Non-vanishing of Rankin-Selberg Convolutions for Hilbert Modular Form

Abstract

In this paper, we study the non-vanishing of the central values of the Rankin-Selberg $L$-function of two adèlic Hilbert primitive forms $\mathbf{f}$ and $\mathbf{g}$, both of which have varying weight parameter $k$. We prove that, for sufficiently large $k$, there are at least $\frac{k}{(\log k{)}^c}$ adèlic Hilbert primitive forms $\mathbf{f}$ of weight $k$ for which $L(\frac{1}{2},\mathbf{f}\otimes \mathbf{g})$ are nonzero.