Related papers: The Functional Equation and Beyond Endoscopy
Langlands' functoriality principle predicts deep relations between the local and automorphic spectra of different reductive groups. This has been generalized by the relative Langlands program to include spherical varieties, among which…
Using Langlands's {\it Beyond Endoscopy} idea and analytic number theory techniques, we study the Asai L-function associated to a real quadratic field $\mathbf{K}/\Q.$ If the Asai L-function associated to an automorphic form over…
In the early 2000's, R. Langlands proposed a strategy called Beyond Endoscopy to attack the principle of functoriality, which is one of the central questions of present day mathematics. A first step was achieved by A. Altug who worked with…
The near-completion of the program of endoscopy poses the question of what lies next. This article takes a broad view of ideas beyond the program of endoscopy, highlighting the connections among them, and emphasizing the relationship…
In recent years L-functions and their analytic properties have assumed a central role in number theory and automorphic forms. In this expository article, we describe the two major methods for proving the analytic continuation and functional…
We try to understand the poles of L-functions via taking a limit in a trace formula. This technique avoids endoscopic and Kim-Shahidi methods. In particular, we investigate the poles of the Rankin-Selberg L-function. Using analytic number…
We establish endoscopic and stable trace formulas whose discrete spectral terms are weighted by automorphic $L$-functions, by the use of basic functions that are incorporated into the global spectral and geometric coefficients. This is a…
According to the Langlands functoriality conjecture, broadened to the setting of spherical varieties (of which reductive groups are special cases), a map between L-groups of spherical varieties should give rise to a functorial transfer of…
At the beginning of this century, Langlands introduced a strategy known as \emph{Beyond Endoscopy} to attack the principle of functoriality. Altu\u{g} studied $\mathsf{GL}_2$ over $\mathbb{Q}$ in the unramified setting. The first step…
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…
We present a "beyond-endoscopic" treatment of the functional equation for the standard $L$-function of a holomorphic cusp form with level and nebentypus. We use Petersson's formula and methods from Venkatesh's thesis and "spectral…
Langlands' beyond endoscopy proposal for establishing functoriality motivates interesting and concrete problems in the representation theory of algebraic groups. We study these problems in a setting related to the Langlands $L$-functions…
This paper is a first attempt at getting information on a symmetric power representation of a $GL_2$ automorphic form via a trace formula that is beyond endoscopic techniques. In particular, we study the symmetric third power representation…
This is an informal note that explains that the classical Langlands theory over function fields can be obtained from the geometric one by taking the trace of Frobenius. The operation of taking the trace of Frobenius takes place at the…
We give a new proof of the converse theorem for Maass forms on ${\rm GL}(3)$ using a technique that is inspired by Langlands' philosophy of "beyond endoscopy", thereby implementing these ideas for the first time in a higher rank setting.
We study some tempered endoscopic cases of Langlands functoriality on the $n$-variable unitary groups via the simple stable trace formula. This extends previous work of Rogawski and Clozel-Harris-Labesse. Ramakrishnan and Kim-Shahidi have…
Following our joint work arXiv:1003.4578 with Robert Langlands, we make the first steps toward developing geometric methods for analyzing trace formulas in the case of the function field of a curve defined over a finite field. We also…
There has been a recent coming together of the Converse Theorem for $\gln$ and the Langlands-Shahidi method of controlling the analytic properties of automorphic $L$-functions which has allowed us to establish a number of new cases of…
We express the discrete noncuspidal terms in the spectral side of the trace formula for GL(2) in terms of orbital integrals, obtaining a geometric expansion for the cuspidal part of the trace formula. Assuming the Ramanujan conjecture for…
We show that the cuspidal component of the stable trace formula of a special odd orthogonal group over a number field, satisfies a weak form of beyond endoscopic decomposition. We also study the $r$-stable trace formula, when $r$ is the…