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 $\mathrm{GL}_n \times \mathrm{GL}_2$ in the unbalanced case, which extends the previous results of Furusawa and Morimoto for $\mathrm{SO}(V) \times \mathrm{GL}_2$. These results are applications of our main theorem on the algebraicity of cross ratios of special values of Rankin–Selberg $L$-functions.
