Arithmetic and Dynamical Degrees on Abelian Varieties

TL;DR

This paper proves that for abelian varieties, the arithmetic degree of a point equals the dynamical degree of a rational map if the orbit is Zariski dense, confirming a key conjecture in arithmetic dynamics.

Contribution

The paper extends the proof of the equality of arithmetic and dynamical degrees to all rational maps on abelian varieties, beyond isogenies.

Findings

Proves the conjecture for all rational maps on abelian varieties.

Confirms that arithmetic and geometric complexities coincide for dense orbits.

Extends previous results from isogenies to general rational maps.

Abstract

Let $\phi:X\dashrightarrow X$ be a dominant rational map of a smooth variety and let $x\in X$, all defined over $\bar{\mathbb Q}$. The dynamical degree $\delta(\phi)$ 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 $\alpha(\phi,x)\le\delta(\phi)$, and it is conjectured that if the $\phi$-orbit of $x$ is Zariski dense in $X$, then $\alpha(\phi,x)=\delta(\phi)$, 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.