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.
Categories: math.NT