On the index of the Stickelberger ideal and the cyclotomic regulator (Q1066190)

For an abelian extension k/\({\mathbb{Q}}\) let \(G=Gal(k/{\mathbb{Q}})\) and \(R={\mathbb{Z}}[G]\), the group ring of G with coefficients in \({\mathbb{Z}}\). Let A be the ideal in R consisting of all \(\alpha\in R\) such that \((1+j)\alpha =a\cdot \sum_{\sigma \in G}\sigma\) for some \(a\in {\mathbb{Z}}\), where \(j\in G\) is complex conjugation. It is well known that a certain explicitly given ideal S in R, the so-called Stickelberger ideal, is contained in A and is of finite index in A. This index [A:S] is intimately related with the relative class number \(h^-\) of k. The first main result in the article says that log [A:S] and log \(h^-\) are asymptotically the same as k ranges over a sequence of imaginary abelian fields for which the discriminant d of k goes to \(\infty\). In addition it is shown that log \(h^-\) and log \(\sqrt{d/d^+}\) are asymptotically the same, where \(d^+\) denotes the discriminant of the maximal real subfield \(k^+\) in k. The second main result concerns the full class number h of k and how it is asymptotically related to the index of the group C of cyclotomic units in the full unit group E of k. It is shown that log([A:S]\(\cdot [E:C])\) and log h are asymptotically the same as d goes to \(\infty\). As a consequence one gets among other things that for any positive r there exists only a finite number of real abelian fields k such that the regulator of C is bounded by \(r^{g(k:{\mathbb{Q}})}\), where g denotes the number of rational primes which ramify in k/\({\mathbb{Q}}\).