[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/JKMS.2012.49.1.049

ℓ-RANKS OF CLASS GROUPS OF FUNCTION FIELDS

Bae, Sung-Han (Department of Mathematics Korea Advanced Institute of Science and Technology)
Jung, Hwan-Yup (Department of Mathematics Education Chungbuk National University)

Publication Information

Journal of the Korean Mathematical Society / v.49, no.1, 2012 , pp. 49-67 More about this Journal

Abstract

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

Keywords

class groups; function fields;

Citations & Related Records

(Related Records In Web of Science)
Times Cited By SCOPUS : 0

Reference

1	F. Gerth, Asymptotic behavior of number fields with prescribed $\ell$ -class numbers, J. Number Theory 17 (1983), no. 2, 191-203. DOI
2	F. Gerth, Counting certain number fields with prescribed $\ell$ -class numbers, J. Reine Angew. Math. 337 (1982), 195-207.
3	F. Gerth, An application of matrices over finite fields to algebraic number theory, Math. Comp. 41 (1983), no. 163, 229-234. DOI ScienceOn
4	F. Gerth, The 4-class ranks of quadratic fields, Invent. Math. 77 (1984), no. 3, 489-515. DOI
5	F. Gerth, Densities for certain $\ell$ -ranks in cyclic fields of degree ${\ell}^n$ , Compositio Math. 60 (1986), no. 3, 295-322.
6	F. Gerth, Densities for ranks of certain parts of p-class groups, Proc. Amer. Math. Soc. 99 (1987), no. 1, 1-8.
7	J. Knopfmacher, Analytic Arithmetic of Algebraic Function Fields, Marcel Decker Inc., New York-Basel, 1979.
8	M. Rosen, Number Theory in Function Fields, GTM vol. 210, Springer, 2002.
9	C. Wittmann, $\ell$ -Class groups of cyclic function fields of degree $\ell$ , Finite Fields Appl. 13 (2007), no. 2, 327-347. DOI ScienceOn
10	B. Angles, On Hilbert class field towers of global function fields, Drinfeld modules, modular schemes and applications (Alden-Biesen, 1996), 261-271, World Sci. Publ., River Edge, NJ, 1997.
11	S. Bae and J. Koo, Genus theory for function fields, J. Austral. Math. Soc. Ser. A 60 (1996), no. 3, 301-310. DOI
12	A. Frohlich, Central Extensions, Galois Groups, and Ideal Class Groups of Number Fields, American Mathematical Society, Providence, RI, 1983.
13	F. Gerth, Number fields with prescribed $\ell$ -class groups, Proc. Amer. Math. Soc. 49 (1975), 284-288.