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

Georges Gras

Published 2020-09-11Version 1

Let $K:=\mathbb{Q}(\ell^n)$, $n \geq 0$, be the $n$th layer in the cyclotomic $\mathbb{Z}_\ell$-extension of $\mathbb{Q}$. It is conjectured that, for all $\ell$ and $n$, $K$ is principal (especially for $\ell=2$, a conjecture due to Weber). Many studies (Ichimura--Morisawa--Nakajima--Okazaki$\,\ldots$) go in this direction, as the Miller use of the Cohen--Lenstra--Martinet heuristics. Nevertheless, we examine in what circumstances a counterexample may be possible. For this, computations show that the $p$-torsion group ${\mathcal T}_K$ of the Galois group of the maximal abelian $p$-ramified pro-$p$-extension of $K$ is not always trivial. This questions the relevance of the conjecture since $\# {\mathcal T}_K = \# {\mathcal C}_K \cdot \# {\mathcal R}_K \cdot \# {\mathcal W}_K$, where ${\mathcal C}_K$ is the $p$-class group of $K$, ${\mathcal R}_K$ its normalized $p$-adic regulator, $\# {\mathcal W}_K = 1$ for $p>2$, $\# {\mathcal W}_K = 2^{\# \{v, \,v \mid 2\}-1}$ for $p=2$; nevertheless, no counterexample has been found so far, even using the reflection theorem giving $p$-ranks equalities between ${\mathcal C}_K$ and a suitable component of ${\mathcal T}_{K(\mu_p)}$. When $n$ increases, some relative components ${\mathcal T}_K^*$ may appear for large $p$. We give a method (Theorem 4.6), for testing $\# {\mathcal T}_K \ne 1$, allowing larger values of $\ell^n$ than those of the literature. Finally, we consider the subfields $K$ of the composite $\widehat{\mathbb{Q}}$ of the $\mathbb{Z}_\ell$-extension and give programs finding again some rare cases of non-trivial class groups (Fukuda--Komatsu--Horie) due to genus theory in connection with a deep link involving ${\mathcal R}_K$ (Theorem 6.2) in relation with Greenberg's conjecture as initiated, via $p$-adic zeta-functions, by Taya. In all attempts, Jaulent's logarithmic class group $\widetilde{{\mathcal C}}_K$, $K \subset \widehat{\mathbb{Q}}$, governing Greenberg's conjecture for $K$ and $p$, was trivial.