3-Class groups of cubic cyclic function fields
Authors : Zhengjun ZHAO
Pages : 2607-2620
View : 12 | Download : 5
Publication Date : 0000-00-00
Article Type : Research Paper
Abstract :Let $F$ be a global function field over the finite constant field $\mathbb{F}_q$ with $3\mid q-1$, and let $K/F$ be a cubic cyclic function fields extension with Galois group $G=$Gal$insert ignore into journalissuearticles values(K/F);=$. Denote by $\mathcal{C}insert ignore into journalissuearticles values(K);$ and $\mathcal{C}insert ignore into journalissuearticles values(K);_3$ the ideal class group of $K$ and its Sylow 3-subgroup, respectively. Let $\mathcal{C}insert ignore into journalissuearticles values(K);_3^G=\{[\fa]\in \mathcal{C}insert ignore into journalissuearticles values(K);_3|\ \sigma[\fa]=[\fa]\}$ and $\mathcal{C}insert ignore into journalissuearticles values(K);_3^{1-\sigma}=\{[\fa]insert ignore into journalissuearticles values(\sigma[\fa]);^{-1}|\ [\fa]\in \mathcal{C}insert ignore into journalissuearticles values(K);_3\}$. In this paper, we present a method for computing the 3-rank of the quotient group $\mathcal{C}insert ignore into journalissuearticles values(K);_3^G\mathcal{C}insert ignore into journalissuearticles values(K);_3^{1-\sigma}/\mathcal{C}insert ignore into journalissuearticles values(K);_3^{1-\sigma}$. Specifically, when $K$ is a cubic Kummer extension of $\mathbb{F}_qinsert ignore into journalissuearticles values(T);$, we determine explicitly the key factors $t$, $x_1,\cdots, x_t$, and $[\mathfrak{A}_1],\cdots, [\mathfrak{A}_t]$ in the process of computing the 3-rank of $\mathcal{C}insert ignore into journalissuearticles values(K);_3^G\mathcal{C}insert ignore into journalissuearticles values(K);_3^{1-\sigma}/\mathcal{C}insert ignore into journalissuearticles values(K);_3^{1-\sigma}$. Combining this deterministic algorithm along with the structure of class groups for cubic Kummer function fields, the 3-rank of the Sylow 3-subgroup of $\mathcal{C}insert ignore into journalissuearticles values(K);$ is determined explicitly in this specific case. Examples are given in the last two sections to elucidate our computational method.Keywords : Class group, cubic function fields, genus theory, Artin reciprocity law map