On the Special Values of Certain Rankin–selberg L-functions and Applications to Odd Symmetric Power L-functions of Modular Forms

We prove an algebraicity result for the central critical value of certain RankinSelberg L-functions for GLn×GLn−1. This is a generalization and refinement of the results of Harder [15], Kazhdan, Mazur and Schmidt [23], Mahnkopf [29], and Kasten and Schmidt [22]. 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. We comment on the compatibility of our results with Deligne’s conjecture on the critical values of motivic L-functions. These results, as in the above mentioned works, are, in general, based on a nonvanishing hypothesis on certain archimedean integrals.