Class numbers in \(\mathbb Z^ d_ p\)-extensions. III (Q1078240)

Let \(L/k\) be a multiple \(\mathbb Z_ p\)-extension of a number field and let \(e_ n=e_ n(L/k)\) be the exponent to which \(p\) appears in the class number of the \(n\)th layer, \(k_ n\), of the extension. In a previous paper [ibid. 191, 377--395 (1986; Zbl 0563.12004)] the author defined non-negative integers \(a=a(L/k)\) and \(\lambda^*=\lambda^*(L/k)\) and showed that \(e_ n=(\lambda^*n+O(1))p^{an}\). The present paper makes a deeper study of the growth of \(e_ n\) when \(a(L/k)=1\). The results are: (a) Let \(H_ n\) be the ideal class group of \(k_ n\). Then for each fixed \(j\geq 0\) the \(\mathbb Z/p\)-dimension of \(p^ jH_ n/p^{j+1}H_ n\) is \(c_ jp^{n-j}+O(1)\) for some integer \(c_ j\). (b) Suppose that \(\lambda^*(L/k)=0\). Then \(\sum^{\infty}_{0}c_ jp^{-j}\) converges to a positive rational \(\alpha\) and \(e_ n\sim \alpha p^ n\). More precisely, \(e_ n\leq \alpha p^ n+O(n^ 2)\) while \(e_ n\geq \alpha p^ n-O(p^{n/2}).\) In fact purely algebraic generalizations of (a) and (b) are established. To this end the author considers a complete local ring (\({\mathcal O},{\mathfrak m})\) whose reduction mod \(p\) has altitude 1, a finitely generated \(\mathbb Z_ p\)-submodule \(E\) of the multiplicative group \(1+{\mathfrak m}\), and a finitely generated \({\mathcal O}\)-module \(M\). He takes \(I_ n\) to be the ideal of \({\mathcal O}\) generated by the \(u^{p^ n}-1\), \(u\in E\). Then analogues of (a) and (b) hold where \(\mathbb Z/p\) is replaced by \({\mathcal O}/{\mathfrak m}\), \(H_ n\) by \(M/I_ nM\) (or some slightly more complicated \({\mathcal O}\)-module), and \(e_ n\) by the \({\mathcal O}\)-length of the \(p\)-power torsion subgroup of \(M/I_ nM\). The \(c_ j\) may be defined as sums of local contributions, \(c_ j({\mathfrak P})\), one for each minimal prime \({\mathfrak P}\) of p\({\mathcal O}\). The definition of \(c_ j({\mathfrak P})\) is made to rest on the following simple facts. First, the integral closure of \({\mathcal O}/{\mathfrak P}\) is a complete discrete valuation ring. Second, if \((A,{\mathfrak m})\) is a complete discrete valuation ring of characteristic \(p\) and \(\Gamma\) is a finitely generated non-trivial \(\mathbb Z_ p\)-submodule of \(1+{\mathfrak m}\) then there is a \(\mathbb Z_ p\)-basis \(u_ 1,...,u_ d\) of \(\Gamma\) with the following property. If \(u=\pi u_ i^{\alpha_ i}\) with \(\alpha_ i\) in \(\mathbb Z_ p\) then \(\mathrm{ord}(u-1)=\min \mathrm{ord}(u_ i^{\alpha_ i}-1).\) The precise definition of the \(c_ j({\mathfrak P})\) and the proofs of (a) and (b) are complicated. But the following periodicity result is crucial to (b) -- for large \(j\), \(c_{j+d}({\mathfrak P})=p c_ j({\mathfrak P})\) where d is the \(\mathbb Z_ p\)-rank of the image of \(E\) in \({\mathcal O}/{\mathfrak P}\). To deduce class number results from the above algebraic results the author takes \({\mathcal O}\) to be \(\Lambda/\mathrm{Ann}\, X\), (where \(\Lambda\) is the complete \(\mathbb Z_ p\)-group algebra of \(G(L/k)\) and \(X\) is the Greenberg-Iwasawa module of \(L/k\)), \(E\) to be the image of \(G(L/k)\) in \({\mathcal O}\), and \(M\) to be \(X\) itself viewed as an \({\mathcal O}\)-module. He promises sharper results for the growth of \(e_ n(L/k)\), both when \(\lambda^*=0\) and \(\lambda^*>0\), in a forthcoming article.