On the Bloch--Kato conjecture for GSp(4) x GL(2)

We prove the Bloch–Kato conjecture for certain critical values of degree 8 $L$-functions associated to cusp forms on $$\mathrm{G}\mathrm{S}\mathrm{p}$_4 \times $\mathrm{G}\mathrm{L}$_2$\mathrm{.}Wealsoconstructa$p$-adic Eichler–Shimura isomorphism in Hida families for ${\mathrm{G}\mathrm{S}\mathrm{p}}_4$, relating $H^2$ of automorphic vector bundles with ${\mathbf{D}}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{s}}$ of a subquotient of 'etale cohomology.