Period relations for automorphic induction and applications. I. (Relations de périodes pour l’induction automorphe et applications, I.) (English. French summary) Zbl 1305.11040
Summary: Let \(K\) be a quadratic imaginary field. Let \(\Pi \) (resp. \(\Pi^{'}\)) be a regular algebraic cuspidal representation of \(\mathrm{GL}_{n}(\mathbb{A}_{K})\) (resp. \(\mathrm{GL}_{n-1}(\mathbb{A}_{K})\), which is moreover cohomological and conjugate self-dual. When \(\Pi\) is a cyclic automorphic induction of a Hecke character \(\chi\) over a CM field, we show relations between automorphic periods of \(\Pi\) defined by M. Harris [J. Reine Angew. Math. 483, 75–161 (1997; Zbl 0859.11032)] and those of \(\chi\). Consequently, we refine a formula given by H. Grobner and {M. Harris} [“Whittaker periods, motivic periods, and special values of tensor product L-functions”, Preprint, arXiv:1308.5090] for critical values of the Rankin-Selberg \(L\)-function \(L(s,\Pi\times\Pi^{'})\). This completes the proof of an automorphic version of Deligne’s conjecture in certain cases.
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 |
Citations:
Zbl 0859.11032References:
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