Holomorphy of Rankin triple \(L\)-functions; special values and root numbers for symmetric cube \(L\)-functions. (English) Zbl 1005.11026
Summary: We prove the holomorphy of Rankin triple \(L\)-functions for three cusp forms on GL(2) on the entire complex plane if at least one of them is nonmonomial. We conclude the paper by proving the equality of our root numbers for the third and the fourth symmetric power \(L\)-functions with those of Artin through the local Langlands correspondence. We also revisit Deligne’s conjecture on special values of symmetric cube \(L\)-functions.
MSC:
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 22E55 | Representations of Lie and linear algebraic groups over global fields and adèle rings |
Keywords:
holomorphy of Rankin triple \(L\)-functions; cusp forms on GL(2); local Langlands correspondence; Deligne’s conjecture; special values of symmetric cube \(L\)-functionsReferences:
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