The eTNC for h^0(Spec L) over Z_p[G]

Let $L/k$ be a finite abelian extension with Galois group $G$, let $C_{L,S,T}$ be the canonical complex constructed from the $(S,T)$-arithmetic data, and let $z_{L/k,S,T}$ be its zeta element in $\mathbb{C}_p\det_{\mathbb{Z}_p[G]}(C_{L,S,T})$.

eTNC for $h^0(\operatorname{Spec}L)$ over $\mathbb{Z}_p[G]$.

$$\mathbb{Z}_p[G]\cdot z_{L/k,S,T}=\det_{\mathbb{Z}_p[G]}(C_{L,S,T}).$$

This is the equivariant Tamagawa number conjecture formulation whose validity is related in the paper to the Rubin–Stark conjecture and Iwasawa main conjectures.