On the asymptotic formulas along the Zl-extensions

Let K-infinity be a Z(l)-extension of a number field K. We consider the Galois group Cl-T(S) (K-n) of the maximal S-ramified and T -split abelian pro-l-extension attached to each layer K-n of the tower K-infinity/K. In this paper we clarify some asymptotic formulas given by Jaulent and Maire (Canad. Math. Bull. 46(2): 178-190, 2003), relating the orders of the l(n)-quotients of the groups Cl-T(S) (K-n) to structural invariants rho(S) (T), mu(S)(T) and.ST of the Iwasawa module X-T(S) := lim(<-)Cl (S)(T) (K-n). We especially show that the lambda invariant lambda(S)(T\) of those quotients can sensibly differ from the structural invariant lambda(S)(T), and we illustrate this fact with explicit examples, where it can be made as large as desired, positive or negative.