## 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

Zbl 1155.11353
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### References:

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