Abstract

We carry out (with technical modifications) some cases of a procedure proposed by R. Langlands in Beyond Endoscopy. This gives a new proof of the classification of “dihedral forms” on GL(2), avoiding endoscopic methods: instead, it uses an infinite limiting process in the trace formula. We also explain the relationship of Langlands' idea to the results of [10] that count automorphic forms associated to Galois representations. We prove results of a similar nature over an arbitrary number field. The main tool is a version of the relative trace formula, namely a slight variant (for PGL(2)) of the Kuznetsov-Bruggeman-Miatello formula.