Realizable Galois module classes over the group ring for non abelian extensions
We say that an extension of number fields is tame if it is at most tamely ramified. Fix a number field \(k\) and a finite group \(\Gamma\). We say that \(N\) is a \(\Gamma\)-extension of \(k\) if \(N/k\) is a Galois extension of number fields equipped with an isomorphism \(\mathrm{Gal}(N/k) \rightarrow \Gamma\). If \(N\) is a tame \(\Gamma\)-extension of \(k\) then the ring of integers \(\mathcal{O}_{N}\) is a locally free module of rank \(1\) over the group ring \(\mathcal{O}_{k}[\Gamma]\), and its structure as such is determined up to stable isomorphism by its class \((\mathcal{O}_{N})_{\mathcal{O}_{k}[\Gamma]}\) in the locally free class group \(\mathrm{Cl}(\mathcal{O}_{k}[\Gamma])\). The set \(\mathcal{R}(\mathcal{O}_{k}[\Gamma])\) of realizable classes is defined to be the subset of \(\mathrm{Cl}(\mathcal{O}_{k}[\Gamma])\) consisting of the classes \((\mathcal{O}_{N})_{\mathcal{O}_{k}[\Gamma]}\) as \(N\) runs through all tame \(\Gamma\)-extensions of \(k\). It is a conjecture that \(\mathcal{R}(\mathcal{O}_{k}[\Gamma])\) is in fact a subgroup of \(\mathrm{Cl}(\mathcal{O}_{k}[\Gamma])\). This has been proven by \textit{L. R. McCulloh} in the case that \(\Gamma\) is abelian see [J. Reine Angew. Math. 375/376, 259--306 (1987; Zbl 0619.12008)]. However, much less is known in the non-abelian case; indeed, a proof that \(\mathcal{R}(\mathcal{O}_{k}[\Gamma])\) is non-empty would solve the inverse Galois problem for \(\Gamma\) over \(k\). The article under review determines \(\mathcal{R}(\mathcal{O}_{k}[\Gamma])\) for certain metabelian groups under a mild restriction on \(k\). In particular, the results show that \(\mathcal{R}(\mathcal{O}_{k}[\Gamma])\) is indeed a subgroup of \(\mathrm{Cl}(\mathcal{O}_{k}[\Gamma])\) in the cases under consideration. Let \(p\) be a prime and assume that \(k\) contains a primitive \(p\)th root of unity. Let \(\Gamma\) be the semidirect product \(V \rtimes C\) of an elementary abelian group \(V\) of order \(p^{r}\) by any cyclic group \(C\) of order \(m>1\) that acts faithfully on \(V\) and makes \(V\) into an irreducible \(\mathbb{F}_{p}[C]\)-module. The authors show that this can occur if and only if \(m\) is a divisor of \(p^{r}-1\) that does not divide \(p^{s}-1\) for any \(s<r\). A family of examples is given by the semidirect products \(\mathbb{F}_{p^{r}} \rtimes \mathbb{F}_{p^{r}}^{\times}\) with the natural action; in particular, \(\mathbb{F}_{3} \rtimes \mathbb{F}_{3}^{\times} \simeq S_{3}\) and \(\mathbb{F}_{4} \rtimes \mathbb{F}_{4}^{\times} \simeq A_{4}\). Viewing \(C\) and \(V\) as subgroups of \(\Gamma\), we have the inclusion homomorphisms \[ \iota^{C}: C \longrightarrow \Gamma, \qquad \iota^{V}: V \longrightarrow \Gamma. \] These induce homomorphisms of class groups \[ \iota^{C}_{*}: \mathrm{Cl}(\mathcal{O}_{k}[C]) \longrightarrow \mathrm{Cl}(\mathcal{O}_{k}[\Gamma]), \qquad \iota^{V}_{*}: \mathrm{Cl}(\mathcal{O}_{k}[V]) \longrightarrow \mathrm{Cl}(\mathcal{O}_{k}[\Gamma]). \] Using McCulloh's aforementioned result that the realizable classes form a subgroup in the abelian case, we therefore have subgroups \(\iota^{C}_{*}\mathcal{R}(\mathcal{O}_{k}[C])\) and \(\iota^{V}_{*}\mathcal{R}(\mathcal{O}_{k}[V])\) of \(\mathcal{R}(\mathcal{O}_{k}[\Gamma])\). The main result of the article under review is as follows. In the situation of the above paragraph, we have \(\mathcal{R}(\mathcal{O}_{k}[\Gamma]) = (\iota^{C}_{*}\mathcal{R}(\mathcal{O}_{k}[C]))(\iota^{V}_{*}\mathcal{R}(\mathcal{O}_{k}[V]))\). It follows immediately that \(\mathcal{R}(\mathcal{O}_{k}[\Gamma])\) is indeed a subgroup of \(\mathrm{Cl}(\mathcal{O}_{k}[\Gamma])\). (The authors also give an alternative description involving \(\iota^{C}_{*}\mathcal{R}(\mathcal{O}_{k}[C])\), a certain Stickelberger ideal, and the kernel of a map between locally free class groups induced by the canonical surjection \(\Gamma \rightarrow C=\Gamma/V\).) Moreover, given any finite set \(S\) of places of \(k\) and any class \(\mathcal{A} \in \mathcal{R}(\mathcal{O}_{k}[\Gamma])\), there are infinitely many \(\Gamma\)-extensions \(N\) of \(k\) with \((\mathcal{O}_{N})_{\mathcal{O}_{k}[\Gamma]}=\mathcal{A}\), and \(N\) can be chosen to satisfy the following properties: \(N/k\) is unramified at all places in \(S\), and every intermediate field \(F \neq k\) of \(N/k\) is ramified at some finite place of \(k\). This extends and refines the results of the authors with \textit{C. Greither} [J. Reine Angew. Math. 601, 1--27 (2006; Zbl 1137.11069)] for \(p=2\), respectively of \textit{C. Bruche} and the second author [J. Number Theory 128, No. 4, 954--978 (2008; Zbl 1189.11051)] for \(p>2\); these only cover the case \(m=p^{r}-1\) and only determine the image \(\mathcal{R}(\mathcal{M})\) of \(\mathcal{R}(\mathcal{O}_{k}[\Gamma])\) under the extension of scalars from \(\mathcal{O}_{k}[\Gamma]\) to a maximal order \(\mathcal{M} \supset \mathcal{O}_{k}[\Gamma]\) in \(k[\Gamma]\). Moreover, this is a generalisation of the calculation of \(\mathcal{R}(\mathcal{O}_{k}[A_{4}])\) (the case \(p=r=2\)) in [the authors, Compos. Math. 141, No. 3, 573--582 (2005; Zbl 1167.11319)]. The proof of this result proceeds via a relative version. Given a tame \(C\)-extension \(E\) of \(k\), we say that \(N\) is a \(\Gamma\)-extension of \(k\) relative to \(E\) if \(E \subset N\) and the isomorphism \(\mathrm{Gal}(N/k) \rightarrow \Gamma\) associated to \(N\) induces the given isomorphism \(\mathrm{Gal}(E/k) \rightarrow C = \Gamma / V\). For a fixed choice of \(E\), there is an analogous version of the result described in the above paragraph, which in particular describes the set \(\mathcal{R}(\mathcal{O}_{k}[\Gamma],E) \subseteq \mathrm{Cl}(\mathcal{O}_{k}[\Gamma])\) of classes \((\mathcal{O}_{N})_{\mathcal{O}_{k}[\Gamma]}\) where \(N\) runs through all tame \(\Gamma\)-extensions of \(k\) relative to \(E\).
scientific article; zbMATH DE number 6177083
scientific article