$\ell$-ranks of class groups of function fields

Sunghan Bae; Hwanyup Jung

doi:10.4134/JKMS.2012.49.1.49

Journal of the
Korean Mathematical Society JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2012; 49(1): 49-67

Printed January 1, 2012

https://doi.org/10.4134/JKMS.2012.49.1.49

$\ell$-ranks of class groups of function fields

Sunghan Bae and Hwanyup Jung

Korea Advanced Institute of Science and Technology, Chungbuk National University

Abstract

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Abstract

In this paper we give asymptotic formulas for the number of $\ell$-cyclic extensions of the rational function field ${\rm k} = \mathbb F_{q}(T)$ with prescribed $\ell$-class numbers inside some cyclotomic function fields, and density results for $\ell$-cyclic extensions of $\rm k$ with certain properties on the ideal class groups.

Keywords: class groups, function fields

MSC numbers: 11R58, 11R29, 11R45