Coates–Sinnott conjecture in cohomological form

Let $(K/k,S,p,n)$ satisfy the hypotheses imposed in the paper, with $n\ge 2$, and let $G={\rm Gal}(K/k)$. Set

$$H^i={\rm H}^i_{\rm et}(\mathcal O_{K,S}[1/p],\mathbb Z_p(n)).$$

Let $\Theta_{S,K/k}(1-n)$ denote the relevant equivariant $L$-function value.

Coates–Sinnott conjecture. One has

$${\rm Ann}_{\mathbb Z_p[G]}(H^1_{\rm tors})\,\Theta_{S,K/k}(1-n)\subseteq {\rm Ann}_{\mathbb Z_p[G]}(H^2).$$

The conjecture gives an annihilator-theoretic refinement of the finiteness of the relevant étale cohomology group and is presented after the paper's preceding lemma and theorem.