no. 1
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.

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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://www.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 | Zbl

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

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

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

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

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

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

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

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

[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 | Zbl

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

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

[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 | Zbl

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

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

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