A generalized PGL(2) Petersson/Bruggeman-Kuznetsov formula for analytic applications

We develop generalized Petersson/Bruggeman-Kuznetsov (PBK) formulas for specified local components at non-archimedean places. In fact, we introduce two hypotheses on non-archimedean test function pairs f ↔ π (f) , called geometric and spectral hypotheses, under which one obtains “nice” PBK formulas by the adelic relative trace function approach. Then, given a supercuspidal representation σ of PGL₂( Qₚ) , we study extensively the case that π (f) is a projection onto the line of the newform if π is isomorphic to σ or its unramified quadratic twist, and π (f) = 0 otherwise. As a first application, we prove an optimal large sieve inequality for families of automorphic representations that arise in our framework.