The explicit reciprocity law for Gross–Prasad Selmer classes

Let $\Delta(\Pi\times\underline{\mathcal G},\heartsuit)$ be the cohomology class supplied by the family Selmer conjecture in a region $\heartsuit$ with global root number $-1$, and let $\mathcal L_p(\Pi\times\underline{\mathcal G},\spadesuit)$ be the p-adic $L$-function in the complementary region with root number $+1$. Explicit reciprocity law. The canonical chain of localisation, projection, Perrin–Riou logarithm, and evaluation maps sends

$$\Delta(\Pi\times\underline{\mathcal G},\heartsuit)$$

to

$$\mathcal L_p(\Pi\times\underline{\mathcal G},\spadesuit).$$

This conjecture links the geometric Selmer class in the sign $-1$ region with the interpolated Gross–Prasad periods in the sign $+1$ region.