Computing the torsion of the $p$-ramified module

Abstract

We fix a prime number $p$ and $\K$ a number field, we denote by $M$ the maximal abelian $p$-extension of $\Ko$ unramified outside $p$. The aim of this paper is to study the $\Z_p$-module $\gal(M/\Ko)$ and to give a method to effectively compute its structure as a $\Z_p$-module. Then we give numerical results, for real quadratic fields, together with interpretations via Cohen-Lenstra's heuristics.