Abstract We study the generalised Iwasawa invariants of ℤ p d {\mathbb{Z}_{p}^{d}} -extensions of a fixed number field K. Based on an inequality between ranks of finitely generated torsion ℤ p ⁢ [ [ T 1 , … , T d ] ] {\mathbb{Z}_{p}[\kern-2.133957pt[T_{1},\dots,T_{d}]\kern-2.133957pt]} -modules and their corresponding elementary modules, we prove that these invariants are locally maximal with respect to a suitable topology on the set of ℤ p d {\mathbb{Z}_{p}^{d}} -extensions of K, i.e., that the generalised Iwasawa invariants of a ℤ p d {\mathbb{Z}_{p}^{d}} -extension 𝕂 {\mathbb{K}} of K bound the invariants of all ℤ p d {\mathbb{Z}_{p}^{d}} -extensions of K in an open neighbourhood of 𝕂 {\mathbb{K}} . Moreover, we prove an asymptotic growth formula for the class numbers of the intermediate fields in certain ℤ p 2 {\mathbb{Z}_{p}^{2}} -extensions, which improves former results of Cuoco and Monsky. We also briefly discuss the impact of generalised Iwasawa invariants on the global boundedness of Iwasawa λ-invariants.

中文翻译：

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广义 Iwasawa 不变量与类数的增长

摘要 我们研究了 ℤ pd {\mathbb{Z}_{p}^{d}} - 定数域 K 的扩展的广义 Iwasawa 不变量。基于有限生成的扭力 ℤ p ⁢ [ [ T 1 , … , T d ] ] {\mathbb{Z}_{p}[\kern-2.133957pt[T_{1},\dots,T_{d}]\kern-2.133957pt]} -modules 和它们对应的基本模块，我们证明这些不变量相对于 K 的 ℤ pd {\mathbb{Z}_{p}^{d}} -扩展集上的合适拓扑是局部最大的，即广义 Iwasawa 不变量的一个 ℤ pd {\mathbb{Z}_{p}^{d}} -extension 𝕂 {\mathbb{K}} 的 K 限制了所有 ℤ pd {\mathbb{Z}_{p}^{ d}} - K 在 𝕂 {\mathbb{K}} 的开放邻域中的扩展。而且，我们证明了某些 ℤ p 2 {\mathbb{Z}_{p}^{2}} -extensions 中中间域的类数的渐近增长公式，它改进了 Cuoco 和 Monsky 的先前结果。我们还简要讨论了广义 Iwasawa 不变量对 Iwasawa λ 不变量的全局有界性的影响。