Infinite families of Artin–Schreier function fields with any prescribed class group rank

Abstract

We study the Galois module structure of the class groups of the Artin–Schreier extensions K over k of extension degree p, where $k:={\mathbb F}_q(T)$ is the rational function field and p is a prime number. The structure of the p-part $Cl_K(p)$ of the ideal class group of K as a finite G-module is determined by the invariant ${\lambda }_n$ , where $G:=\operatorname {\mathrm {Gal}}(K/k)=\langle {\sigma } \rangle $ is the Galois group of K over k, and ${\lambda }_n = \dim _{{\mathbb F}_p}(Cl_K(p)^{({\sigma }-1)^{n-1}}/Cl_K(p)^{({\sigma }-1)^{n}})$ . We find infinite families of the Artin–Schreier extensions over k whose ideal class groups have guaranteed prescribed ${\lambda }_n$ -rank for $1 \leq n \leq 3$ . We find an algorithm for computing ${\lambda }_3$ -rank of $Cl_K(p)$ . Using this algorithm, for a given integer $t \ge 2$ , we get infinite families of the Artin–Schreier extensions over k whose ${\lambda }_1$ -rank is t, ${\lambda }_2$ -rank is $t-1$ , and ${\lambda }_3$ -rank is $t-2$ . In particular, in the case where $p=2$ , for a given positive integer $t \ge 2$ , we obtain an infinite family of the Artin–Schreier quadratic extensions over k whose $2$ -class group rank (resp. $2^2$ -class group rank and $2^3$ -class group rank) is exactly t (resp. $t-1$ and $t-2$ ). Furthermore, we also obtain a similar result on the $2^n$ -ranks of the divisor class groups of the Artin–Schreier quadratic extensions over k.