Numerical verification of the Cohen-Lenstra-Martinet heuristics and of Greenberg's $p$-rationality conjecture

TL;DR

This paper provides numerical evidence supporting Greenberg's $p$-rationality conjecture, introduces new examples of $p$-rational fields, and explores the conjecture's connection to Cohen-Lenstra-Martinet heuristics and other conjectures.

Contribution

It presents new $p$-rational field examples, supports conjectures with numerical data, and proposes improvements to computational methods.

Findings

Numerical support for Greenberg's $p$-rationality conjecture.

Discovery of new $p$-rational multiquadratic fields.

Connection established between conjectures and heuristic models.

Abstract

In this paper we make a series of numerical experiments to support Greenberg's $p$-rationality conjecture, we present a family of $p$-rational biquadratic fields and we find new examples of $p$-rational multiquadratic fields. In the case of multiquadratic and multicubic fields we show that the conjecture is a consequence of the Cohen-Lenstra-Martinet heuristic and of the conjecture of Hofmann and Zhang on the $p$-adic regulator, and we bring new numerical data to support the extensions of these conjectures. We compare the known algorithmic tools and propose some improvements.