On the special values of certain Rankin-Selberg L-functions and applications to odd symmetric power L-functions of modular forms

Abstract

We prove an algebraicity result for the central critical value of certain Rankin–Selberg L-functions for GL_n × GL_n−1. This is a generalization and refinement of the results of Harder [14], Kazhdan, Mazur, and Schmidt [23], and Mahnkopf [29]. As an application of this result, we prove algebraicity results for certain critical values of the fifth and the seventh symmetric power L-functions attached to a holomorphic cusp form. Assuming Langlands' functoriality, one can prove similar algebraicity results for the special values of any odd symmetric power L-function. We also prove a conjecture of Blasius and Panchishkin on twisted L-values in some cases. These results, as in the above works, are, in general, based on a nonvanishing hypothesis on certain archimedean integrals.