The Petersson/Kuznetsov trace formula with prescribed local ramifications

abstract: In this paper we derive refined Petersson/Kuznetsov trace formulae with prescribed local ramifications. The spectral side of these formulae \yh{picks out newforms whose associated local components come from specific sub-families of representations of given level, and are much shorter compared with the classical versions.} We use them to study the first moment and the subconvexity bound of certain Rankin-Selberg $L$-function in a hybrid setting, \yh{obtaining Weyl bound in a wider range compared to previous works}.