Sur la dimension cohomologique des pro- p -extensions des corps de nombres

Christian Maire[1]

  • [1] GRIMM Université Toulouse le Mirail 5, allées A. Machado 31058 Toulouse cédex

Journal de Théorie des Nombres de Bordeaux (2005)

  • Volume: 17, Issue: 2, page 575-606
  • ISSN: 1246-7405

Abstract

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In this paper, we study the cohomological dimension of groups G S : = Gal ( 𝕂 S / 𝕂 ) , where 𝕂 S is the maximal pro- p -extension of a number field 𝕂 , unramified outside a finite set S of places of 𝕂 . This dimension is well-understood only when S contains all places above p ; in the case where only some of the places above p are contained in S , one can still obtain some results if 𝕂 S / 𝕂 contains at least one ℤ p -extension 𝕂 ∞ / 𝕂 . Indeed, in that case, the study of the ℤ p [ [ Gal ( 𝕂 ∞ / 𝕂 ) ] ] -module Gal ( 𝕂 S / 𝕂 ∞ ) a b allows one to give sufficient conditions for the pro- p -group Gal ( 𝕂 S / 𝕂 ∞ ) to be free. Under the latter condition, the dimension of G S is at most 2. Here, we develop an explicit strategy for realizing these conditions so as to produce numerical examples for which we effectively compute this cohomological dimension.

How to cite

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Maire, Christian. "Sur la dimension cohomologique des pro-$p$-extensions des corps de nombres." Journal de Théorie des Nombres de Bordeaux 17.2 (2005): 575-606. <http://eudml.org/doc/249422>.

@article{Maire2005,
abstract = {In this paper, we study the cohomological dimension of groups $G_S:=\operatorname\{Gal\}(\mathbb\{K\}_S/\mathbb\{K\})$, where $\mathbb\{K\}_S$ is the maximal pro-$p$-extension of a number field $\mathbb\{K\}$, unramified outside a finite set $S$ of places of $\mathbb\{K\}$. This dimension is well-understood only when $S$ contains all places above $p$; in the case where only some of the places above $p$ are contained in $S$, one can still obtain some results if $\mathbb\{K\}_S/\mathbb\{K\}$ contains at least one $\mathbb\{Z\}_p$-extension $\mathbb\{K\}_\infty /\mathbb\{K\}$. Indeed, in that case, the study of the $\mathbb\{Z\}_p[[\operatorname\{Gal\}(\mathbb\{K\}_\infty /\mathbb\{K\})]]$-module $\operatorname\{Gal\}(\mathbb\{K\}_S/\mathbb\{K\}_\infty )^\{ab\}$ allows one to give sufficient conditions for the pro-$p$-group $\operatorname\{Gal\}(\mathbb\{K\}_S/\mathbb\{K\}_\infty )$ to be free. Under the latter condition, the dimension of $G_S$ is at most 2. Here, we develop an explicit strategy for realizing these conditions so as to produce numerical examples for which we effectively compute this cohomological dimension.},
affiliation = {GRIMM Université Toulouse le Mirail 5, allées A. Machado 31058 Toulouse cédex},
author = {Maire, Christian},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {conditions of Leopoldt type; Spiegelungssatz; -extensions; Gras logarithm},
language = {eng},
number = {2},
pages = {575-606},
publisher = {Université Bordeaux 1},
title = {Sur la dimension cohomologique des pro-$p$-extensions des corps de nombres},
url = {http://eudml.org/doc/249422},
volume = {17},
year = {2005},
}

TY - JOUR
AU - Maire, Christian
TI - Sur la dimension cohomologique des pro-$p$-extensions des corps de nombres
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2005
PB - Université Bordeaux 1
VL - 17
IS - 2
SP - 575
EP - 606
AB - In this paper, we study the cohomological dimension of groups $G_S:=\operatorname{Gal}(\mathbb{K}_S/\mathbb{K})$, where $\mathbb{K}_S$ is the maximal pro-$p$-extension of a number field $\mathbb{K}$, unramified outside a finite set $S$ of places of $\mathbb{K}$. This dimension is well-understood only when $S$ contains all places above $p$; in the case where only some of the places above $p$ are contained in $S$, one can still obtain some results if $\mathbb{K}_S/\mathbb{K}$ contains at least one $\mathbb{Z}_p$-extension $\mathbb{K}_\infty /\mathbb{K}$. Indeed, in that case, the study of the $\mathbb{Z}_p[[\operatorname{Gal}(\mathbb{K}_\infty /\mathbb{K})]]$-module $\operatorname{Gal}(\mathbb{K}_S/\mathbb{K}_\infty )^{ab}$ allows one to give sufficient conditions for the pro-$p$-group $\operatorname{Gal}(\mathbb{K}_S/\mathbb{K}_\infty )$ to be free. Under the latter condition, the dimension of $G_S$ is at most 2. Here, we develop an explicit strategy for realizing these conditions so as to produce numerical examples for which we effectively compute this cohomological dimension.
LA - eng
KW - conditions of Leopoldt type; Spiegelungssatz; -extensions; Gras logarithm
UR - http://eudml.org/doc/249422
ER -

