Existence and properties of the Bloch–Kato submodule

Let $R$ be the deformation ring, let $\mathbb{T}$ be the corresponding representation, and let $\varphi:R\rightarrow\frak{O}$ be a pseudo-geometric specialization into a finite, flat, normal extension $\frak{O}$ of $\mathbb{Z}_p$. Write $T=\mathbb{T}\otimes_{\varphi}\frak{O}$ and $V=T\otimes\mathbb{Q}_p$. Under the hypotheses $($\textbf{H.nA}$)$ and $\chi(\bar{T})=1$, define the Bloch–Kato subgroup at a specialization by

$$H^1_{\mathcal{F}_{\mathrm{BK}}}(\mathbb{Q}_p,T):=\ker\left(H^1(\mathbb{Q}_p,T)\rightarrow H^1(\mathbb{Q}_p,V\otimes B_{\mathrm{cris}})\right).$$

Bloch–Kato submodule conjecture. There is an $R$-submodule, called the Bloch–Kato submodule,

$$H^1_{\mathcal{F}_{\mathrm{BK}}}(\mathbb{Q}_p,\mathbb{T})\subset H^1(\mathbb{Q}_p,\mathbb{T}),$$

such that: (i) for every pseudo-geometric specialization $\varphi$, it maps into $H^1_{\mathcal{F}_{\mathrm{BK}}}(\mathbb{Q}_p,T)$ with finite cokernel; and (ii) if the Pančiškin condition holds, namely $H^1_{\mathcal{F}_{\mathrm{BK}}}(\mathbb{Q}_p,\mathbb{T})$ is a free $R$-module of rank $d-1$, where $d=\operatorname{rank}_R(\mathbb{T})$, then

$$H^1_{\mathcal{F}_{\mathrm{BK}}}(\mathbb{Q},\mathbb{T}):=\ker\left(H^1_{\mathcal{F}_{\mathrm{can}}}(\mathbb{Q},\mathbb{T})\longrightarrow \frac{H^1(\mathbb{Q}_p,\mathbb{T})}{H^1_{\mathcal{F}_{\mathrm{BK}}}(\mathbb{Q}_p,\mathbb{T})}\right)=0.$$

This conjectural construction is intended to identify the correct Selmer structure for deformations of motives and to formulate a Bloch–Kato refinement of the canonical Selmer group. The source presents the assertion as conjectural and supplies no resolution.