Infinite families of Artin-Schreier function fields with any prescribed class group rank (Q6633512)

The main goal of this paper is the study of the Galois module structure of the class groups of the Artin-Schreier extensions \(K\) over \(k= {\mathbb F}_q(T)\) of degree \(p\). The structure of the \(p\)-part \(Cl_K(p)\) of the ideal class group of \(K\) is determined by the filtration, originally introduced by M. Madan,\N\[\N\lambda_n=\dim_{ {\mathbb F}_p}\big(Cl_K(p)^{(\sigma-1)^{n-1}}/Cl_K(p)^{(\sigma-1)^n} \big),\quad n\geq 1,\N\]\Nwhere \(G={\mathrm{Gal}}(K/k)=\langle\sigma\rangle\).\N\NIn Section 3, the authors find infinite families of Artin-Schreier extensions over \(k\) with any prescribed class groups of \(\lambda_1\)-rank, Theorems 3.2, 3.3 and 3.4. In Section 4, the authors compute the \(\lambda_3\)-rank of class groups of Artin-Schreier function fields. It is introduced Algorithm 1 and it is proved, Theorem 4.3, that if \(K/k\) is an Artin-Schreier extension of degree \(p\), then the \(\lambda_3\)-rank of the ideal class group of \(K\) can be computed by Algorithm 1.\N\NIn Section 5, it is presented in Theorem 5.1 an infinite family of Artin-Schreier function fields with higher \(\lambda_n\)-rank. In the last section, the paper presents implementation results.