×

On asymptotic formulas on \(\mathbb Z_\ell\)-extensions. (Sur les formules asymptotiques le long des \(\mathbb Z_\ell\)-extensions.) (French. English summary) Zbl 1360.11113

Summary: Let \(K_\infty\) be a \(\mathbb Z_\ell\)-extension of a number field \(K\) . We consider the Galois group \(\mathcal C\ell_T^S(K_n)\) of the maximal \(S\)-ramified and \(T\)-split abelian pro-\(\ell\)-extension attached to each layer \(K_n\) of the tower \(K_\infty/K\). In this paper we clarify some asymptotic formulas given by the first two authors [Can. Math. Bull. 46, No. 2, 178–190 (2003; Zbl 1155.11353)], relating the orders of the \(\ell^n\)-quotients of the groups \(\mathcal C\ell_T^S(K_n)\) to structural invariants \(\rho_T^S\), \(\mu^S_T\) and \(\lambda^S_T\) of the Iwasawa module \(\chi_S^T:=\underset\leftarrow\lim\,\mathcal C\ell^S_T(K_n)\). We especially show that the lambda invariant \(\tilde\lambda^S_T\) of those quotients can sensibly differ from the structural invariant \(\lambda_T^S\), and we illustrate this fact with explicit examples, where it can be made as large as desired, positive or negative.

MSC:

11R23 Iwasawa theory
11R37 Class field theory

Citations:

Zbl 1155.11353
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Gras, G.: Théorèmes de réflexion. J. Théor. Nombres Bordx. 10(2), 399–499 (1998) · Zbl 0949.11058
[2] Jaulent, J.-F.: L’arithmétique des $$l$$ l -extensions (Thèse d’état). Pub. Math. Fac. Sci. Besançon Théor. Nombres, 1985–86 (1986)
[3] Jaulent, J.-F.: Théorie $$\(\backslash\)ell $$ -adique globale du corps de classes. J. Théor. Nombres Bordx. 10(2), 355–397 (1998) · Zbl 0938.11052
[4] Jaulent, J.-F.: Généralisation d’un théorème d’Iwasawa. J. Théor. Nombres Bordx. 17(2), 527–553 (2005) · Zbl 1176.11052
[5] Jaulent, J.-F., Maire, C.: Sur les invariants d’Iwasawa des tours cyclotomiques. Canad Math. Bull. 46(2), 178–190 (2003) · Zbl 1155.11353
[6] Neukirch, J., Schmidt, A., Wingberg, K.: Cohomology of number fields, 2nd edn. Grundlehren der Mathematischen Wissenschaften, vol. 323, pp xvi+825. Springer, Berlin (2008) · Zbl 1136.11001
[7] Salle, L.: On maximal tamely ramified pro-2-extensions over the cyclotomic $$\(\backslash\)mathbb{Z}_2$$ Z 2 -extension of an imaginary quadratic field. Osaka J. Math. 47(4), 921–942 (2010) · Zbl 1263.11097
[8] Serre, J.-P.: Classes des corps cyclotomiques (d’après Iwasawa), Séminaire Bourbaki, vol. 5, Exp. No. 174, pp. 83–93, Soc. Math. France, Paris (1995)
[9] Washington, L.: Introduction to cyclotomic fields, 2nd edn. Graduate Texts in Mathematics, vol. 83, pp. xiv+487. Springer, New York (1997) · Zbl 0966.11047
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.