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

Abstract

Let $K$ be the $N$th layer in the cyclotomic $\hat{\mathbb{Z}}$-extension $\hat{\mathbb{Q}}$. Many authors (Aoki, Fukuda, Horie, Ichimura, Inatomi, Komatsu, Miller, Morisawa, Nakajima, Okazaki, Washington,\,$\dots$) 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 $\hat{\mathbb{Q}}$ 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.

arXiv commentNew redaction with improvements and corrections

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Cited by in corpus (2)

• The $p$-adic limits of class numbers in ${\mathbb{Z}}_p$-towers
• Unlimited lists of fundamental units of quadratic fields -- Applications to some arithmetic properties