The Tamagawa number conjecture for $h^0(K)(j)$

Let $K$ be a number field, let $S$ contain the archimedean places and the places above $p$, and let $j$ be an integer. Let $\vartheta_{\mathbb{Z}_p(j)}$ denote the period isomorphism attached to the compactly supported complex $\operatorname{R}\Gamma_c(\mathcal{O}_{K,S},\mathbb{Z}_p(j))$, and let $\zeta_{K,S}^\ast(j)$ be the leading term of the truncated Dedekind zeta function at $s=j$.

The Tamagawa number conjecture. There exists a unique $\mathbb{Z}_p$-basis

$$z_{K,S} \in {\det}_{\mathbb{Z}_p}^{-1}(\operatorname{R}\Gamma_c(\mathcal{O}_{K,S},\mathbb{Z}_p(j)))$$

with

$$\vartheta_{\mathbb{Z}_p(j)}(z_{K,S})=\zeta_{K,S}^\ast(j).$$

The text presents this as the precise Tamagawa number conjecture and later records evidence in several cases, but does not claim a general resolution.