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

Abstract: In this paper, we prove Deligne's conjecture on the algebraicity of critical values of symmetric power $L$-functions associated to modular forms of weight greater than four. We also prove new cases of Blasius' conjecture on the algebraicity of critical values of tensor product $L$-functions associated to modular forms, and an algebraicity result on critical values of Rankin-Selberg $L$-functions for ${\rm GL}_n \times {\rm GL}_2$ in the unbalanced case, which extends the previous results of Furusawa and Morimoto for ${\rm SO}(V) \times {\rm GL}_2$. These are applications of our main result on the algebraicity of ratios of special values of Rankin-Selberg $L$-functions.