×

A survey of computational class field theory. (English) Zbl 0949.11063

This paper immediately preceded the author’s new definitive book on class field theory [Advanced topics in computational number theory, Graduate Texts in Mathematics. 193. Springer (2000)]. So it is essentially an extended abstract (or outline), which also highlights certain current topics in that book, e.g., Stark’s units [X. F. Roblot, C. R. Acad. Sci., Paris, Sér. I 323, 1165-1168 (1996; Zbl 0871.11080)] and complex multiplication [R. Schertz, J. Théor. Nombres Bordx. 9, 383-394 (1997; Zbl 0902.11047)].

MSC:

11Y40 Algebraic number theory computations
11R37 Class field theory
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11G15 Complex multiplication and moduli of abelian varieties
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML EMIS

References:

[1] Bach, E., Explicit bounds for primality testing and related problems, Math. Comp.55 (1990), p. 355-380. · Zbl 0701.11075
[2] Birkhoff, G., Subgroups of Abelian groups, Proc. Lond. Math. Soc. (2) 38 (1934-5), p. 385-401. · JFM 60.0893.03
[3] Butler, L., Subgroup Lattices and Symmetric Functions, Memoirs of the A.M.S. 539 (1994). · Zbl 0813.05067
[4] Cohen, H., A Course in Computational Algebraic Number Theory, GTM 138, Springer-Verlag, Berlin, Heidelberg, New-York (1993). · Zbl 0786.11071
[5] Cohen, H., Hermite and Smith normal form algorithms over Dedekind domains, Math. Comp.65 (1996), p. 1681-1699. · Zbl 0853.11100
[6] Cohen, H. and Diaz y Diaz, F.A polynomial reduction algorithm, Sém. Th. des Nombres Bordeaux (série 2), 3 (1991), p. 351-360. · Zbl 0758.11053
[7] Cohen, H., Diaz y Diaz, F. and Olivier, M., Subexponential algorithms for class and unit group computations, J. Symb. Comp.24 (1997), p. 433-441. · Zbl 0880.68067
[8] Cohen, H., Diaz y Diaz, F. and Olivier, M., Algorithmic methods for finitely generated Abelian groups, J. Symb. Comp., to appear. · Zbl 1007.20031
[9] Cohen, H., Diaz y Diaz, F. and Olivier, M., Computing ray class groups, conductors and discriminants, Math. Comp.67 (1998), p. 773-795. · Zbl 0929.11064
[10] Cohen, H. and Roblot, X., Computing the Hilbert class field of real quadratic fields, Math. Comp., to appear. · Zbl 1042.11075
[11] Fieker, C., Computing class fields via the Artin map, J. Symb. Comput., to appear. · Zbl 0982.11074
[12] Gee, A., Class invariants by Shimura’s reciprocity law, J. Théor. Nombres Bordeaux11 (1999), 45-72. · Zbl 0957.11048
[13] Hecke, E., Lectures on the theory of algebraic numbers GTM 77, Springer-Verlag, Berlin, Heidelberg, New York (1981). · Zbl 0504.12001
[14] Leutbecher, A., Euclidean fields having a large Lenstra constant, Ann. Inst. Fourier35, 2 (1985), p. 83-106. · Zbl 0546.12005
[15] Leutbecher, A. and Niklasch, G., On cliques of exceptional units and Lenstra’s construction of Euclidean fields, TUM Math. Inst. preprint M8705 (1987).
[16] Martinet, J., Petits discriminants des corps de nombres, Journées arithmétiques1980 (J.V. Armitage, Ed.), (1982), p. 151-193. · Zbl 0491.12005
[17] Nakagoshi, N., The structure of the multiplicative group of residue classes modulo PN+1, Nagoya Math. J.73 (1979), p. 41-60. · Zbl 0393.12023
[18] Roblot, X.-F., Unités de Stark et corps de classes de Hilbert, C. R. Acad. Sci. Paris323 (1996), p. 1165-1168. · Zbl 0871.11080
[19] Roblot, X.-F., Stark’s Conjectures and Hilbert’s Twelfth Problem, J. Number Theory, submitted, and Algorithmes de Factorisation dans les Extensions Relatives et Applications de la Conjecture de Stark à la Construction des Corps de Classes de Rayon, Thesis, Université Bordeaux I (1997).
[20] Schertz, R., Zur expliciten Berechnung von Ganzheitbasen in Strahlklassenkörpern über einem imaginär-quadratischen Zahlkörper, J. Number Theory34 (1990), p. 41-53. · Zbl 0701.11059
[21] Schertz, R., Problèmes de Construction en Multiplication Complexe, Sém. Th. des Nombres Bordeaux (Séries 2), 4 (1992), p. 239-262. · Zbl 0797.11083
[22] Schertz, R., Construction of ray class fields by elliptic units, J. Th. des Nombres Bordeaux9 (1997), p. 383-394. · Zbl 0902.11047
[23] Yui, N. and Zagier, D., On the singular values of Weber modular functions, Math. Comp.66 (1997), p. 1645-1662. · Zbl 0892.11022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.