \left(\phi \right)$. 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.

Let $\phi :X⤏X$ be a dominant rational map of a smooth variety and let $x\in X$, all defined over $\overline{\mathbb{Q}}$. The dynamical degree $\delta \left(\phi \right)$ measures the geometric complexity of the iterates of $\phi$, and the arithmetic degree $\alpha (\phi ,x)$ measures the arithmetic complexity of the forward $\phi$-orbit of $x$. It is known that $\alpha (\phi ,x)\le \delta \left(\phi \right)$, and it is conjectured that if the $\phi$-orbit of $x$ is Zariski dense in $X$, then $\alpha (\phi ,x)=\delta \left(\phi \right)$, 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.

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

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