Unités cyclotomiques, unités semi-locales et ℤ ℓ -extensions. II
Gillard, Roland
Annales de l'Institut Fourier, Tome 29 (1979), p. 1-15 / Harvested from Numdam
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Soient K un corps abélien réel, ℓ un nombre premier, premier à [K:Q] et Y n le quotient du groupe des unités semi-locales de K(1 ℓ n ) par celui des unités cyclotomiques : on donne la structure galoisienne de la limite projective des Y n , généralisant un théorème d’Iwasawa, et on applique ceci à la comparaison de conjecture classique sur la limite projective des groupes de classes.

Let K an abelian number field, ℓ a prime number, prime to [K:Q], Y n the quotient of the group of semi-local units in K(1 ℓ n ) by the group of cyclotomic units. By giving the Galois structure of lim ← Y n , we generalise a theorem of Iwasawa and use this result for comparing classical conjectures about projective limits of class groups.

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     author = {Gillard, Roland},
     title = {Unit\'es cyclotomiques, unit\'es semi-locales et ${\mathbb {Z}}\_\ell $-extensions. II},
     journal = {Annales de l'Institut Fourier},
     volume = {29},
     year = {1979},
     pages = {1-15},
     doi = {10.5802/aif.763},
     mrnumber = {81e:12005b},
     zbl = {0403.12006},
     language = {fr},
     url = {http://dml.mathdoc.fr/item/AIF_1979__29_4_1_0}
}

Gillard, Roland. Unités cyclotomiques, unités semi-locales et ${\mathbb {Z}}_\ell $-extensions. II. Annales de l'Institut Fourier, Tome 29 (1979) pp. 1-15. doi : 10.5802/aif.763. http://gdmltest.u-ga.fr/item/AIF_1979__29_4_1_0/

[1] N. Bourbaki, Éléments de mathématique, Algèbre commutative, Chap. 7, Hermann, Paris, 1965. | Zbl 0141.03501

[2] J. Coates, p-adic L-functions and Iwasawa's theory, Durham conference on Algebraic Number Theory, edited by A. Fröhlich, Academic Press, Londres, 1977. | MR 57 #276 | Zbl 0393.12027

[3] J. Coates et S. Lichtenbaum, On l-adic zeta functions, Ann. of Math., 98 (1973), 498-550. | MR 48 #8445 | Zbl 0279.12005

[4] J. Coates et A. Wiles, On p-adic L-functions and elliptic units, J. Austral. Math. Soc., series A 26 (1978), 1-25. | MR 80a:12007 | Zbl 0442.12007

[5] R. Coleman, Some modules attached to Lubin-Tate groups, à paraître.

[6] B. Ferrero, Iwasawa invariants of abelian number fields, Math. Ann., 234 (1978), 9-24. | MR 58 #5584 | Zbl 0347.12004

[7] B. Ferrero et L. Washington, The Iwasawa invariant µp vanishes for abelian number fields, Ann. of Math., à paraître. | Zbl 0443.12001

[8] R. Gillard, Unités cyclotomiques, unités semi-locales et Zl-extensions, Ann. Inst. Fourier, t. 29, fasc. 1 (1979), 49-79. | Numdam | MR 81e:12005a | Zbl 0387.12002

[9] R. Gillard, Remarques sur les unités cyclotomiques et les unités elliptiques, J. of Numbers Theory, 11, 1 (1979), 21-48. | MR 80j:12004 | Zbl 0405.12008

[10] R. Greenberg, On p-adic L-functions and cyclotomic fields, Nagoya Math. J., 56 (1974), 61-77. | MR 50 #12984 | Zbl 0315.12008

[11] R. Greenberg, On p-adic L-functions and cyclotomic fields II, Nagoya Math. J., 67 (1977), 139-158. | MR 56 #2964 | Zbl 0373.12007

[12] R. Greenberg, On 2-adic L-functions and cyclotomic invariants, Math. Zeit., 159 (1978), 37-45. | MR 57 #16263 | Zbl 0354.12014

[13] K. Iwasawa, On some modules in the theory of cyclotomic fields, J. Math. Soc. Japan, vol. 16, n° 1, (1964), 42-82. | MR 35 #6646 | Zbl 0125.29207

[14] K. Iwasawa, On Zl-extensions of algebraic number fields, Ann. of Math., 98 (1973), 246-326. | MR 50 #2120 | Zbl 0285.12008

[15] K. Iwasawa, Lectures on p-adic L-functions, Ann. Math. Studies, 74, Princeton Univ. Press, 1972. | MR 50 #12974 | Zbl 0236.12001

[16] S. Lang, Cyclotomic fields, Springer Verlag, 1978. | MR 58 #5578 | Zbl 0395.12005