Bloch–Kato conjecture for quadratic Hilbert modular forms

Let $K$ be a real quadratic field, and let $\pi$ be a cuspidal automorphic representation of $\operatorname{GL}_2/K$ of weight $(k_1,k_2,t_1,t_2)$, where $k_i,t_i$ are integers with $k_i\geqslant 2$ and $k_1+2t_1=k_2+2t_2$. Let $L(\pi,s)$ be its $L$-function, and let $\rho_{\pi,v}:G_K\to\operatorname{GL}_2(E_v)$ be the associated Galois representation.

Bloch–Kato conjecture. If $L(\pi,1+j)\ne 0$ for some integer $j$ with $t_i\leqslant j\leqslant k_i+t_i-2$ for all $i$, then

$$H^1_{\mathrm{f}}(K,\rho_{\pi,v}^*(-j))=0.$$

This predicts the vanishing of the Bloch–Kato Selmer group at nonvanishing critical $L$-values. The paper proves new cases, while the general assertion in this setting remains open.