Critical values of Rankin-Selberg \(L\)-functions for \(\mathrm{GL}_n \times \mathrm{GL}_{n-1}\) and the symmetric cube \(L\)-functions for \(\mathrm{GL}_2\). (English) Zbl 1417.11082
Summary: In a previous article we had proved an algebraicity result for the central critical value for \(L\)-functions for \(\mathrm{GL}_n \times \mathrm{GL}_{n-1}\) over \(\mathbb{Q}\) assuming the validity of a nonvanishing hypothesis involving archimedean integrals. The purpose of this article is to generalize that result for all critical values for \(L\)-functions for \(\mathrm{GL}_n \times \mathrm{GL}_{n-1}\) over any number field \(F\) while using certain period relations proved by F. Shahidi and the author, and some additional inputs as will be explained below. Thanks to some recent work of Binyong Sun, the nonvanishing hypothesis has now been proved. The results of this article are unconditional. Applying this to \(\mathrm{GL}_3 \times \mathrm{GL}_2\), new unconditional algebraicity results for the special values of symmetric cube \(L\)-functions for \(\mathrm{GL}_2\) over \(F\) have been proved. Previously, algebraicity results for the critical values of symmetric cube \(L\)-functions for \(\mathrm{GL}_2\) have been known only in special cases by the works of Garrett-Harris, Kim-Shahidi, Grobner-Raghuram, and Januszewski.
MSC:
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 11F41 | Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces |
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 11F75 | Cohomology of arithmetic groups |
| 22E55 | Representations of Lie and linear algebraic groups over global fields and adèle rings |
