Limiting forms of the trace formula

Abstract

We carry out the first nontrivial cases of the limiting process proposed by Langlands in his manuscript Beyond Endoscopy, with technical variations that enable us to treat the limit unconditionally. This gives an elementary proof, on GL(2), of the classification of forms such that the symmetric square L-function has a pole (including, implicitly, the construction of these forms). The result of this may be seen as one of the simplest cases of the “pipe-dream” Langlands proposes. We also apply similar methods to derive a converse theorem, and to produce a result that generalizes Duke's estimate on the dimension of weight 1 forms to arbitrary number fields—but is sharper, even over [special characters omitted], than Duke's original estimate.