Abstract.
Let p be an odd prime. For any CM number field K containing a primitive pth-root of unity, class field theory and Kummer theory put together yield the well known reflection inequality λ+≤λ− between the “plus” and “minus” parts of the λ-invariant of K. Greenberg’s conjecture asserts that λ+ is always trivial. We study here a weak form of this conjecture, namely λ+=λ− if and only if λ+=λ−=0.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anglès, B.: Units and norm residue symbol. Acta Arithmetica 98 (1), 33–51 (2001)
Assim, J. et, Nguyen Quang Do, T.: Sur la constante de Kummer-Leopoldt d’un corps de nombres. à paraître dans Manuscripta Math
Badino, R.: “Sur les égalités du miroir en théorie d’Iwasawa”. thèse, Besançon, 2003
Brinkhuis, J.: Normal integral bases problem and the Spiegelungssatz of Scholz. Acta Arithmetica 69 (1), (1995) 1–9
Belliard, J.-R. et, Nguyen Quang Do, T.: On modified circular units and annihilation of real classes. à paraître dans Nagoya Math. J.
Coates, J.: p-adic L-functions and Iwasawa’s Theory. In: “Algebraic number fields” (Durham Symp. 1975) A. Fröhlich (ed.), Academic Press, London, 1977, pp. 269–353
Federer, L.J., Gross, B.H.: Regulators and Iwasawa modules. With an appendix by W. Sinnott, Inventiones Math. 62, (1981) (3), 443–457
Gras, G.: χ-composantes. GROTA, Besançon, 1992
Gras, G.: Théorèmes de réflexion. J. Théorie des Nombres de Bordeaux 10 (2), 399–499 (1998)
Greenberg, R.: On the Iwasawa invariants of totally real number fields. Am. J. Math. 98 (1), 263–284 (1976)
Greenberg, R.: A note on K2 and the theory of ℤ p -extensions. Am. J. Math. 100 (6), 1235–1245 (1978)
Ichimura, H.: A note on the Iwasawa λ-invariant of real quadratic fields. Proceedings Japan Acad. Ser. A. Math. Sci. 72 (1), 28–30 (1996).
Ichimura, H., Sakaguchi, K.: The non-vanishing of a certain Kummer character χ m (after C. Soulé), and related topics. In: “Galois Representations and Algebraic Arithmetic Geometry”, Y. Ihara (ed.), Advanced Studies in Pure Math. 12, 53–64 (1987)
Ichimura, H., Sumida, H.: On the Iwasawa invariants of certain real abelian fields II. Internat. J. Math. 7 (6), 721–744 (1996)
Iwasawa, K.: On ℤ l -extensions of algebraic number fields. Ann. Math. 98, 246–326 (1973)
Jaulent, J.-F.: Dualité dans les corps surcirculaires. Sém. Théorie des Nombres, Paris, 1986-87, Progress in Math. 75, Birkhäuser, 1988, pp. 183–220
Kraft, J.S.: Iwasawa invariants of CM fields. J. Number Theory 32 (1), 65–77 (1989)
Kolster, M.: Remarks on étale K-theory and Leopoldt’s conjecture. Sém. Théorie des Nombres, Paris, 1991-92, Progress in Math. 116, Birkhäuser, 1993, pp. 37–62
Kurihara, M.: Some remarks on conjectures about cyclotomic fields and K-groups of ℤ. Compositio Math. 81 (2), 223–236 (1992)
Kuz’min, L.V.: The Tate module of algebraic number fields. Izv. Akad. Nauk SSSR Ser. Mat. 36, 267–327 (1972)
Kolster, M., Nguyen Quang Do, T., Fleckinger, V.: Twisted S-units, p-adic class number formulas and the Lichtenbaum conjectures. Duke Math. J. 84 (3), 679–717 (1996)
Kraft, J.S., Schoof, R.: Computing Iwasawa modules of real quadratic number fields. Compositio Math. 97 (1–2), 135–155 (1995)
Le Floc’h, M., Movahhedi, A., Nguyen Quang Do, T.: On capitulation cokernels in Iwasawa theory. à paraître dans Am. J. Math.
Movahhedi, A. et, Nguyen Quang Do, T.: Sur l’arithmétique des corps de nombres p-rationnels. Séminaire de Théorie des Nombres, Paris, 1987–1988, Progress in Math. 81, Birkhaüser, 1990, pp. 155–200
Nguyen Quang Do, T.: Sur la ℤ p -torsion de certains modules galoisiens. Ann. Inst. Fourier, Grenoble 36 (2), 27–46 (1986)
Nguyen Quang Do, T.: Sur la torsion de certains modules galoisiens II. Séminaire de Théorie des Nombres, Paris, 1986–1987, Progress in Math. 75, Birkhaüser, 1988, pp. 271–297
Ozaki, M.: An application of Iwasawa theory to constructing fields
which have class group with large p-rank. Nagoya Math. J. 169, 179–190 (2003)Ozaki, M., Taya, H.: A note on Greenberg’s conjecture for real abelian number fields. Manuscripta Math. 88 (3), 311–320 (1995)
Schneider, P.: Über gewisse Galoiscohomologiegruppen. Math. Zeit. 168 (2), 181–205 (1979)
Soulé, C.: Perfect forms and the Vandiver conjecture. J. Reine Angew. Math. 517, 209–221 (1999)
Taya, H.: On p-adic zêta functions and ℤ p -extensions of certain totally real number fields. Tohoku Math. J. 51 (1), 21–33 (1999)
Washington, L.C.: “Introduction to cyclotomic fields”. Grad. Texts in Math. 83, Springer-Verlag, second edition, 1997
Wingberg, K.: Duality theorems for Γ-extensions of algebraic number fields. Compositio Math. 55 (3), 333–381 (1985)
which have class group with large p-rank. Nagoya Math. J. 169, 179–190 (2003)Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Badino, R., Do, T. Sur les égalités du miroir et certaines formes faibles de la conjecture de Greenberg. manuscripta math. 116, 323–340 (2005). https://doi.org/10.1007/s00229-004-0531-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-004-0531-4