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

TL;DR

This paper proves algebraicity results for critical values of various automorphic $L$-functions, confirming conjectures and extending known cases, with applications to Deligne's conjecture.

Contribution

It establishes the algebraicity of ratios of Rankin-Selberg $L$-functions and applies these results to prove cases of Deligne's and Blasius' conjectures.

Findings

Proved Deligne's conjecture for symmetric power $L$-functions of modular forms.

Established algebraicity of critical values of tensor product $L$-functions.

Extended algebraicity results for Rankin-Selberg $L$-functions in the unbalanced case.

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.