On the maximal unramified pro-2-extension over the cyclotomic 2 -extension of an imaginary quadratic field
Journal de Théorie des Nombres de Bordeaux, Tome 22 (2010) no. 1, pp. 115-138.

Pour k quadratique imaginaire, nous étudions le groupe de Galois G(k ) de la pro-2-extension non ramifiée maximale au-dessus de la 2 -extension cyclotomique k de k. Nous déterminons des familles de tels corps imaginaires k pour lesquels G(k ) est un pro-2-groupe métabélien et en donnons une présentation explicite ; nous précisons de même des familles pour lesquelles G(k ) est un pro-2-groupe métacyclique non abélien. Nous calculons enfin en termes de Théorie d’Iwasawa les groupes de Galois de 2-tours de corps de classes de certaines 2-extensions cyclotomiques.

For the cyclotomic 2 -extension k of an imaginary quadratic field k, we consider the Galois group G(k ) of the maximal unramified pro-2-extension over k . In this paper, we give some families of k for which G(k ) is a metabelian pro-2-group with the explicit presentation, and determine the case that G(k ) becomes a nonabelian metacyclic pro-2-group. We also calculate Iwasawa theoretically the Galois groups of 2-class field towers of certain cyclotomic 2-extensions.

Reçu le :
Publié le :
DOI : https://doi.org/10.5802/jtnb.707
@article{JTNB_2010__22_1_115_0,
     author = {Yasushi Mizusawa},
     title = {On the maximal unramified pro-2-extension over the cyclotomic $\mathbb{Z}_2$-extension of an imaginary quadratic field},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {115--138},
     publisher = {Universit\'e Bordeaux 1},
     volume = {22},
     number = {1},
     year = {2010},
     doi = {10.5802/jtnb.707},
     zbl = {1221.11215},
     mrnumber = {2675876},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.707/}
}
Yasushi Mizusawa. On the maximal unramified pro-2-extension over the cyclotomic $\mathbb{Z}_2$-extension of an imaginary quadratic field. Journal de Théorie des Nombres de Bordeaux, Tome 22 (2010) no. 1, pp. 115-138. doi : 10.5802/jtnb.707. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.707/

[1] E. Benjamin, F. Lemmermeyer and C. Snyder, Imaginary quadratic fields k with cyclic Cl 2 (k 1 ). J. Number Theory 67 (1997), no. 2, 229–245. | MR 1486501 | Zbl 0919.11074

[2] E. Benjamin, F. Lemmermeyer and C. Snyder, Real quadratic fields with abelian 2-class field tower. J. Number Theory 73 (1998), no. 2, 182–194. | MR 1658015 | Zbl 0919.11073

[3] E. Benjamin, F. Lemmermeyer and C. Snyder, Imaginary quadratic fields k with Cl 2 (k)(2,2 m ) and rank Cl 2 (k 1 )=2. Pacific J. Math. 198 (2001), no. 1, 15–31. | MR 1831970 | Zbl 1063.11038

[4] E. Benjamin, F. Lemmermeyer and C. Snyder, Imaginary quadratic fields with Cl 2 (k)(2,2,2). J. Number Theory 103 (2003), no. 1, 38–70. | MR 2008065 | Zbl 1045.11077

[5] N. Boston, Galois groups of tamely ramified p-extensions. J. Théor. Nombres Bordeaux 19 (2007), no. 1, 59–70. | Numdam | MR 2332053 | Zbl 1123.11038

[6] M. R. Bush, Computation of Galois groups associated to the 2-class towers of some quadratic fields. J. Number Theory 100 (2003), no. 2, 313–325. | MR 1978459 | Zbl 1039.11091

[7] J. D. Dixon, M. P. F. du Sautoy, A. Mann and D. Segal, Analytic pro-p groups. Second edition. Cambridge Studies in Advanced Mathematics 61, Cambridge University Press, Cambridge, 1999. | MR 1720368 | Zbl 0934.20001

[8] B. Ferrero, The cyclotomic 2 -extension of imaginary quadratic fields. Amer. J. Math. 102 (1980), no. 3, 447–459. | MR 573095 | Zbl 0463.12002

[9] B. Ferrero and L. C. Washington, The Iwasawa invariant μ p vanishes for abelian number fields. Ann. of Math. 109 (1979), no. 2, 377–395. | MR 528968 | Zbl 0443.12001

[10] S. Fujii, On a higher class number formula of p -extensions. Tokyo J. Math. 28 (2005), no. 1, 55–61. | MR 2149622 | Zbl 1074.11060

[11] S. Fujii, Non-abelian Iwasawa theory of cyclotomic p -extensions. The COE Seminar on Mathematical Sciences 2007, 85–97, Sem. Math. Sci. 37, Keio Univ., Yokohama, 2008. | MR 2478430 | Zbl 1159.11041

[12] S. Fujii and K. Okano, Some problems on p-class field towers. Tokyo J. Math. 30 (2007), no. 1, 211–222. | MR 2328064 | Zbl pre05202540

[13] E. S. Golod and I. R. Shafarevich, On the class field tower. Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 261–272. | MR 161852 | Zbl 0136.02602

