Etudiant : BELLAL MOHAMED

Filière : Master Mathématiques Pures (MMP)

Encadrant : Pr. CHEMS-EDDINE MOHAMED MAHMOUD

Annèe : 2026

Résumé : Iwasawa theory, initiated by Kenkichi Iwasawa in the 1950s and 1960s, provides a profound framework for studying the arithmetic of number fields through infinite towers of field extensions known as $\mathbb{Z}_p$-extensions. Given a number field $K$ and a prime $p$, a $\mathbb{Z}_p$-extension $K_\infty/K$ is an infinite Galois extension whose Galois group is topologically isomorphic to $(\mathbb{Z}_p, +)$. Such an extension gives rise to a tower of intermediate fields \[ K = K_0 \subset K_1 \subset K_2 \subset \cdots \subset K_\infty, \] with $\mathrm{Gal}(K_n/K) \cong \mathbb{Z}/p^n\mathbb{Z}$ for every $n \geq 0$. Iwasawa's foundational observation was that the $p$-primary part of the ideal class group of $K_n$ exhibits a remarkably regular growth pattern as $n$ tends to infinity. More precisely, he proved that the exponent $e_n$ of the exact power of $p$ dividing the class number of $K_n$ satisfies a formula of the type \[ e_n = \lambda n + \mu p^n + \nu \qquad \text{for } n \gg 0, \] where $\lambda, \mu, \nu \in \mathbb{Z}$ are integers depending only on the extension $K_\infty/K$, with $\lambda, \mu \geq 0$. These integers, known as the \emph{Iwasawa invariants}, encode deep arithmetic information about the tower and are central objects of study throughout this memoir. The proof of this theorem relies on realizing the inverse limit of the $p$-parts of the class groups along the tower as a module over the \emph{Iwasawa algebra} $\Lambda = \mathbb{Z}_p[\![T]\!]$, and on a structure theorem classifying finitely generated modules over this ring up to pseudo-isomorphism. This algebraic machinery is what ultimately produces the asymptotic formula for $e_n$ and allows one to extract the invariants $\lambda$ and $\mu$ as intrinsic data of the associated Iwasawa module. This memoir aims to give a self-contained introduction to the algebraic and number-theoretic foundations needed to establish Iwasawa's theorem on the growth of class numbers in $\mathbb{Z}_p$-extensions, and to present \textbf{Greenberg's conjecture}, one of the central open problems arising naturally from this theory. The exposition is structured as follows: \textbf{Chapter 1} introduces the Iwasawa algebra $\Lambda = \mathbb{Z}_p[\![T]\!]$ and develops the structure theory for finitely generated $\Lambda$-modules. We prove the crucial Structure Theorem, which shows that every finitely generated $\Lambda$-module is pseudo-isomorphic to a direct sum of a free part and torsion parts corresponding to prime ideals of height one. This theorem provides the algebraic framework for defining the $\lambda$ and $\mu$ invariants and the characteristic ideal of an Iwasawa module.