Explicit units and the equivariant Tamagawa number conjecture

Let L/K be a finite abelian extension of number fields of group G. We study the Tamagawa number T Omega (L/K) of h(0)(Spec L), considered as a motive defined over K and with coefficients Q[G]. For a large class of extensions L/K we interpret the conjectural vanishing of T Omega (L/K) in terms of the existence of S-units in L satisfying a variety of explicit conditions. The se explicit conditions are in the same spirit as, but are in general much f iner than, the conditions studied by Rubin and Popescu. By using this appro ach we are able to complete the proof that T Omega (L/K) vanishes for all o f the genus field extensions considered by Frohlich. In the course of provi ng this result we find that for certain extensions the vanishing of T Omega (L/K) is a refinement of the ma-in result of Solomon concerning "wild Eule r systems."