Algebraicity of ratios of Rankin-Selberg $L$-functions and applications to Deligne's conjecture

In this paper, we prove Deligne’s conjecture on the algebraicity of the critical values of symmetric power $L$-functions associated with modular forms of weight at least 5. We also establish new cases of Blasius’ conjecture on the algebraicity of the critical values of tensor product $L$-functions associated with modular forms. Additionally, we prove an algebraicity result for the critical values of Rankin–Selberg $L$-functions for ${\text{GL}}_n\times {\text{GL}}_2$ in the unbalanced case, which extends the previous results of Furusawa and Morimoto for $\mathrm{S}\mathrm{O}(V)\times {\text{GL}}_2$. These results are applications of our main theorem on the algebraicity of cross ratios of Rankin–Selberg $L$-functions at critical points.