The critical height is a moduli height

Abstract

Silverman defined the critical height of a rational function $f\left(z\right)$ of degree $d\ge 2$ in terms of the asymptotic rate of growth of the Weil height along the critical orbits of $f$. He also conjectured that this quantity was commensurate to an ample Weil height on the moduli space of rational functions degree $d$. We prove that conjecture.