Bloch–Kato dimension formula for étale cohomology Selmer groups

Let $X$ be as above, let $p$ be a prime of good reduction, and let $m<2r-1$. Write $H^m_{\operatorname{\acute{e}t}}(X_{\overline{\mathbb{Q}}},\mathbb{Q}_p(r))$ for the $p$-adic étale cohomology representation, and let $H^1_f$ and $H^0$ denote the corresponding Bloch–Kato Selmer and invariant groups. Let $V^*$ denote the dual representation. Bloch–Kato dimension formula.

$$\begin{aligned} \dim H^1_f(\mathbb{Q},H^m_{\operatorname{\acute{e}t}}(X_{\overline{\mathbb{Q}}},\mathbb{Q}_p(r))) ={}&\dim H^1_f(\mathbb{Q}_p,H^m_{\operatorname{\acute{e}t}}(X_{\overline{\mathbb{Q}}},\mathbb{Q}_p(r)))\\ &-\dim H^0(\mathbb{R},H^m(X_{\overline{\mathbb{Q}}},\mathbb{Q}_p(r)))\\ &-\dim H^0(\mathbb{Q},H^m(X_{\overline{\mathbb{Q}}},\mathbb{Q}_p(r-1))^*). \end{aligned}$$

This is presented as a form of part of the Bloch–Kato conjectures, obtained using Poitou–Tate duality. Its validity is not established in general.