A survey of computational class field theory
Journal de Théorie des Nombres de Bordeaux, Tome 11 (1999) no. 1, pp. 1-13.

Le but de cet article est de décrire les avancées récentes dans la théorie algorithmique du corps de classes. Nous expliquons comment calculer les groupes de classes de rayon ainsi que les discriminants des corps de classes correspondants. Nous donnons ensuite les trois méthodes principales utilisées pour le calcul des équations des corps de classes : la théorie de Kummer, les unités de Stark et la multiplication complexe. En utilisant ces techniques, nous avons pu construire de nombreux nouveaux corps de nombres intéressants, en particulier ayant un discriminant très proche des bornes d'odlyzko.

We give a survey of computational class field theory. We first explain how to compute ray class groups and discriminants of the corresponding ray class fields. We then explain the three main methods in use for computing an equation for the class fields themselves: Kummer theory, Stark units and complex multiplication. Using these techniques we can construct many new number fields, including fields of very small root discriminant.

@article{JTNB_1999__11_1_1_0,
     author = {Cohen, Henri},
     title = {A survey of computational class field theory},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {1--13},
     publisher = {Universit\'e Bordeaux I},
     volume = {11},
     number = {1},
     year = {1999},
     zbl = {0949.11063},
     mrnumber = {1730429},
     language = {en},
     url = {http://archive.numdam.org/item/JTNB_1999__11_1_1_0/}
}
Cohen, Henri. A survey of computational class field theory. Journal de Théorie des Nombres de Bordeaux, Tome 11 (1999) no. 1, pp. 1-13. http://archive.numdam.org/item/JTNB_1999__11_1_1_0/

[1] E. Bach, Explicit bounds for primality testing and related problems, Math. Comp. 55 (1990), p. 355-380. | MR 1023756 | Zbl 0701.11075

[2] G. Birkhoff, Subgroups of Abelian groups, Proc. Lond. Math. Soc. (2) 38 (1934-5), p. 385-401. | JFM 60.0893.03 | Zbl 0010.34304

[3] L. Butler, Subgroup Lattices and Symmetric Functions, Memoirs of the A.M.S. 539 (1994). | MR 1223236 | Zbl 0813.05067

[4] H. Cohen, A Course in Computational Algebraic Number Theory, GTM 138, Springer-Verlag, Berlin, Heidelberg, New-York (1993). | MR 1228206 | Zbl 0786.11071

[5] H. Cohen, Hermite and Smith normal form algorithms over Dedekind domains, Math. Comp. 65 (1996), p. 1681-1699. | MR 1361805 | Zbl 0853.11100

[6] H. Cohen and F. Diaz Y Diaz A polynomial reduction algorithm, Sém. Th. des Nombres Bordeaux (série 2), 3 (1991), p. 351-360. | Numdam | MR 1149802 | Zbl 0758.11053

[7] H. Cohen, F. Diaz Y Diaz and M. Olivier, Subexponential algorithms for class and unit group computations, J. Symb. Comp. 24 (1997), p. 433-441. | MR 1484490 | Zbl 0880.68067

[8] H. Cohen, F. Diaz Y Diaz and M. Olivier, Algorithmic methods for finitely generated Abelian groups, J. Symb. Comp., to appear. | Zbl 1007.20031

[9] H. Cohen, F. Diaz Y Diaz and M. Olivier, Computing ray class groups, conductors and discriminants, Math. Comp. 67 (1998), p. 773-795. | MR 1443117 | Zbl 0929.11064

[10] H. Cohen and X. Roblot, Computing the Hilbert class field of real quadratic fields, Math. Comp., to appear. | MR 1651747 | Zbl 1042.11075

[11] C. Fieker, Computing class fields via the Artin map, J. Symb. Comput., to appear. | MR 1826583

[12] A. Gee, Class invariants by Shimura's reciprocity law, J. Théor. Nombres Bordeaux 11 (1999), 45-72. | Numdam | MR 1730432 | Zbl 0957.11048

[13] E. Hecke, Lectures on the theory of algebraic numbers GTM 77, Springer-Verlag, Berlin, Heidelberg, New York (1981). | MR 638719 | Zbl 0504.12001

[14] A. Leutbecher, Euclidean fields having a large Lenstra constant, Ann. Inst. Fourier 35, 2 (1985), p. 83-106. | Numdam | MR 786536 | Zbl 0546.12005

[15] A. Leutbecher and G. Niklasch, On cliques of exceptional units and Lenstra's construction of Euclidean fields, TUM Math. Inst. preprint M8705 (1987).

[16] J. Martinet, Petits discriminants des corps de nombres, Journées arithmétiques 1980 (J.V. Armitage, Ed.), London Math. Soc. Lecture Notes Ser. 56 (1982), p. 151-193. | MR 697261 | Zbl 0491.12005

[17] N. Nakagoshi, The structure of the multiplicative group of residue classes modulo PN+1, Nagoya Math. J. 73 (1979), p. 41-60. | MR 524007 | Zbl 0393.12023

[18] X.-F. Roblot, Unités de Stark et corps de classes de Hilbert, C. R. Acad. Sci. Paris 323 (1996), p. 1165-1168. | MR 1423444 | Zbl 0871.11080

[19] X.-F. Roblot, 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] R. Schertz, Zur expliciten Berechnung von Ganzheitbasen in Strahlklassenkörpern über einem imaginär-quadratischen Zahlkörper, J. Number Theory 34 (1990), p. 41-53. | MR 1039766 | Zbl 0701.11059

[21] R. Schertz, Problèmes de Construction en Multiplication Complexe, Sém. Th. des Nombres Bordeaux (Séries 2), 4 (1992), p. 239-262. | Numdam | MR 1208864 | Zbl 0797.11083

[22] R. Schertz, Construction of ray class fields by elliptic units, J. Th. des Nombres Bordeaux 9 (1997), p. 383-394. | Numdam | MR 1617405 | Zbl 0902.11047

[23] N. Yui and D. Zagier, On the singular values of Weber modular functions, Math. Comp. 66 (1997), p. 1645-1662. | MR 1415803 | Zbl 0892.11022