The class number one problem for some non-abelian normal CM-fields of degree $24$

Lemmermeyer, F.; Louboutin, S.; Okazaki, R.

The class number one problem for some non-abelian normal CM-fields of degree

24

Lemmermeyer, F. ; Louboutin, S. ; Okazaki, R.

Journal de théorie des nombres de Bordeaux, Tome 11 (1999) no. 2, pp. 387-406.

Résumé
Abstract

Nous déterminons tous les corps de nombres de degré $24$ , galoisiens mais non-abéliens, à multiplication complexe et tels que les groupes de Galois de leurs sous-corps totalement réels maximaux soient isomorphes à $𝒜_{4}$ (le groupe alterné de degré $4$ et d’ordre $12$ ) qui sont de nombres de classes d’idéaux égaux à $1$ . Nous prouvons $(𝑖)$ qu’il y a deux tels corps de nombres de groupes de Galois $𝒜_{4} \times 𝒞_{2}$ (voir Théorème $14$ ), $(𝑖𝑖)$ qu’il y a au plus un tel corps de nombres de groupe de Galois $S L_{2} (𝔽_{3})$ (voir Théorème 18), et $(𝑖𝑖𝑖)$ que sous l’hypothèse de Riemann généralisée ce dernier corps candidat est effectivement de nombre de classes d’idéaux égal à $1$ .

We determine all the non-abelian normal CM-fields of degree 24 with class number one, provided that the Galois group of their maximal real subfields is isomorphic to $𝒜_{4}$ , the alternating group of degree $4$ and order $12$ . There are two such fields with Galois group $𝒜_{4} \times 𝒞_{2}$ (see Theorem 14) and at most one with Galois group SL $_{2} (𝔽_{3})$ (see Theorem 18); if the generalized Riemann hypothesis is true, then this last field has class number $1$ .

MR Zbl

@article{JTNB_1999__11_2_387_0,
     author = {Lemmermeyer, F. and Louboutin, S. and Okazaki, R.},
     title = {The class number one problem for some non-abelian normal {CM-fields} of degree $24$},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {387--406},
     publisher = {Universit\'e Bordeaux I},
     volume = {11},
     number = {2},
     year = {1999},
     mrnumber = {1745886},
     zbl = {1010.11063},
     language = {en},
     url = {https://www.numdam.org/item/JTNB_1999__11_2_387_0/}
}

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JO  - Journal de théorie des nombres de Bordeaux
PY  - 1999
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EP  - 406
VL  - 11
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PB  - Université Bordeaux I
UR  - https://www.numdam.org/item/JTNB_1999__11_2_387_0/
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%A Louboutin, S.
%A Okazaki, R.
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%J Journal de théorie des nombres de Bordeaux
%D 1999
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%G en
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Lemmermeyer, F.; Louboutin, S.; Okazaki, R. The class number one problem for some non-abelian normal CM-fields of degree $24$. Journal de théorie des nombres de Bordeaux, Tome 11 (1999) no. 2, pp. 387-406. https://www.numdam.org/item/JTNB_1999__11_2_387_0/

Bibliographie
Cité par

[1] J.V. Armitage, A. Frohlich, Classnumbers and unit signatures. Mathematika 14 (1967), 94-98. | MR | Zbl

[2] C. Bachoc, S.-H. Kwon, Sur les extensions de groupe de Galois Ã4. Acta Arith. 62 (1992), 1-10. | MR | Zbl

[3] H. Cohen A course in computational algebraic number theory. Grad. Texts Math. 138, Springer-Verlag, Berlin, Heidelberg, New York (1993). | MR | Zbl

[4] H. Cohen, F. Diaz Y Diaz, M. Olivier, Tables of octic fields with a quartic subfield. Math. Comp., à paraître. | Zbl

[5] A. Derhem, Capitulation dans les extensions quadratiques non ramifiées de corps de nombres cubiques cycliques. thèse, Université Laval, Québec (1988).

[6] Ph. Furtwängler, Über das Verhalten der Ideale des Grundkörpers im Klassenkörper. Monatsh. Math. Phys. 27 (1916), 1-15. | JFM | MR

[7] H. Heilbronn, Zeta-functions and L-functions. Algebraic Number Theory, Chapter VIII, J.W.S. Cassels and A. Fröhlich, Academic Press, (1967). | MR

[8] G. James, M. Liebeck, Representations and Characters of groups. Cambridge University Press, (1993). | MR | Zbl

[9] KANT V4, by M. Daberkow, C. Fieker, J. Klüners, M. Pohst, K. Roegner, M. Schörnig, and K. Wildanger, J. Symbolic Computation 24 (1997),267-283. | MR | Zbl

[10] H. Koch, Über den 2-Klassenkörperturm eines quadratischen Zahlkörpers. I, J. Reine Angew. Math. 214/215 (1963), 201-206. | MR | Zbl

[11] P. Kolvenbach, Zur arithmetischen Theorie der SL(2, 3)-Erweiterungen. Diss. Köln, 1982.

[12] S.-H. Kwon, Corps de nombres de degré 4 de type alterné. C. R. Acad. Sci. Paris 299 (1984), 41-43. | MR | Zbl

[13] S. Lang, Cyclotomic Fields II. Springer-Verlag 1980 | MR | Zbl

[14] F. Lemmermeyer, Ideal class groups of cyclotomic number fields I. Acta Arith. 72 (1995), 347-359. | MR | Zbl

[15] F. Lemmermeyer, Unramified quaternion extensions of quadratic number fields. J. Théor. N. Bordeaux 9 (1997), 51-68. | Numdam | MR | Zbl

[16] Y. Lefeuvre, Corps diédraux à multiplication complexe principaux. Ann. Inst. Fourier, à paraitre. | Numdam | Zbl

[17] 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

[18] 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, No.3 (1998), 523-548. | MR | Zbl

[19] 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

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

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

[22] S. Louboutin, The class number one problem for the non-abelian normal CM-fields of degree 16. Acta Arith. 82 (1997), 173-196. | MR | Zbl

[23] S. Louboutin, Upper bounds on |L(1, χ)| and applications. Canad. J. Math., 50 (4), (1998), 794-815. | Zbl

[24] R. Okazaki, Inclusion of CM-fields and divisibility of class numbers. Acta Arith., à paraître. | Zbl

[25] User's Guide to PARI-GP, by C. Batut,K. Belabas,D. Bernardi,H. CohenandM. Olivierlast updated Nov. 14, 1997; Laboratoire A2X, Univ. Bordeaux I.

[26] Y.-H. Park, S.-H. Kwon, Determination of all imaginary abelian sextic number fields with class number < 11. Acta Arith. 82 (1997), 27-43. | MR | Zbl

[27] A. Rio, Dyadic exercises for octahedral extensions. preprint 1997; URL: http://www-ma2. upc. as/~rio/papers. html

[28] H.M. Stark, Some effective cases of the Brauer-Siegel theorem. Invent. Math. 23 (1974), 135-152. | MR | Zbl

[29] O. Taussky, A remark on the class field tower. J. London Math. Soc. 12 (1937), 82-85. | JFM | Zbl

[30] A.D. Thomas, G.V. Wood, Group Tables, Shiva Publishing Ltd, Kent, UK 1980. | MR | Zbl

[31] L.C. Washington, Introduction to Cyclotomic Fields. Grad. Texts Math. 83, Springer-Verlag 1982; 2nd edition 1997. | MR | Zbl

[32] A. Weil, Exercices dyadiques. Invent. Math. 27 (1974), 1-22. | MR | Zbl