The weight two and opposite sign cases for the Fourier relative trace formulas

We provide an adelic relative trace formula proof to the Petersson/Bruggeman-Kuznetsov (PBK) formulas, specifically in the holomorphic case for $\kappa =2$ and the non-holomorphic case for $m_1{m}_2<0$. Given two sets of hypothesis on the non archimedean test function $f$, called the geometric and spectral assumptions, this approach allows us to obtain refined PBK formulas.