On the maximal unramified pro-2-extension over the cyclotomic ℤ 2 -extension of an imaginary quadratic field

Yasushi Mizusawa[1]

  • [1] Department of Mathematics Nagoya Institute of Technology Gokiso, Showa, Nagoya, Aichi 466-8555, JAPAN

Journal de Théorie des Nombres de Bordeaux (2010)

  • Volume: 22, Issue: 1, page 115-138
  • ISSN: 1246-7405

Abstract

top
For the cyclotomic ℤ 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.

How to cite

top

Mizusawa, Yasushi. "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 22.1 (2010): 115-138. <http://eudml.org/doc/116391>.

@article{Mizusawa2010,
abstract = {For the cyclotomic $\mathbb\{Z\}_2$-extension $k_\{\infty \}$ of an imaginary quadratic field $k$, we consider the Galois group $G(k_\{\infty \})$ of the maximal unramified pro-$2$-extension over $k_\{\infty \}$. In this paper, we give some families of $k$ for which $G(k_\{\infty \})$ is a metabelian pro-$2$-group with the explicit presentation, and determine the case that $G(k_\{\infty \})$ 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.},
affiliation = {Department of Mathematics Nagoya Institute of Technology Gokiso, Showa, Nagoya, Aichi 466-8555, JAPAN},
author = {Mizusawa, Yasushi},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Iwasawa theory; unramified abelian extensions; pro-p-groups},
language = {eng},
number = {1},
pages = {115-138},
publisher = {Université Bordeaux 1},
title = {On the maximal unramified pro-2-extension over the cyclotomic $\mathbb\{Z\}_2$-extension of an imaginary quadratic field},
url = {http://eudml.org/doc/116391},
volume = {22},
year = {2010},
}

TY - JOUR
AU - Mizusawa, Yasushi
TI - On the maximal unramified pro-2-extension over the cyclotomic $\mathbb{Z}_2$-extension of an imaginary quadratic field
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2010
PB - Université Bordeaux 1
VL - 22
IS - 1
SP - 115
EP - 138
AB - For the cyclotomic $\mathbb{Z}_2$-extension $k_{\infty }$ of an imaginary quadratic field $k$, we consider the Galois group $G(k_{\infty })$ of the maximal unramified pro-$2$-extension over $k_{\infty }$. In this paper, we give some families of $k$ for which $G(k_{\infty })$ is a metabelian pro-$2$-group with the explicit presentation, and determine the case that $G(k_{\infty })$ 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.
LA - eng
KW - Iwasawa theory; unramified abelian extensions; pro-p-groups
UR - http://eudml.org/doc/116391
ER -

