Multiplicator freeness for restricted-ramification Galois groups over global function fields

Abstract

Let $K$ be a global function field with full constant field ${\mathbb{F}}_q$, let $T$ be a finite set of finite places, and let $\infty$ be a distinguished place. We study the maximal pro-$\ell$ extension of $K$ which is unramified outside $T$ and in which $\infty$ splits completely. Using factorisation results for $\ell$-primary idele class characters, we prove that its Galois group is multiplicator-free if $\ell \nmid q-1$ or $T\ne \varnothing$. In particular, if $T\ne \varnothing$, the conclusion holds for every prime $\ell$.