A Bogomolov property for moduli spaces of polynomials over abelian extensions

Abstract

Let $d\ge 2$ be an integer, and let ${\mathrm{MPoly}}^d$ be the moduli space of degree-$d$ polynomials. For every number field $K$, we prove that there exists ${\epsilon }_{K,d}>0$ such that \[ \left\lbraceα\in\operatorname{MPoly}^d(K^{\mathrm{ab}})\colon h_{\mathrm{crit}}(α)<ε_{K,d}\right\rbrace=\left\lbraceα\in\operatorname{MPoly}^d(K^{\mathrm{ab}})\colon h_{\mathrm{crit}}(α)=0\right\rbrace \] is finite, where $h_{\mathrm{crit}}$ is the critical height. In particular, only finitely many $K^{\mathrm{ab}}$-rational points of ${\mathrm{MPoly}}^d$ are postcritically finite. The proof combines the correspondence method of Ji--Song--Xie with local ramification estimates and the uniform finiteness of intertwined classes in ${\mathrm{MPoly}}^d$. We also prove an analogous finiteness theorem for non-preperiodic marked points in non-isotrivial one-parameter polynomial families.

arXiv comment44 pages. A theorem on polynomial dynamical pairs added