\(\mathcal S\)-units and \(\mathcal S\)-class groups of a cyclic number field of prime degree
If \(L/K\) is a Galois extension of number fields with group \(G\), then the \(G\)-structures of the unit group \(E_L\) and the class group \(Cl_K\) are (conjecturally) linked. This is made more precise in Chinburg's Third Conjecture, and in the yet sharper conjecture dubbed Lifted Root Number Conjecture, or also Equivariant Tamagawa Number Conjecture. These conjectures are far-reaching and fascinate by their generality; it can be hard, however, to extract concrete isomorphism statements from them. The merit of the present article is that it presents such concrete results, with a moderate expenditure of technique. To do so, the author makes the (of course rather restrictive) assumption that \(K\) is the field of rationals and \(G\) is of prime order \(l\). This assumption has the advantage that torsion-free \(ZG\)-modules (``lattices'') are completely classified by the theorem of Diederichsen and Reiner. (The first of these two names is persistently misspelled in the paper.) These lattices are given by some discrete invariants (three natural numbers) and one element in the class group of \(\mathbb Q(\zeta_l)\), the so-called class invariant. Theorem 3.1 gives the discrete invariants not only for \(E_K\) but for all \(S\)-unit groups \(E_{K,S}\) as well. (Actually for \(E_K\) itself the result is not surprising.) The class invariant of \(E_{K,S}\) is subtler and cannot be calculated right off. Instead, the author deduces from the Gras ``conjecture'' (which is a theorem, proved from the Main Conjecture) that it equals the class invariant of the \(S\)-class group of \(K\). The tools used include cohomology and the theory of ambiguous classes. The article contains enlightening remarks concerning the question whether we are able to exhibit an example in the given context where \(U_K = E_K/\{\pm 1\}\) is not free over \(Z[\zeta_l]\). Here \(l\) would have to be at least 23, and in spite of an optimistic remark of Brumer, even with the aid of computers no certified counterexample seems to be available right now, although it is believed to exist.
scientific article; zbMATH DE number 1547372
scientific article