Beyond Endoscopy via the trace formula - III: The standard representation

We finalize the analysis of the trace formula initiated in \cite{Altug:2015aa} and developed in \cite{Altug:2015ab}, and calculate the asymptotic expansion of the beyond endoscopic averages for the standard $L$-functions attached to weight $k\ge 3$ cusp forms on $GL(2)$ (cf. Theorem \ref{mainthm}). This, in particular, constitutes the first example of beyond endoscopy executed via the Arthur-Selberg trace formula, as originally proposed in \cite{Langlands:2004aa}. As an application we also give a new proof of the analytic continuation of the $L$-function attached to Ramanujan’s $\mathrm{\Delta }$-function.