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

Iwasawa theory and zeta elements for $\mathbb{G}_m$