On the density of abelian l-extensions

Abstract

We derive an asymptotic formula which counts the number of abelian extensions of prime degrees over rational function fields. Specifically, let $\ell$ be a rational prime and $K$ a rational function field ${\mathbb{F}}_q(t)$ with $\ell \nmid q$. Let ${\text{Disc}}_f\left(F/K\right)$ denote the finite discriminant of $F$ over $K$. Denote the number of abelian $\ell$-extensions $F/K$ with $\text{deg}\left({\text{Disc}}_f(F/K)\right)=(\ell -1)\alpha n$ by $a_{\ell }(n)$, where $\alpha =\alpha (q,\ell )$ is the order of $q$ in the multiplicative group ${\left(\mathbb{Z}/\ell \mathbb{Z}\right)}^{\times }$. We give a explicit asymptotic formula for $a_{\ell }(n)$. In the case of cubic extensions with $q\equiv 2(\mathrm{mod}3)$, our formula gives an exact analogue of Cohn's classical formula.