Beyond Endoscopy via the trace formula, II: Asymptotic expansions of Fourier transforms and bounds towards the Ramanujan conjecture

We continue the analysis of the elliptic part of the trace formula for $GL(2)$ initiated in the earlier paper of the author with the same title. In that paper Poisson summation was applied to the elliptic part and the dominant term was analyzed. The main aim of this paper is to study the remaining terms after Poisson summation. We analyze the Fourier transforms of (smoothed) orbital integrals and obtain exact asymptotic expansions. As an application we recover, using the Arthur-Selberg trace formula, Kuznetsov's result that the trace of the $p$th Hecke operator on cuspidal automorphic representations is bounded by $p^{\frac14}$.