  We describe an explicit `higher rank' Iwasawa theory for zeta elements
associated to the multiplicative group over abelian extensions of general
number fields. We then show that this theory leads to a concrete new strategy
for proving special cases of the equivariant Tamagawa number conjecture. As a
first application of this approach, we use it to prove new cases of the
conjecture for Tate motives over natural families of abelian CM extensions of
totally real fields for which the relevant $p$-adic $L$-functions possess
trivial zeroes.