References

top
  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. Zbl0919.11074MR1486501
  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. Zbl0919.11073MR1658015
  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. Zbl1063.11038MR1831970
  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. Zbl1045.11077MR2008065
  5. N. Boston, Galois groups of tamely ramified p -extensions. J. Théor. Nombres Bordeaux 19 (2007), no. 1, 59–70. Zbl1123.11038MR2332053
  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. Zbl1039.11091MR1978459
  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. Zbl0934.20001MR1720368
  8. B. Ferrero, The cyclotomic ℤ 2 -extension of imaginary quadratic fields. Amer. J. Math. 102 (1980), no. 3, 447–459. Zbl0463.12002MR573095
  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. Zbl0443.12001MR528968
  10. S. Fujii, On a higher class number formula of ℤ p -extensions. Tokyo J. Math. 28 (2005), no. 1, 55–61. Zbl1074.11060MR2149622
  11. S. Fujii, Non-abelian Iwasawa theory of cyclotomic ℤ p -extensions. The COE Seminar on Mathematical Sciences 2007, 85–97, Sem. Math. Sci. 37, Keio Univ., Yokohama, 2008. Zbl1159.11041MR2478430
  12. S. Fujii and K. Okano, Some problems on p -class field towers. Tokyo J. Math. 30 (2007), no. 1, 211–222. Zbl1245.11110MR2328064
  13. E. S. Golod and I. R. Shafarevich, On the class field tower. Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 261–272. Zbl0136.02602MR161852
  14. R. Greenberg, On the Iwasawa invariants of totally real number fields. Amer. J. Math. 98 (1976), no. 1, 263–284. Zbl0334.12013MR401702
  15. F. Hajir, On a theorem of Koch. Pacific J. Math. 176 (1996), no. 1, 15–18. Correction: 196 (2000), no. 2, 507–508. Zbl0879.11066MR1433980
  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. Zbl0881.11075MR1417782
  17. T. Itoh, Pseudo-null Iwasawa modules for ℤ 2 2 -extensions. Tokyo J. Math. 30 (2007), no. 1, 199–209. Zbl1234.11147MR2328063
  18. K. Iwasawa, On ℤ l -extensions of algebraic number fields. Ann. of Math. (2) 98 (1973), 246–326. Zbl0285.12008MR349627
  19. Y. Kida, On cyclotomic ℤ 2 -extensions of imaginary quadratic fields. Tohoku Math. J. (2) 31 (1979), no. 1, 91–96. Zbl0408.12006MR526512
  20. Y. Kida, Cyclotomic ℤ 2 -extensions of J -fields. J. Number Theory 14 (1982), no. 3, 340–352. Zbl0493.12015MR660379
  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. Zbl0334.12019MR417128
  22. M. Lazard, Groupes analytiques p -adiques. Inst. Hautes Études Sci. Publ. Math. 26 (1965), 389–603. Zbl0139.02302MR209286
  23. F. Lemmermeyer, On 2 -class field towers of imaginary quadratic number fields. J. Théor. Nombres Bordeaux 6 (1994), no. 2, 261–272. Zbl0826.11052MR1360645
  24. F. Lemmermeyer, Ideal class groups of cyclotomic number fields I. Acta Arith. 72 (1995), no. 4, 347–359. Zbl0837.11059MR1348202
  25. B. Mazur and A. Wiles, Class fields of abelian extensions of ℚ . Invent. Math. 76 (1984), no. 2, 179–330. Zbl0545.12005MR742853
  26. Y. Mizusawa, On the maximal unramified pro- 2 -extension of ℤ 2 -extensions of certain real quadratic fields II. Acta Arith. 119 (2005), no. 1, 93–107. Zbl1151.11055MR2163520
  27. Y. Mizusawa and M. Ozaki, Abelian 2 -class field towers over the cyclotomic ℤ 2 -extensions of imaginary quadratic fields. Math. Ann. 347 (2010, no. 2, 437–453. Zbl1239.11121MR2606944
  28. K. Okano, Abelian p -class field towers over the cyclotomic ℤ p -extensions of imaginary quadratic fields. Acta Arith. 125 (2006), no. 4, 363–381. Zbl1155.11051MR2271283
  29. M. Ozaki, Non-Abelian Iwasawa theory of ℤ p -extensions. (Japanese) Young philosophers in number theory (Kyoto, 2001), RIMS Kôkyûroku 1256 (2002), 25–37. MR1924192
  30. M. Ozaki, Non-Abelian Iwasawa theory of ℤ p -extensions. J. Reine Angew. Math. 602 (2007), 59–94. Zbl1123.11034MR2300452
  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. Zbl0935.11040MR1484637
  32. J.-P. Serre, Galois cohomology. Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002. Zbl1004.12003MR1867431
  33. R. T. Sharifi, On Galois groups of unramified pro- p extensions. Math. Ann. 342 (2008) 297–308. Zbl1165.11078MR2425144
  34. L. C. Washington, Introduction to Cyclotomic Fields (2nd edition). Graduate Texts in Math. vol. 83, Springer, 1997. Zbl0966.11047MR1421575
  35. A. Wiles, The Iwasawa conjecture for totally real fields. Ann. of Math. (2) 131 (1990), no. 3, 493–540. Zbl0719.11071MR1053488
  36. K. Wingberg, On the Fontaine-Mazur conjecture for CM-fields. Compositio Math. 131 (2002), no. 3, 341–354. Zbl1038.11074MR1905027
  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. Zbl0535.12002MR736966
  38. K. Yamamura, Maximal unramified extensions of imaginary quadratic number fields of small conductors. J. Théor. Nombres Bordeaux 9 (1997), no. 2, 405–448. Zbl0905.11048MR1617407

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.