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.