Determining Hilbert modular forms by central values of Rankin-Selberg convolutions: the weight aspect. (English) Zbl 1426.11049
Summary: The purpose of this paper is to prove that a primitive Hilbert cusp form \(\mathbf{g}\) is uniquely determined by the central values of the Rankin-Selberg \(L\)-functions \(L(\mathbf{f}\otimes \mathbf{g}, \frac{1}{2})\), where \(\mathbf{f}\) runs through all primitive Hilbert cusp forms of weight \(k\) for infinitely many weight vectors \(k\). This result is a generalization of the work of S. Ganguly et al. [Math. Ann. 345, No. 4, 843–857 (2009; Zbl 1234.11065)] to the setting of totally real number fields, and it is a weight aspect analogue of our previous work [Trans. Am. Math. Soc. 369, No. 12, 8781–8797 (2017; Zbl 1426.11048)].
MSC:
| 11F41 | Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces |
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 11F30 | Fourier coefficients of automorphic forms |
| 11F11 | Holomorphic modular forms of integral weight |
| 11F12 | Automorphic forms, one variable |
| 11N75 | Applications of automorphic functions and forms to multiplicative problems |
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