Let $K=\mathbb{Q}(N)$ be the Nth layer in the cyclotomic $\hat{\mathbb{Z}}$-extension. Many authors (Aoki, Fukuda, Horie, Ichimura, Inatomi, Komatsu, Miller, Morisawa, Nakajima, Okazaki, Washington, …) prove results on the p-class groups ${\mathcal{C}}_K$. We enlarge “Weber's problem” to the Tate–Shafarevich groups ${\mathrm{III}}_K^1\simeq {\mathcal{C}}_K^{S_p}$ ($S_p$-class group) and ${\mathrm{III}}_K^2$ having same p-rank as the more easily computable torsion group, ${\mathcal{T}}_K$, of the Galois group of the maximal abelian p-ramified pro-p-extension of K; but ${\mathcal{T}}_K$ is often non-trivial, which raises questions for class groups since

, 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 in Appendix A.7) and characterize the fields $K_1=K\mathbb{Q}(p)$ with ${\mathcal{C}}_{K_1}\ne 1$ (Main Theorem 1.1 affirming, for short, that ${\mathcal{C}}_{K_1}\ne 1$ if and only if p totally splits in K and ${\mathcal{T}}_K\ne 1$); this highlights the analytical results and justifies the eight known examples. All PARI/GP programs are given for further investigations.

中文翻译：

---

圈分中的 Tate-Shafarevich 群 $\widehat{\mathbb{Z}}$-扩展和韦伯的班级编号问题

让 $钾=\mathbb{问}(N)$是圈层中的第N层$\widehat{\mathbb{Z}}$-扩大。许多作者（青木、福田、堀江、市村、稻美、小松、米勒、森泽、中岛、冈崎、华盛顿 ……）证明了p级组的结果${\mathcal{C}}_{钾}$. 我们将“韦伯问题”扩大到泰特-沙法列维奇集团${\mathrm{三}}_{钾}^1\simeq {\mathcal{C}}_{钾}^{{秒}_{磷}}$ (${秒}_{磷}$-类组）和 ${\mathrm{三}}_{钾}^2$与更容易计算的扭转组具有相同的p等级，${\mathcal{吨}}_{钾}$，伽罗瓦组的极大交换的p -ramified亲p的-extension ķ ; 但${\mathcal{吨}}_{钾}$ 通常是不平凡的，这给班级小组提出了问题，因为

， 在哪里 ${\mathcal{电阻}}_{钾}$是归一化p- adic 调节器。我们给出了一种新的方法测试${\mathcal{吨}}_{钾}\ne 1$ （定理 4.6，附录 A.7 中的表）并表征字段 ${钾}_1=钾\mathbb{问}(磷)$ 和 ${\mathcal{C}}_{{钾}_1}\ne 1$ （主定理 1.1 简言之，肯定 ${\mathcal{C}}_{{钾}_1}\ne 1$当且仅当p在K 中完全分裂并且${\mathcal{吨}}_{钾}\ne 1$); 这突出了分析结果并证明了八个已知示例的合理性。所有 PARI/GP 计划都用于进一步调查。