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 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 $\GL_n \times \GL_2$ in the unbalanced case, which extends the previous results of Furusawa and Morimoto for ${\rm SO}(V) \times \GL_2$. These results are applications of our main theorem on the algebraicity of cross ratios of Rankin--Selberg $L$-functions at critical points.