Rationality for isobaric automorphic representations: the CM-case. (English) Zbl 1451.11040
Summary: In this note, we prove a simultaneous extension of the author’s joint result with M. Harrisfor critical values of Rankin-Selberg \(L\)-functions \(L(s,\Pi \times \Pi ')\) [J. Inst. Math. Jussieu 15, No. 4, 711–769 (2016; Zbl 1423.11096), Theorem 3.9] to (i) general CM-fields \(F\) and (ii) cohomological automorphic representations \(\Pi '=\Pi _1\boxplus \dots \boxplus \Pi _k\) which are the isobaric sum of unitary cuspidal automorphic representations \(\Pi _i\) of general linear groups of arbitrary rank over \(F\). In this sense, the main result of these notes, cf. Theorem 1.9, is a generalization, as well as a complement, of the main results of A. Raghuram [“Critical values of Rankin-Selberg \(L\)-functions for \(GL_n \times GL_{n-1}\) and the symmetric cube \(L\)-functions for \(GL_2\)”, Forum Math. 28, No. 3 (2016; doi:10.1515/forum-2014-0043); Int. Math. Res. Not. 2010, No. 2, 334–372 (2010; Zbl 1221.11128)] and J. Mahnkopf [J. Inst. Math. Jussieu 4, No. 4, 553–637 (2005; Zbl 1086.14019)].
MSC:
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 11G18 | Arithmetic aspects of modular and Shimura varieties |
| 11R39 | Langlands-Weil conjectures, nonabelian class field theory |
| 22E55 | Representations of Lie and linear algebraic groups over global fields and adèle rings |
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