Bloch–Kato conjecture for Hilbert modular forms

Let $g$ be a normalised cuspidal Hilbert modular newform of weight $(k_1,k_2,\ldots,k_d,w=2)$ and level $\Gamma_0(\mathfrak{n})$ over a totally real number field $F$, with each $k_i$ even and with associated automorphic representation having trivial central character. Let $E$ be the totally real number field generated by the Hecke eigenvalues $t_v(g)$. For a finite place $\lambda$ of $E$ with residue characteristic $p$, let $V_{g,\lambda}$ be the associated two-dimensional $E_\lambda$-representation, and let $H^1_f(F,V_{g,\lambda})$ denote its Bloch–Kato Selmer group. The $L$-functions are normalised so that, at good places,

$$L_v(V_{g,\lambda},s)=L_v(g,s)=\left(1-t_vq_v^{-s-1}+q_v^{-2s-1}\right)^{-1}.$$

Bloch–Kato conjecture.

$$\dim_{E_\lambda}H^1_f(F,V_{g,\lambda})=\operatorname{ord}_{s=0}L(V_{g,\lambda},s),$$

equivalently,

$$\dim_{E_\lambda}H^1_f(F,V_{g,\lambda})=\operatorname{ord}_{s=0}L(g,s).$$

This predicts that the dimension of the Bloch–Kato Selmer group equals the central order of vanishing of the relevant $L$-function. The statement is presented as a conjecture in the source; no resolution evidence is supplied here.