American Journal of Mathematics

Let L/K be a finite abelian extension of number fields of group G. We study the Tamagawa number TΩ(L/K) of h⁰(Spec L), considered as a motive defined over K and with coefficients [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /][G]. For a large class of extensions L/K we interpret the conjectural vanishing of TΩ(L/K) in terms of the existence of S-units in L satisfying a variety of explicit conditions. These explicit conditions are in the same spirit as, but are in general much finer than, the conditions studied by Rubin and Popescu. By using this approach we are able to complete the proof that TΩ(L/K) vanishes for all of the genus field extensions considered by Fröhlich. In the course of proving this result we find that for certain extensions the vanishing of TΩ(L/K) is a refinement of the main result of Solomon concerning "wild Euler systems."