On the density of abelian l-extensions

TL;DR

This paper derives an explicit asymptotic formula for counting abelian extensions of prime degree over rational function fields, extending classical results and providing new insights into their distribution.

Contribution

It presents a novel asymptotic formula for the number of abelian -extensions over rational function fields, generalizing and extending classical results.

Findings

Explicit asymptotic formula for abelian -extensions

Analogue of Cohn's classical formula for cubic extensions

Provides distribution estimates for abelian 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 $\Bbb F_q(t)$ with $\ell \nmid q$. Let $\textup{Disc}_f\left(F/K\right)$ denote the finite discriminant of $F$ over $K$. Denote the number of abelian $\ell$-extensions $F/K$ with $\textup{deg}\left(\textup{Disc}_f(F/K)\right) = (\ell-1)\alpha n$ by $a_{\ell}(n)$, where $\alpha=\alpha(q, \ell)$ is the order of $q$ in the multiplicative group $\left(\Bbb Z/\ell \Bbb Z\right)^\times$. We give a explicit asymptotic formula for $a_\ell(n)$. In the case of cubic extensions with $q\equiv 2 \pmod 3$, our formula gives an exact analogue of Cohn's classical formula.