Corps diédraux à multiplication complexe principaux
Annales de l'Institut Fourier, Volume 50 (2000) no. 1, pp. 67-103.

We determine all the dihedral CM fields with relative class number one, then all of them with class number one: there are 32 such non-abelian fields with class number one. This is the first example of resolution of the class number one problem for non-abelian normal CM-fields of a given Galois group.

Nous déterminons tous les corps diédraux à multiplication complexe de nombres de classes relatif un, puis ceux de nombre de classes un : il y a 32 tels corps non-abéliens principaux. C’est le premier exemple, dans ce cadre assez général, de résolution du problème de nombre de classes un pour les corps galoisiens à multiplication complexe avec un type de groupe de Galois non-abélien fixé.

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     title = {Corps di\'edraux \`a multiplication complexe principaux},
     journal = {Annales de l'Institut Fourier},
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     doi = {10.5802/aif.1747},
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Lefeuvre, Yann. Corps diédraux à multiplication complexe principaux. Annales de l'Institut Fourier, Volume 50 (2000) no. 1, pp. 67-103. doi : 10.5802/aif.1747. http://www.numdam.org/articles/10.5802/aif.1747/

[1] H. Cohen, F. Diaz Y Diaz, M. Olivier, Computing ray class groups, conductors and discriminants, Actes du Colloque ANTS II, Talence, 1996. | Zbl

[2] S. Chowla, Note on Dirichlet's L-functions, Acta Arith., 1 (1936), 113-114. | JFM | Zbl

[3] S. Chowla, M.J. Deleon, P. Hartung, On a hypothesis implying the non-vanishing of Dirichlet's L-series L(s,χ) for s > 0, J. reine angew. Math., 262/263 (1973), 415-419. | MR | Zbl

[4] A. Fröhlich, J. Queyrut, On the functional equation of the Artin L-function for characters of real representations, Invent. Math., 20 (1973), 125-138. | MR | Zbl

[5] H. Hida, Elementary theory of L-functions and Eisenstein series, London Math. Soc., Student Texts, Cambridge University Press, 26 (1993). | MR | Zbl

[6] J. Hoffstein, Some analytic bounds for zeta functions and class numbers, Invent. Math., 55 (1979), 37-47. | MR | Zbl

[7] Y. Lefeuvre, Corps à multiplication complexe diédraux principaux, Thèse, Univ. Caen, soutenue le 28 juin 1999.

[8] Y. Lefeuvre, S. Louboutin, The class number one problem for the dihedral CM-fields, to appear in the Proceedings of Conference on Algebraic Number Theory and Diophantine Analysis, Gras, 1998. | Zbl

[9] S. Louboutin, Lower bounds for relative class numbers of CM-fields, Proc. Amer. Math. Soc., 120 (1994), 425-434. | MR | Zbl

[10] S. Louboutin, Corps quadratiques principaux à corps de classes de Hilbert principaux et à multiplication complexe, Acta Arith., 74 (1996), 121-140. | MR | Zbl

[11] S. Louboutin, Majorations explicites du résidu au point 1 des fonctions zêta de certains corps de nombres, J. Math. Soc. Japan, 50 (1998), 57-69. | MR | Zbl

[12] S. Louboutin, Upper bounds on |L(1,χ)| and applications, Canad. J. Math., 50 (1998), 795-815. | Zbl

[13] S. Louboutin, Computation of relative class numbers of CM-fields by using Hecke L-functions, Math. Comp., 69 (1999), 371-393. | MR | Zbl

[14] S. Louboutin, Computation of L(0,χ) and of relative class numbers of CM-fields, Preprint Univ. Caen, 1998. | Zbl

[15] S. Louboutin, R. Okazaki, Determination of all non-normal quartic CM-fields and of all non-abelian normal octic CM-fields with class number one, Acta Arith., 67 (1994), 47-62. | MR | Zbl

[16] S. Louboutin, R. Okazaki, The class number one problem for some non-abelian normal CM-fields of 2-power degrees, Proc. London Math. Soc., 76 (3) (1998), 523-548. | MR | Zbl

[17] S. Louboutin, R. Okazaki, M. Olivier, The class number one problem for some non-abelian normal CM-fields, Trans. Amer. Math. Soc., 349 (1997), 3657-3678. | MR | Zbl

[18] S. Louboutin, Y.H. Park, Y. Lefeuvre, Construction of the real dihedral number fields of degree 2p. Applications, Acta Arith., 89 (1999), 201-215. | MR | Zbl

[19] J. Martinet, Sur l'arithmétique des extensions galoisiennes à groupe de Galois diédral d'ordre 2p, Ann. Inst. Fourier Grenoble, 19, 1 (1969), 1-80. | Numdam | MR | Zbl

[20] A.M. Odlyzko, Some analytic estimates of class numbers and discriminants, Invent. Math., 29 (1975), 279-286. | MR | Zbl

[21] A.M. Odlyzko, On conductors and discriminants, Algebraic number fields, Durham Symposium, 1975, A. Fröhlich, éd., Academic Press (1977), 377-407. | MR | Zbl

[22] X.F. Roblot, Algorithmes de factorisation dans les extensions relatives et applications de la conjecture de Stark à la construction des corps de classes de rayon, Thèse, Univ. Bordeaux, 1997.

[23] L.C. Washington, Introduction to cyclotomic fields, Springer-Verlag, Graduate Texts in Mathematics 83, second edition, 1997. | MR | Zbl

[24] K. Yamamura, The determination of the imaginary abelian number fields with class number one, Math. Comp., 62 (1994), 899-921. | MR | Zbl

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