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International Mathematics Research Notices Advance Access published online on August 24, 2009
International Mathematics Research Notices, doi:10.1093/imrn/rnp127
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© The Author 2009. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oxfordjournals.org

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

A. Raghuram

Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK 74078, USA

Correspondence: Correspondence to be sent to: araghur{at}math.okstate.edu

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 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.

Communicated by Prof. Jim Cogdell


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