Formules de genres et conjecture de Greenberg
Tóm tắt
Greenberg’s well known conjecture, (GC) for short, asserts that the Iwasawa invariants
$$\lambda $$
and
$$\mu $$
associated to the cyclotomic
$${\mathbb {Z}}_p$$
-extension of any totally real number field F should vanish. In his foundational 1976 paper, Greenberg has shown two necessary and sufficient conditions for (GC) to hold, in two seemingly opposite cases, when p is undecomposed, resp. totally decomposed in F. In this article we present an encompassing approach covering both cases and resting only on “ genus formulas ”, that is (roughly speaking) on formulas which express the order of the Galois (co-)invariants of certain modules along the cyclotomic tower. These modules are akin to class groups, and in the end we obtain several unified criteria, which naturally contain the particular conditions given by Greenberg.
Tài liệu tham khảo
Belliard, J.-R.: Sous-modules d’unités en théorie d’Iwasawa, pp. 1–12. Publ. Math., Besançon (2002)
Belabas, K., Jaulent, J.-F.: The logarithmic class group package in PARI/GB. Publ. Math, Bordeaux (2016)
Badino, R., Nguyen Quang Do, T.: Sur les égalités du miroir et certaines formes faibles de la conjecture de Greenberg. Manuscr. Math. 116, 323–340 (2005)
Belliard, J.-R., Nguyen Quang Do, T.: On modified circular units and annihilation of real classes. Nagoya Math. J. 117, 77–115 (2005)
Federer, L., Gross, B.: Regulator and Iwasawa modules. Invent. Math. 62, 443–457 (1981). (with an appendix by W. Sinnott)
Gillard, R.: Unités cyclotomiques, unités semi-locales et \({\mathbb{Z}}_l\)-extensions. Ann. Inst. Fourier 29(4), 1–15 (1979)
Gras, G.: Class Field Theory: from theory to practice, 2nd edn. Springer Monographs in Math. (2003)
Gras, G.: Approche \(p\)-adique de la conjecture de Greenberg, à paraître dans Annales Math. Blaise Pascal (2017)
Greenberg, R.: On the Iwasawa invariants of totally real number fields. Am. J. Math. 98(1), 263–284 (1976)
Ichimura, H.: A note on the ideal class group of the cyclotomic \({\mathbb{Z}}_p\)-extension of a totally real field. Acta Arithmetica 105(1), 29–34 (2002)
Iwasawa, K.: On \(\mathbb{Z}_l\)-extensions of algebraic number fields. Ann. Math. 98, 246–326 (1973)
Iwasawa, K.: On cohomology groups of units for \({\mathbb{Z}}_p\)-extensions. Am. J. Math. 105(1), 189–2000 (1983)
Jaulent, J.-F.: Classes logarithmiques de corps de nombres. J. Th. des Nombres Bordeaux 6, 301–325 (1994)
Jaulent, J.-F.: Sur les normes cyclotomiques et les conjectures de Leopoldt et de Gross-Kuz’min. Annales Math. Québec, à paraître (2017)
Jaulent, J.-F.: Note sur la conjecture de Greenberg. prépublication (2016)
Iwasawa, K.: On cohomology groups of units for \({\mathbb{Z}}_p\)-extensions. Am. J. Math. 105, 189–200 (1983)
Kuz’min, L.V.: The Tate module for algebraic number fields. Math. USSR Izv. 6(2), 263–321 (1972)
Le Floc’h, M., Movahhedi, A., Nguyen Quang Do, T.: On capitulation cokernels in Iwasawa theory. Am. J. Math. 127, 851–877 (2003)
Movahhedi, A., Nguyen Quang Do, T.: Sur l’arithmétique des corps \(p\)-rationnels. In: Sém. Théorie des Nombres, Paris 1987–1988, vol. 81, pp. 155–200. Birkhaüser Progress in Math
Nguyen Quang Do, T.: Sur la conjecture faible de Greenberg dans le cas abélien \(p\)-décomposé. Int. J. Number Theory 2(1), 1–16 (2006)
Nguyen Quang Do, T.: Sur une forme faible de la conjecture de Greenberg II. Int. J. Number Theory, à paraître (2017)
Nguyen Quang Do, T.: On Greenberg’s generalized conjecture for infinite classes of number fields (2016, preprint)
Nguyen Quang Do, T., Lescop, M.: Iwasawa descent and codescent for units modulo circular units. Pure Appl. Math. Q. 2(2), 465–496 (2006). (with an appendix by J.-R. Belliard)
Ozaki, M.: A note on the capitulation in \(\mathbb{Z}_p\)-extensions. Proc. Jpn. Acad 71(A), 218–219 (1995)
Ozaki, M.: The class group of \({\mathbb{Z}}_p\)-extensions over totally real number fields. Tohoku Math. J. 49, 431–435 (1997)
Sinnott, W.: On the Stickelberger ideal and the circular units of an abelian field. Invent. Math. 62(2), 181–234(1980/81)
Sumida, H.: On capitulation of S-ideals in \({\mathbb{Z}}_p\)-extensions. J. Number Theory 86, 163–174 (2001)
Taya, H.: On p-adic zeta-functions and \({\mathbb{Z}}_p\)-extensions of certain totally real number fields. Tohoku Math. J. 51(1), 21–33 (1999)