BEYOND ENDOSCOPY VIA THE TRACE FORMULA - III THE STANDARD REPRESENTATION

摘要

We finalize the analysis of the trace formula initiated in S. A. Altug [Beyond endoscopy via the trace formula-I: Poisson summation and isolation of special representations, Compos. Math. 151(10) (2015), 1791{1820] and developed in S. A. Altug [Beyond endoscopy via the trace formula-II: asymptotic expansions of Fourier transforms and bounds toward the Ramanujan conjecture. Submitted, preprint, 2015, Available at: arXiv:1506.08911.pdf], and calculate the asymptotic expansion of the beyond endoscopic averages for the standard L-functions attached to weight k >= 3 cusp forms on GL.2 / (cf. Theorem 1.1). This, in particular, constitutes the first example of beyond endoscopy executed via the Arthur{Selberg trace formula, as originally proposed in R. P. Langlands [Beyond endoscopy, in Contributions to Automorphic Forms, Geometry, and Number Theory, pp. 611{698 (The Johns Hopkins University Press, Baltimore, MD, 2004), chapter 22]. As an application we also give a new proof of the analytic continuation of the L-function attached to Ramanujan's Delta-function.