Sur les formules asymptotiques le long des Zl-extensions

In this paper we clarify some asymptotic formulas given by Jaulent-Maire, which relate orders of finite quotients of S-infinitesimal T-classes l-groups $C{l}_T^S(K_n)$ associated to finite layers $K_n$ of a Zl-extension $K_{\mathrm{\infty }}\mathrm{/}K$ over a number field to the structural invariants of the Iwasawa module $X_T^S:=\mathrm{}{\text{Cl}}_T^S(K_n)$. We especially show that the lambda invariant ${\tilde{\lambda }}_T^S$ of those quotients sensibly differs 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.