DOI QR코드

DOI QR Code

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)
  • Received : 2010.08.10
  • Published : 2012.01.01

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

References

  1. 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.
  2. S. Bae and J. Koo, Genus theory for function fields, J. Austral. Math. Soc. Ser. A 60 (1996), no. 3, 301-310. https://doi.org/10.1017/S1446788700037824
  3. A. Frohlich, Central Extensions, Galois Groups, and Ideal Class Groups of Number Fields, American Mathematical Society, Providence, RI, 1983.
  4. F. Gerth, Number fields with prescribed $\ell$-class groups, Proc. Amer. Math. Soc. 49 (1975), 284-288.
  5. F. Gerth, Asymptotic behavior of number fields with prescribed $\ell$-class numbers, J. Number Theory 17 (1983), no. 2, 191-203. https://doi.org/10.1016/0022-314X(83)90020-3
  6. F. Gerth, Counting certain number fields with prescribed $\ell$-class numbers, J. Reine Angew. Math. 337 (1982), 195-207.
  7. F. Gerth, An application of matrices over finite fields to algebraic number theory, Math. Comp. 41 (1983), no. 163, 229-234. https://doi.org/10.1090/S0025-5718-1983-0701637-0
  8. F. Gerth, The 4-class ranks of quadratic fields, Invent. Math. 77 (1984), no. 3, 489-515. https://doi.org/10.1007/BF01388835
  9. F. Gerth, Densities for certain $\ell$-ranks in cyclic fields of degree ${\ell}^n$, Compositio Math. 60 (1986), no. 3, 295-322.
  10. F. Gerth, Densities for ranks of certain parts of p-class groups, Proc. Amer. Math. Soc. 99 (1987), no. 1, 1-8.
  11. J. Knopfmacher, Analytic Arithmetic of Algebraic Function Fields, Marcel Decker Inc., New York-Basel, 1979.
  12. M. Rosen, Number Theory in Function Fields, GTM vol. 210, Springer, 2002.
  13. C. Wittmann, $\ell$-Class groups of cyclic function fields of degree $\ell$, Finite Fields Appl. 13 (2007), no. 2, 327-347. https://doi.org/10.1016/j.ffa.2005.09.001

Cited by

  1. DENSITIES FOR 4-RANKS OF REAL QUADRATIC FUNCTION FIELDS vol.27, pp.4, 2014, https://doi.org/10.14403/jcms.2014.27.4.553