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

In this paper, we prove new cases of Blasius' and Deligne's conjectures on the algebraicity of critical values of tensor product $L$-functions and symmetric odd power $L$-functions associated to modular forms. We also prove 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 results are applications of our main result on the algebraicity of ratios of special values of Rankin-Selberg $L$-functions. Let $\mathit{\Sigma},\mathit{\Sigma}',\mathit{\Pi},\mathit{\Pi}'$ be algebraic automorphic representations of general linear groups over $\mathbb Q$ such that $\mathit{\Sigma}_\infty = \mathit{\Sigma}_\infty'$ and $\mathit{\Pi}_\infty = \mathit{\Pi}_\infty'$. Based on conjectures of Clozel and Deligne, and Yoshida's computation of motivic periods, we expect the ratio \[ \frac{L(s,\mathit{\Sigma} \times \mathit{\Pi})\cdot L(s,\mathit{\Sigma}' \times \mathit{\Pi}')}{L(s,\mathit{\Sigma} \times \mathit{\Pi}')\cdot L(s,\mathit{\Sigma}' \times \mathit{\Pi})} \] to be algebraic and Galois-equivariant at critical points. We show that this assertion holds under certain parity and regularity assumptions on the archimedean components. Our second main result is to prove an automorphic analogue of Blasius' conjecture on the behavior of critical values of motivic $L$-functions upon twisting by Artin motives. We consider Rankin-Selberg $L$-functions twisted by finite order Hecke characters