Sur les formules asymptotiques le long des Zl-extensions

TL;DR

This paper refines asymptotic formulas in Iwasawa theory for Zl-extensions, highlighting differences between lambda invariants of finite quotients and the Iwasawa module, with explicit examples illustrating the variability.

Contribution

It clarifies the relationship between asymptotic formulas and Iwasawa invariants, especially the lambda invariant, providing explicit examples of their divergence.

Findings

Lambda invariant of quotients differs from the Iwasawa module invariant.

Explicit examples show the lambda invariant can be arbitrarily large or negative.

Refined understanding of asymptotic behavior in Zl-extensions.

Abstract

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 $Cl^S_T(K_n)$ associated to finite layers $K_n$ of a Zl-extension $K_\infty/K$ over a number field to the structural invariants of the Iwasawa module $X^S_T:=\varprojlim \Cl^S_T(K_n)$. We especially show that the lambda invariant $\tilde\lambda^S_T$ of those quotients sensibly differs 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.