Determining Hilbert Modular Forms by Central Values of Rankin-Selberg Convolutions: The Weight Aspect

Abstract: 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 work is a generalization of a result of Ganguly, Hoffstein, and Sengupta to the setting of totally real number fields, and it is a weight aspect analogue of the authors recent work.