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Stickelberger elements, Fitting ideals of class groups of CM-fields, and dualisation. (English) Zbl 1159.11042

For an abelian extension \(K/k\) such that \(k\) is totally real and \(K\) is CM, Brumer’s conjecture asserts that \(\text{Ann} (\mu(K)) \ldotp {\mathcal O}_K \subset \text{Ann} (C\ell_K),\) where \(\mu (K)\) denotes the group of roots of unity in \(K,\) \(\text{Ann} (\ldotp)\) the annihilator over the group algebra of \(\text{Gal} (K/k),\) and \({\mathcal O}_K\) the Stickelberger element of \(K/k\) (for \(k = {\mathbb Q},\) this is just Stickelberger’s theorem). It is natural to ask if a stronger version holds when replacing the annihialtor ideal by the Fitting ideal. This refined conjecture can be tackled \(p\)-component by \(p\)-component in the framework of Iwasawa theory. The authors give a negative answer by showing that under suitable hypotheses, the standard (unramified) Iwasawa module does not contain the relevant Stickelberger element at infinite level. This already implies the existence of similar counterexamples at finite level, since projective limits commute with Fitting ideals (as shown in detail by the authors). Certain classes of cases (one case is studied fully with \([k : {\mathbb Q}]\) = 36) are presented, for which the relevant Stickelberger element is not in the Fitting ideal of the class group but in the Fitting ideal of the dual, which has a tendency to behave better functorially. In a sense, this better behaviour is hardly surprising: as a \({\mathbb Z}_p\)-extension is unramified outside \(p,\) étale cohomology (or Galois cohomology with restricted ramification) provides the most powerful tool to study the standard \(p\)-ramified Iwasawa module (denoted by \(X_p\) at the end of the paper) ; in this cohomological setting, the \(H^2\)’s with twisted coefficients \({\mathbb Z}_p(m),\) \(m \not= 0,\) can be dealt with by means of (various versions of) the ETNC, whereas \(H^2\) of \({\mathbb Z}_p\) itself, modulo Leopoldt’s conjecture, is dual to the torsion of \(X_p,\) which in turn is related to the class-group by “Spiegelung”. Upstairs, Spiegelung is expressed by the isomorphism (invoked at the end of the paper) between the module denoted \(X_{du}\) and a certain Iwasawa adjoint.

MSC:

11R23 Iwasawa theory
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