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.

New redaction with improvements and corrections

References in corpus (3)

• Algorithms for Finding Almost Irreducible and Almost Primitive Trinomials
• Class numbers of totally real fields and applications to the Weber class number problem
• Algorithmic complexity of Greenberg's conjecture

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