[14] R. Greenberg, On the Iwasawa invariants of totally real number fields. Amer. J. Math. 98 (1976), no. 1, 263–284. | MR 401702 | Zbl 0334.12013

[15] F. Hajir, On a theorem of Koch. Pacific J. Math. 176 (1996), no. 1, 15–18. Correction: 196 (2000), no. 2, 507–508. | MR 1433980

[16] H. Ichimura and H. Sumida, On the Iwasawa invariants of certain real abelian fields II. Inter. J. Math. 7 (1996), no. 6, 721–744. | MR 1417782 | Zbl 0881.11075

[17] T. Itoh, Pseudo-null Iwasawa modules for 2 2 -extensions. Tokyo J. Math. 30 (2007), no. 1, 199–209. | MR 2328063 | Zbl pre05202539

[18] K. Iwasawa, On l -extensions of algebraic number fields. Ann. of Math. (2) 98 (1973), 246–326. | MR 349627 | Zbl 0285.12008

[19] Y. Kida, On cyclotomic 2 -extensions of imaginary quadratic fields. Tohoku Math. J. (2) 31 (1979), no. 1, 91–96. | MR 526512 | Zbl 0408.12006

[20] Y. Kida, Cyclotomic 2 -extensions of J-fields. J. Number Theory 14 (1982), no. 3, 340–352. | MR 660379 | Zbl 0493.12015

[21] H. Kisilevsky, Number fields with class number congruent to 4 mod 8 and Hilbert’s theorem 94. J. Number Theory 8 (1976), no. 3, 271–279. | MR 417128 | Zbl 0334.12019

[22] M. Lazard, Groupes analytiques p-adiques. Inst. Hautes Études Sci. Publ. Math. 26 (1965), 389–603. | Numdam | MR 209286 | Zbl 0139.02302

[23] F. Lemmermeyer, On 2-class field towers of imaginary quadratic number fields. J. Théor. Nombres Bordeaux 6 (1994), no. 2, 261–272. | Numdam | MR 1360645 | Zbl 0826.11052

[24] F. Lemmermeyer, Ideal class groups of cyclotomic number fields I. Acta Arith. 72 (1995), no. 4, 347–359. | MR 1348202 | Zbl 0837.11059

[25] B. Mazur and A. Wiles, Class fields of abelian extensions of . Invent. Math. 76 (1984), no. 2, 179–330. | MR 742853 | Zbl 0545.12005

[26] Y. Mizusawa, On the maximal unramified pro-2-extension of 2 -extensions of certain real quadratic fields II. Acta Arith. 119 (2005), no. 1, 93–107. | MR 2163520 | Zbl 1151.11055

[27] Y. Mizusawa and M. Ozaki, Abelian 2-class field towers over the cyclotomic 2 -extensions of imaginary quadratic fields. Math. Ann. 347 (2010, no. 2, 437–453. | MR 2606944 | Zbl pre05700555

[28] K. Okano, Abelian p-class field towers over the cyclotomic p -extensions of imaginary quadratic fields. Acta Arith. 125 (2006), no. 4, 363–381. | MR 2271283 | Zbl 1155.11051

[29] M. Ozaki, Non-Abelian Iwasawa theory of p -extensions. (Japanese) Young philosophers in number theory (Kyoto, 2001), RIMS Kôkyûroku 1256 (2002), 25–37. | MR 1924192

[30] M. Ozaki, Non-Abelian Iwasawa theory of p -extensions. J. Reine Angew. Math. 602 (2007), 59–94. | MR 2300452 | Zbl 1123.11034

[31] M. Ozaki and H. Taya, On the Iwasawa λ 2 -invariants of certain families of real quadratic fields. Manuscripta Math. 94 (1997), no. 4, 437–444. | MR 1484637 | Zbl 0935.11040

[32] J.-P. Serre, Galois cohomology. Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002. | MR 1867431 | Zbl 1004.12003

[33] R. T. Sharifi, On Galois groups of unramified pro-p extensions. Math. Ann. 342 (2008) 297–308. | MR 2425144 | Zbl 1165.11078

[34] L. C. Washington, Introduction to Cyclotomic Fields (2nd edition). Graduate Texts in Math. vol. 83, Springer, 1997. | MR 1421575 | Zbl 0966.11047

[35] A. Wiles, The Iwasawa conjecture for totally real fields. Ann. of Math. (2) 131 (1990), no. 3, 493–540. | MR 1053488 | Zbl 0719.11071

[36] K. Wingberg, On the Fontaine-Mazur conjecture for CM-fields. Compositio Math. 131 (2002), no. 3, 341–354. | MR 1905027 | Zbl 1038.11074

[37] Y. Yamamoto, Divisibility by 16 of class number of quadratic fields whose 2-class groups are cyclic. Osaka J. Math. 21 (1984), no. 1, 1–22. | MR 736966 | Zbl 0535.12002

[38] K. Yamamura, Maximal unramified extensions of imaginary quadratic number fields of small conductors. J. Théor. Nombres Bordeaux 9 (1997), no. 2, 405–448. | Numdam | MR 1617407 | Zbl 0905.11048