Computing the torsion of the $p$-ramified module

TL;DR

This paper investigates the structure of the Galois group of the maximal abelian p-extension unramified outside p over a number field, providing methods for computation and numerical results for real quadratic fields.

Contribution

It introduces an effective method to compute the structure of the p-ramified module's Galois group and offers numerical data with heuristic interpretations.

Findings

Computed Galois groups for real quadratic fields.

Numerical evidence supporting Cohen-Lenstra heuristics.

Method improves understanding of p-ramified modules.

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.