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Turkish Journal of Mathematics

ZHENGJUN ZHAO

DOI

10.3906/mat-1710-30

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$(K/F)=$. Denote by $\mathcal{C}(K)$ and $\mathcal{C}(K)_3$ the ideal class group of $K$ and its Sylow 3-subgroup, respectively. Let $\mathcal{C}(K)_3^G=\{[\fa]\in \mathcal{C}(K)_3 \ \sigma[\fa]=[\fa]\}$ and $\mathcal{C}(K)_3^{1-\sigma}=\{[\fa](\sigma[\fa])^{-1} \ [\fa]\in \mathcal{C}(K)_3\}$. In this paper, we present a method for computing the 3-rank of the quotient group $\mathcal{C}(K)_3^G\mathcal{C}(K)_3^{1-\sigma}/\mathcal{C}(K)_3^{1-\sigma}$. Specifically, when $K$ is a cubic Kummer extension of $\mathbb{F}_q(T)$, 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 $\mathcal{C}(K)_3^G\mathcal{C}(K)_3^{1-\sigma}/\mathcal{C}(K)_3^{1-\sigma}$. 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 $\mathcal{C}(K)$ 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

First Page

2607

Last Page

2620

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