Infinitely wildly ramified arboreal representations for postcritically finite polynomials with potential good reduction

Abstract

Let $K$ be a number field, let $v$ be a finite place of $K$, let $f\in K[z]$ be a degree $d\geqslant2$ polynomial with $v|d$, and let $a\in K$. We show that if $f$ is postcritically bounded and has potential good reduction with respect to $v$, then the arboreal representation associated to the pair $(f,a)$ is either finite or infinitely wildly ramified above $v$.