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

In this paper, we study the non-vanishing of the central values of the Rankin-Selberg $L$-function of two ad\`elic Hilbert primitive forms ${\bf f}$ and ${\bf g}$, both of which have varying weight parameter $k$. We prove that, for sufficiently large $k$, there are at least $\frac{k}{(\log k)^{c}}$ ad\`elic Hilbert primitive forms ${\bf f}$ of weight $k$ for which $L(\frac12, {\bf f}\otimes{\bf g})$ are nonzero.