Finite Λ-submodules of Iwasawa modules for a CM-field for p=2
Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 1017-1035.

Let p be a prime, X F - the minus quotient of the Iwasawa module, which we define to be the Galois group of the maximal unramified abelian pro-p-extension over the cyclotomic p -extension over a CM field F. If p is an odd prime, it is well known that X F - has no non-trivial finite p Gal(F /F)-submodule. But X F - has non-trivial finite p Gal(F /F)-submodule in some cases for p=2. In this paper, we study the maximal finite p Gal(F /F)-submodule of X F - for p=2. We determine the size of the maximal finite 2 Gal(F /F)-submodule of X F - under some mild assumptions.

Soit F un corps CM et p un nombre premier. Soit X F - le quotient “moins” du groupe de Galois de la pro-p-extension abélienne non ramifiée maximale de la p -extension cyclotomique de F. Si p ne vaut pas 2, il est bien connu que X F - n’a pas de sous-module fini non-trivial. Mais pour p=2, il peut arriver que X F - contient un sous-module fini non-trivial. Dans cet article, nous étudions le sous-module fini maximal de X F - pour p=2, et nous déterminons ce module sous certaines légères hypothèses.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1063
Classification: 11N56,  14G42
Keywords: Iwasawa theory, Iwasawa module, Galois module structure
Mahiro Atsuta 1

1 Department of Mathematics Keio University 3-14-1 Hiyoshi, Kohoku-ku Yokohama, 223-8522, Japan
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{JTNB_2018__30_3_1017_0,
     author = {Mahiro Atsuta},
     title = {Finite $\Lambda $-submodules of {Iwasawa} modules for a {CM-field} for $p=2$},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {1017--1035},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {30},
     number = {3},
     year = {2018},
     doi = {10.5802/jtnb.1063},
     zbl = {1435.11137},
     mrnumber = {3938640},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1063/}
}
TY  - JOUR
AU  - Mahiro Atsuta
TI  - Finite $\Lambda $-submodules of Iwasawa modules for a CM-field for $p=2$
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2018
DA  - 2018///
SP  - 1017
EP  - 1035
VL  - 30
IS  - 3
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1063/
UR  - https://zbmath.org/?q=an%3A1435.11137
UR  - https://www.ams.org/mathscinet-getitem?mr=3938640
UR  - https://doi.org/10.5802/jtnb.1063
DO  - 10.5802/jtnb.1063
LA  - en
ID  - JTNB_2018__30_3_1017_0
ER  - 
%0 Journal Article
%A Mahiro Atsuta
%T Finite $\Lambda $-submodules of Iwasawa modules for a CM-field for $p=2$
%J Journal de théorie des nombres de Bordeaux
%D 2018
%P 1017-1035
%V 30
%N 3
%I Société Arithmétique de Bordeaux
%U https://doi.org/10.5802/jtnb.1063
%R 10.5802/jtnb.1063
%G en
%F JTNB_2018__30_3_1017_0
Mahiro Atsuta. Finite $\Lambda $-submodules of Iwasawa modules for a CM-field for $p=2$. Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 1017-1035. doi : 10.5802/jtnb.1063. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1063/

[1] Bruce Ferrero The cyclotomic 2 -extension of imaginary quadratic fields, Am. J. Math., Volume 102 (1980), pp. 447-459 | DOI | MR | Zbl

[2] Ralph Greenberg On the Iwasawa invariants of totally real number fields, Am. J. Math., Volume 98 (1976), pp. 263-284 | DOI | MR | Zbl

[3] Ralph Greenberg On the structure of certain Galois cohomology groups, Doc. Math., Volume Extra Volume (2006), pp. 357-413 | MR | Zbl

[4] Ralph Greenberg On the structure of Selmer groups, Elliptic curves, modular forms and Iwasawa theory. In honour of John H. Coates’ 70th birthday (Springer Proceedings in Mathematics & Statistics), Volume 188, Springer, 2016, pp. 225-252 | DOI | MR | Zbl

[5] Helmut Hasse Über die Klassenzahl abelscher Zahlkörper, Mathematische Lehrbücher und Monographien, 1, Akademie-Verlag, 1952 | Zbl

[6] Kenkichi Iwasawa On l -extensions of algebraic number fields, Ann. Math., Volume 98 (1973), pp. 246-326 | DOI | Zbl

[7] Franz Lemmermeyer Ideal class groups of cyclotomic number fields I, Acta Arith., Volume 72 (1984) no. 2, pp. 347-359 | MR | Zbl

[8] Jürgen Neukirch; Alexander Schmidt; Kay Wingberg Cohomology of number fields, Grundlehren der Mathematischen Wissenschaften, 323, Springer, 2008 | MR | Zbl

[9] Lawrence C. Washington Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, 83, Springer, 1997 | MR | Zbl

Cited by Sources: