Variation of the canonical height for a family of polynomials

TL;DR

This paper extends Tate's theorem from elliptic surfaces to one-parameter families of polynomial dynamical systems, analyzing the variation of canonical heights and their local contributions.

Contribution

It proves an analogous height variation theorem for polynomial families and compares local canonical heights with local height contributions, showing analyticity near the divisor.

Findings

Canonical height variation is bounded by a divisor on the parameter space.

Local canonical heights differ from local height contributions by an analytic function.

Results generalize Silverman's work from elliptic surfaces to polynomial dynamics.

Abstract

A theorem of Tate asserts that, for an elliptic surface E/X defined over a number field k, and a section P of E, there exists a divisor D on X such that the canonical height of the specialization of P to the fibre above t differs from the height of t relative to D by at most a bounded amount. We prove the analogous statement for a one-parameter family of polynomial dynamical systems. Moreover, we compare, at each place of k, the local canonical height with the local contribution to the height relative to D, and show that the difference is analytic near the support of D, a result which is analogous to results of Silverman in the elliptic surface context.