On the density of abelian l-extensions

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