Generalised Iwasawa invariants and the growth of class numbers (Q1998918)

Let \(K\) be a number field, let \(p\) be a rational prime and let \(d\) be a positive integer. Let \(\mathbb{K}\) be a \(\mathbb{Z}_{p}^{d}\)-extension of \(K\), that is, a Galois extension with \(\Gamma := \mathrm{Gal}(\mathbb{K}/K)\) topologically isomorphic to a direct product of \(d\) copies of the \(p\)-adic integers. For each positive integer \(n\), let \(\mathbb{K}_{n}\) be the subfield of \(\mathbb{K}\) fixed by \(\Gamma^{p^{n}}\). Let \(A_{n}\) denote the Sylow \(p\)-subgroup of the class group of the ring of integers of \(\mathbb{K}_{n}\) and define \(e_{n}\) by \(|A_{n}|=p^{e_{n}}\). In the case \(d=1\), \textit{K. Iwasawa} [Bull. Am. Math. Soc. 65, 183--226 (1959; Zbl 0089.02402)] showed that the growth of the order of \(A_{n}\) can be described in a very explicit manner: there exist integers \(n_{0},\lambda,\mu \geq 0\) and \(\nu\) such that for every \(n \geq n_{0}\), we have \(e_{n}=\mu p^{n} + \lambda n + \nu\). \textit{A. A. Cuoco} and \textit{P. Monsky} [Math. Ann. 255, 235--258 (1981; Zbl 0437.12003)] generalised this result to include the case \(d \geq 2\). They showed that there exist integers \(m_{0}, l_{0} \geq 0\), called the generalised Iwasawa invariants of \(\mathbb{K}/K\), such that \(e_{n}=(m_{0}p^{n} + l_{0}n + O(1))p^{(d-1)n}\). In the article under review, the author considers the local behaviour of generalised Iwasawa invariants on the set \(\mathcal{E}^{d}(K)\) of \(\mathbb{Z}_{p}^{d}\)-extensions of \(K\), with respect to a suitable topology. The main result is as follows. Let \(\mathbb{K}/K\) be a \(\mathbb{Z}_{p}^{d}\)-extension. Assume that there exists a prime of \(K\) that is totally ramified in \(\mathbb{K}/K\). Then with respect to a suitable topology on \(\mathcal{E}^{d}(K)\), there exists a neighbourhood \(\mathcal{U} \subseteq \mathcal{E}^{d}(K)\) of \(\mathbb{K}\) such that: \begin{itemize} \item[(i)] \(m_{0}(\mathbb{L}/K) \leq m_{0}(\mathbb{K}/K)\) for every \(\mathbb{L} \in \mathcal{U}\), and \item[(ii)] there exists a constant \(k \in \mathbb{N}\) such that \(l_{0}(\mathbb{L}/K) \leq k\) for each \(\mathbb{L} \in \mathcal{U}\) satisfying \(m_{0}(\mathbb{L}/K) = m_{0}(\mathbb{K}/K)\). \end{itemize} Moreover, the author gives a condition that ensures that \(l_{0}(\mathbb{K}/K)\) is locally maximal, that is, \(k=l_{0}(\mathbb{K}/K)\) in (ii). In previous work of the same author [Ann. Math. Qué. 43, No. 2, 305--339 (2019; Zbl 1470.11281)], the same results were proven, but under a strong technical assumption, which is now proven in the article under review. In the case that \(\mathbb{K}/K\) is a \(\mathbb{Z}_{p}^{2}\)-extension such that exactly one prime \(\mathfrak{p}\) of \(K\) ramifies in \(\mathbb{K}\) and, moreover, \(\mathfrak{p}\) is totally ramified in \(\mathbb{K}/K\), the author proves an asymptotic growth formula for the class numbers of the intermediate fields, which improves the aforementioned results of Cuoco and Monsky in this situation. The author also briefly discusses the impact of generalised Iwasawa invariants on the global boundedness of Iwasawa \(\lambda\)-invariants.