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

We study the Galois module structure of the class groups of the Artin–Schreier extensions K over k of extension degree p, where k ∶= Fq(T) is the rational function field and p is a prime number. The structure of the p-part ClK(p) of the ideal class group of K as a finite G-module is determined by the invariant λn, where G ∶= Gal(K/k) = 〈σ〉 is the Galois group of K over k, and λn = dim_Fp (ClK(p)(σ^-1)n-¹ /ClK(p)(σ^-1)n). We find infinite families of the Artin–Schreier extensions over k whose ideal class groups have guaranteed prescribed λn-rank for 1 ≤ n ≤ 3. We find an algorithm for computing λ3-rank of ClK(p). Using this algorithm, for a given integer t ≥ 2, we get infinite families of the Artin–Schreier extensions over k whose λ1-rank is t, λ2-rank is t - 1, and λ3-rank is t - 2. In particular, in the case where p = 2, for a given positive integer t ≥ 2, we obtain an infinite family of the Artin–Schreier quadratic extensions over k whose 2-class group rank (resp. 2²-class group rank and 2³-class group rank) is exactly t (resp. t - 1 and t - 2). Furthermore, we also obtain a similar result on the 2ⁿ-ranks of the divisor class groups of the Artin–Schreier quadratic extensions over k.