ℓ-RANKS OF CLASS GROUPS OF FUNCTION FIELDS

Title & Authors
ℓ-RANKS OF CLASS GROUPS OF FUNCTION FIELDS
Bae, Sung-Han; Jung, Hwan-Yup;

Abstract
In this paper we give asymptotic formulas for the number of $\small{{\ell}}$-cyclic extensions of the rational function field $k Keywords class groups;function fields; Language English Cited by 1. DENSITIES FOR 4-RANKS OF REAL QUADRATIC FUNCTION FIELDS,; 충청수학회지, 2014. vol.27. 4, pp.553-562 1. DENSITIES FOR 4-RANKS OF REAL QUADRATIC FUNCTION FIELDS, Journal of the Chungcheong Mathematical Society, 2014, 27, 4, 553 References 1. B. Angles, On Hilbert class field towers of global function fields, Drinfeld modules, modular schemes and applications (Alden-Biesen, 1996), 261-271, World Sci. Publ., River Edge, NJ, 1997. 2. S. Bae and J. Koo, Genus theory for function fields, J. Austral. Math. Soc. Ser. A 60 (1996), no. 3, 301-310. 3. A. Frohlich, Central Extensions, Galois Groups, and Ideal Class Groups of Number Fields, American Mathematical Society, Providence, RI, 1983. 4. F. Gerth, Number fields with prescribed$\ell$-class groups, Proc. Amer. Math. Soc. 49 (1975), 284-288. 5. F. Gerth, Asymptotic behavior of number fields with prescribed$\ell$-class numbers, J. Number Theory 17 (1983), no. 2, 191-203. 6. F. Gerth, Counting certain number fields with prescribed$\ell$-class numbers, J. Reine Angew. Math. 337 (1982), 195-207. 7. F. Gerth, An application of matrices over finite fields to algebraic number theory, Math. Comp. 41 (1983), no. 163, 229-234. 8. F. Gerth, The 4-class ranks of quadratic fields, Invent. Math. 77 (1984), no. 3, 489-515. 9. F. Gerth, Densities for certain$\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.