Imaginary quadratic function fields with ideal class group of prime exponent

Bautista-Ancona, Victor; Diaz-Vargas, Javier; Rodríguez, José Alejandro Lara

doi:10.1016/j.jnt.2016.09.021

Cited by 1 publication

(2 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…∎ Theorem 3. 4 Let t be a positive integer. Let K = k(α D t ) be the Artin-Schreier extension over the rational function field k = F q (T) of extension degree p, where…”

Section: Lemma 31mentioning

confidence: 99%

See 1 more Smart Citation

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

Yoo,

Lee

2023

Can. J. Math.-J. Can. Math.

View full text Add to dashboard Cite

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.

show abstract

“…∎ Theorem 3. 4 Let t be a positive integer. Let K = k(α D t ) be the Artin-Schreier extension over the rational function field k = F q (T) of extension degree p, where…”

Section: Lemma 31mentioning

confidence: 99%

“…∎ Corollary 4. 4 Let K be the Artin-Schreier quadratic extension over k, and let the λ 3rank of Cl K be computed by Algorithm 1. Then the 2 3 -rank of Cl K is exactly λ 3 : that is, Cl K (2) has a subgroup isomorphic to (Z/2 3 Z) λ3 .…”

Section: Theorem 43 Let K Be the Artin-schreier Extension Over The Ra...mentioning

confidence: 99%

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

Yoo,

Lee

2023

Can. J. Math.-J. Can. Math.

View full text Add to dashboard Cite

show abstract

Imaginary quadratic function fields with ideal class group of prime exponent

Cited by 1 publication

References 11 publications

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

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

Contact Info

Product

Resources

About