On Deligne's conjecture for certain automorphic $L$-functions for $\mathrm{G}\mathrm{L}(3)\times \mathrm{G}\mathrm{L}(2)$

We prove Deligne’s conjecture for central critical values of certain automorphic $L$-functions for $\mathrm{G}\mathrm{L}(3)\times \mathrm{G}\mathrm{L}(2)$. The proof is base on rationality results for central critical values of triple product $L$-functions, which follow from establishing explicit Ichino’s formulae for trilinear period integrals for Hilbert cusp forms on totally real etale cubic algebras over $\mathbb{Q}$.