Fontaine–Perrin-Riou extension of the Bloch–Kato conjecture for Rankin–Selberg motives

Let $f$ and $g$ be the modular forms, with coefficient field $K$, set of primes $S$, and associated objects $L_{f\otimes g}(t)$, $\Omega(t)$, $A(t)$, $\check{A}(1-t)$, Tamagawa factors $c_p(t)$, and Shafarevich–Tate group ${\text{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}(t)$ defined in the paper. For an integer $t$ satisfying

$$k\leq t\leq k'-1,$$

Fontaine–Perrin-Riou extension of the Bloch–Kato conjecture. The following equality of fractional ideals of $O_K[1/S]$ holds:

$$\frac{L_{f\otimes g}(t)}{\Omega(t)}=\frac{\prod_{p\leq \infty}c_p(t)~{\text{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}(t)}{\#H^0(\mathbb{Q},A(t))\#H^0(\mathbb{Q},\check{A}(1-t))}.$$

This is the relevant case of the Fontaine–Perrin-Riou extension of the Bloch–Kato conjecture to arbitrary weights and not-necessarily-rational coefficients. The source presents the equality as equivalent to that conjectural extension; no resolution is supplied here.