Explicit units and the equivariant Tamagawa number conjecture (Q2762640)

This paper is related by its methods to the authors' previous paper [\textit{W. Bley} and \textit{D. Burns}, Compos. Math. 126, 213--247 (2001; Zbl 0987.11069)], but the results go further. Some context: The lifted Omega invariant \(T\Omega(L/K)\) is defined for any Galois extension \(L/K\) of number fields; it is a special case of a much more general theory of ``Equivariant Tamagawa Numbers'' developed by Burns and Flach. For \(L/K\) abelian, one conjectures that \(T\Omega(L/K)\) vanishes, and the authors treat the case \(K=\mathbb Q\), \(L\) of prime power degree \(p^e\), and they impose the natural and ``classical'' condition that \(p\) is coprime to \(h_L\). Under the assumption that not more than two primes ramify in \(L\), this leads to three cases (found by Fröhlich) numbered I--III. \textit{D. Burns} and \textit{M. Flach} [Equivariant Tamagawa numbers of motives (preprint 1998)] had proved the vanishing of \(T\Omega(L/\mathbb Q)\) in cases I and II. Case III was open, and is taken care of in the paper under review. Very recently, D.~Burns and the reviewer proved \(T\Omega(L/\mathbb Q)=0\) up to 2-primary parts for all abelian fields \(L\) using much stronger machinery. NEWLINENEWLINEThe main point of the paper is to reduce the a priori quite unwieldy statement \(T\Omega(L/\mathbb Q)=0\) to a statement concerning existence of explicit units. In order to achieve this concretisation, one needs, among other things, a tight grasp on the Tate canonical class, a little homological algebra, and the basic theory of the first Fitting ideal. NEWLINENEWLINESimilarly as in the Compos. Math. paper mentioned above, the exposition is again very readable; the paper is recommended for a comparatively easy approach to this subject.