References

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  1. C. Batut, K. Belabas, H. Cohen, M. Olivier, User’s guide to PARI-GP. A2X, Université Bordeaux I, 1999. 
  2. N. Boston, Explicit Galois groups of infinite p -extensions unramified outside p , preprint 2003. 
  3. N. Boston, Some cases of the Fontaine-Mazur conjecture. Journal of Number Theory 42 (1992), 285–291. Zbl0768.11044MR1189506
  4. N. Boston, Some cases of the Fontaine-Mazur conjecture II. Journal of Number Theory 75 (1999), 161–169. Zbl0928.11050MR1681626
  5. J.-M. Fontaine, B. Mazur, Geometric Galois representations. Elliptic curves, modular forms and Fermat’s last theorem, Internat. Press, Cambridge, MA, 1995. Zbl0839.14011
  6. R. Gillard, Fonctions L p -adiques des corps quadratiques imaginaires et de leurs extensions abéliennes. Crelle 358 (1985), 76–91. Zbl0551.12011MR797675
  7. G. Gras, Class Field Theory. SMM, Springer 2003. Zbl1019.11032MR1941965
  8. G. Gras, Groupe de Galois de la p -extension abélienne p -ramifiée maximale d’un corps de nombres. Crelle 333 (1982), 86–132. Zbl0477.12009
  9. G. Gras, Plongements kummériens dans les ℤ p -extensions. Compositio Math. 55 (1985), 383–396. Zbl0584.12004MR799822
  10. G. Gras, J.-F. Jaulent, Sur les corps de nombres réguliers. Math. Z. 202 (1989), 343–365. Zbl0704.11040MR1017575
  11. K. Haberland, Galois cohomology of algebraic number fields, With two appendices by Helmut Koch and Thomas Zink. VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. Zbl0418.12004MR519872
  12. F. Hajir, C. Maire, Extensions of number fields with ramification of bounded depth. International Math. Research Notices 13 (2002), 667–696. Zbl1160.11358MR1890847
  13. H. Hasse, Number Theory. Classics in Mathematics, Springer, 1980. Zbl0991.11001MR562104
  14. J.-F. Jaulent, L’arithmétique des l -extensions. Publications Math. de Besançon, 1986. Zbl0601.12002
  15. J.-F. Jaulent, C. Maire, Sur les invariants d’Iwasawa des tours cyclotomiques. Canadian Math. Bulletin 46 (2003), 178–190. Zbl1155.11353
  16. J.-F. Jaulent, T. Nguyen Quang Do, Corps p -rationnels, corps réguliers et ramification restreinte. J. Théorie des Nombres de Bordeaux 5 (1993), 343–363. Zbl0957.11046MR1265910
  17. J.-F. Jaulent, O. Sauzet, Pro- l -extensions de corps de nombres l -rationnels. Jour. Number Theory 65 (1997), 240–267. Zbl0896.11043MR1462840
  18. H. Koch, Galoissche Theorie der p -Erweiterungen. VEB, Berlin, 1970. Zbl0216.04704MR291139
  19. J. Labute, Mild pro- p -groups and Galois groups of p -extensions of ℚ . J. Reine Angew. Math., à paraître. Zbl1122.11076
  20. C. Maire, On the ℤ l -rank of abelian extensions with restricted ramification. Journal of Number Theory 92 (2002), 376–404. Zbl1026.11084MR1884709
  21. A. Movahhedi, Sur les p -extensions des corps p -rationnels. Math. Nach. 149 (1990), 163–176. Zbl0723.11054MR1124802
  22. A. Movahhedi, T. Nguyen Quang Do, Sur l’arithmétique des corps de nombres p -rationnels. Sém. Théorie des Nombres, Paris (1987/89), Prog. Math 81, 155–200. Zbl0703.11059
  23. J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields. Grundlehren 323, Springer-Verlag 2002. Zbl0948.11001MR1737196
  24. T. Nguyen Quang Do, Formations de classes et modules d’Iwasawa. Number Theory, Noordwijkerhout 1983, Lecture Notes in Math. 1068, 167–185. Zbl0543.12007
  25. A. Schmidt, On the relation between 2 and ∞ in Galois Cohomology of Number Fields. Compositio Math. 133 (2002), 267–288. Zbl1021.11029MR1930978
  26. A. Schmidt, Circular sets of prime numbers and p -extensions of the rationals, preprint, 2005. MR2254808
  27. A. Schmidt, Bounded defect in partial Euler characteristics. Bull. London Math. Soc. 28 (1996), 463–464. Zbl0866.20019MR1396144
  28. L. Schneps, On the μ -invariant of p -adic L -functions attached to elliptic curves with complex multiplication. Journal of Number Theory 25 (1987), 20–33. Zbl0615.12018MR871166
  29. J.-P. Serre, Galois Cohomology. Lecture Notes in Math., Springer-Verlag, Berlin, 1994. Zbl1004.12003MR1324577
  30. J.-P. Serre, Corps locaux. Publications de l’Université de Nancago, Hermann, Paris, 1968. Zbl0423.12017
  31. L.C. Washington, Introduction to cyclotomic fields. GTM 83, Springer 1997. Zbl0966.11047MR1421575
  32. K. Wingberg, Galois groups of number fields generated by torsion points of elliptic curves. Nagoya Math. J 104 (1986), 43–53. Zbl0621.12011MR868436

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