A Bogomolov property for moduli spaces of polynomials over abelian extensions

Abstract

Let $d\geq2$ be an integer, and let $\operatorname{MPoly}^d$ be the moduli space of degree-$d$ polynomials. For every number field $K$, we prove that there exists $ε_{K,d}>0$ 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 $\operatorname{MPoly}^d$ are postcritically finite. The proof uses a universal critical-divisor family, a nef adelic line bundle with an explicit orbit-height formula, the intertwined relation, local ramification estimates, and the correspondence method of Ji--Song--Xie.