We use relative trace formula to prove a non-vanishing result and a subconvexity result for the twisted base change $L$-functions associated to Hilbert modular forms whose local components at ramified places are some supercuspidal representations. This generalizes the work of Feigon and Whitehouse.
Non-vanishing of central values of certain $L$-functions on ${\rm GL}(2)\times {\rm GL}(3)$
Averages and nonvanishing of central values of triple product $L$-functions
Central values of L-functions over CM fields
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