Beyond Endoscopy via the Trace Formula

Abstract

Motivated by the paper \cite{L1} of Langlands, we show how one can study the elliptic part of the trace formula in analytic applications. The thesis consists of several parts. First, by using an appropriate approximate functional equation we show how to rewrite the elliptic part of the trace formula which allows one to control the Artin $L$-functions that that appear in the formula and to smooth out the singularities of orbital integrals. In the second part we analyze the resulting formula, isolate contributions of special representations an develop asymptotic expansions of various functions that are involved in the analysis which are needed in various analytic applications (in particular for the one suggested in \cite{L1}). In the final part we apply the theory developed in the previous parts to give a new proof of the $\tfrac14$-bound of Kuznetsov (\cite{Kuz}) towards the Ramanujan conjecture, and to carry out the analysis suggested in \cite{L1} for the standard representation. This last part in particular establishes the first case of the method as suggested in \cite{L1}.