Automorphic spectral identities and applications to automorphic L-functions on GL₂

We prove an automorphic spectral identity on GL₂ involving second moments. From it we obtain an asymptotic, with powersaving error term, for (non-archimedean) conductor-aspect integral moments, twisting by GL₁ characters ramifying at a fixed finite place. The strength of the spectral identity, and of the resulting asymptotics, is illustrated by extracting a subconvex bound in conductor aspect at a fixed finite prime.