Weber's class number problem and $p$-rationality in the cyclotomic $\widehat{\mathbb{Z}}$-extension of $\mathbb{Q}$

TL;DR

This paper investigates the behavior of p-class groups and p-torsion in the cyclotomic Z-hat extension of Q, introducing new methods to test p-rationality and analyze p-extensions, supported by computational tools.

Contribution

It presents a new conceptual approach to analyze p-torsion groups in cyclotomic extensions and provides computational tools to test p-rationality and identify non-trivial p-class groups.

Findings

p-torsion groups often non-trivial in cyclotomic layers

New method to test p-rationality of fields

Characterization of p-extensions with non-trivial class groups

Abstract

Let $K$ be the $N$th layer in the cyclotomic $\widehat{\mathbb{Z}}$-extension $\widehat{\mathbb{Q}}$. Many authors (Aoki, Fukuda, Horie, Ichimura, Inatomi, Komatsu, Miller, Morisawa, Nakajima, Okazaki, Washington,\,$\ldots$) analyse the behavior of the $p$-class groups ${\mathcal C}_K$. We revisit this problem, in a more conceptual form, since computations show that the $p$-torsion group ${\mathcal T}_K$ of the Galois groups of the maximal abelian $p$-ramified pro-$p$-extension of $K$ (Tate--Shafarevich group of $K$) is often non-trivial; this raises questions since $\# {\mathcal T}_K = \# {\mathcal C}_K\, \# {\mathcal R}_K$ where ${\mathcal R}_K$ is the normalized $p$-adic regulator. We give a new method testing ${\mathcal T}_K \ne 1$ (Theorem 4.6, Table 6.2) and characterize the $p$-extensions $F$ of $K$ in $\widehat{\mathbb{Q}}$ with ${\mathcal C}_F \ne 1$ (Theorem 7.5 and Corollary 7.6). We publish easy to use programs, justifying again the eight known examples, and allowing further extensive computations.