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

摘要

We prove an algebraicity result for the central critical value of certainRankin-Selberg L-functions for GL(n) x GL(n-1). This is a generalization andrefinement of some results of Harder, Kazhdan-Mazur-Schmidt, Mahnkopf, andKasten-Schmidt. As an application of this result, we prove algebraicity resultsfor certain critical values of the fifth and the seventh symmetric powerL-functions attached to a holomorphic cusp form. Assuming Langlandsfunctoriality one can prove similar algebraicity results for the special valuesof any odd symmetric power L-function. We also prove a conjecture of Blasiusand Panchishkin on twisted L-values in some cases. We comment on thecompatibility of our results with Deligne's conjecture on the critical valuesof motivic L-functions. These results, as in the above mentioned works, are, ingeneral, based on a nonvanishing hypothesis on certain archimedean integrals.