Variation of the canonical height in a family of polarized dynamical systems

TL;DR

This paper refines the understanding of how the canonical height varies in families of polarized dynamical systems, providing sharper asymptotics and explicit bounds for special parameters.

Contribution

It improves the asymptotic formula for height variation in families of polarized dynamical systems, reducing the error term and deriving explicit bounds for parameters with finite orbits.

Findings

Sharper asymptotic formula for height variation

Explicit bounds on parameters with finite orbits

Improved error term in height asymptotics

Abstract

Call and Silverman introduced the canonical height associated to a polarized dynamical system, that is, an endomorphism of a projective variety and an ample line bundle which pulls back to a tensor power of itself. They also presented an asymptotic for the variation of this height in a family over a one-dimensional base in terms of the height on the generic fibre and the height of the parameter. Here we improve this asymptotic, saving a power in the error term. As a corollary, we give an explicit bound on the height of parameters at which the dynamical system specialized to a finite orbit, in the case of endomorphisms of projective space over the projective line.