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

TL;DR

This paper completes the analysis of the trace formula for standard L-functions on GL(2), providing the first example of beyond endoscopy via the Arthur-Selberg trace formula and offering a new proof of the Ramanujan Delta function's L-function continuation.

Contribution

It finalizes the trace formula analysis for beyond endoscopy on GL(2), demonstrating its execution and deriving asymptotic expansions for standard L-functions.

Findings

Calculated asymptotic expansion of beyond endoscopic averages

First example of beyond endoscopy via the Arthur-Selberg trace formula

Provided a new proof of the analytic continuation of the Ramanujan Delta L-function

Abstract

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 $k\geq3$ 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 $\Delta$-function.