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

TL;DR

This paper proves algebraicity of special values of Rankin-Selberg and symmetric power L-functions for modular forms, extending previous results and confirming conjectures under certain hypotheses.

Contribution

It generalizes and refines algebraicity results for Rankin-Selberg L-functions and applies these to symmetric power L-functions, including cases of odd powers.

Findings

Proves algebraicity of central critical values for Rankin-Selberg L-functions.

Establishes algebraicity for fifth and seventh symmetric power L-functions.

Confirms a conjecture of Blasius and Panchishkin in specific cases.

Abstract

We prove an algebraicity result for the central critical value of certain Rankin-Selberg L-functions for GL(n) x GL(n-1). This is a generalization and refinement of some results of Harder, Kazhdan-Mazur-Schmidt, Mahnkopf, and Kasten-Schmidt. 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.