Class numbers in \(\mathbb Z^ d_ p\)-extensions. IV (Q1096670)

[For part III see ibid. 193, 491--514 (1986; Zbl 0596.12005).] Let \(L\) be a Galois extension of a number field \(k\), with Galois group \(E\) isomorphic to a product of \(d\quad copies\) of the \(p\)-adic integers. Let \(k_ n\) be the fixed field of \(E^{p^ n}\), \(H_ n\) the \(p\)-primary component of the ideal class group of \(k_ n\) and \(p^{e_ n}\) the order of \(H_ n.\) In an earlier paper [Math. Ann. 263, 509--514 (1983; Zbl 0514.12008)] the author defined an integer \(a=a(L/k)\) and proved: (1) If \(a=0\), the \(\mathbb Z/p\)-dimension of \(H_ n/pH_ n\) is eventually constant. (2) If \(a>0\), the \(\mathbb Z/p\)-dimension of \(H_ n/pH_ n\) is \(c_ 0p^{an}+O(p^{(a-1)n})\) for some positive real \(c_ 0\). The present paper fixes a \(t\) and shows that the \(\mathbb Z/p\)-dimension of \(p^ tH_ n/p^{t+1}H_ n\) is \(c_ tp^{an}+O(p^{(a-1)n})\) for some real \(c_ t\). (In the case \(a=1\) it was known that this is true and that each \(c_ t\) is rational; the rationality remains an open question for \(2\leq a\leq d-1\). The paper goes on to verify a conjecture [cf. part II of this paper in Math. Z. 191, 377--395 (1986; Zbl 0563.12004)] concerning the asymptotic growth of the \(e_ n\). When \(a=0\) it is known that \(e_ n\) is an eventually linear function on \(n\). When \(a>0\) the following is proved. Either \(e_ n=(\lambda^*n+O(1))p^{an}\) for some positive integer \(\lambda^*\) or \(p^{-an}e_ n\to\) a finite positive limit \(\alpha\) as \(n\to \infty\). When \(a=d-1\) and \(\lambda^*=0\) the more precise estimate \(e_ n=\alpha p^{an}+O(np^{(a-1)n})\) is obtained.