Related papers: Beyond Endoscopy via the trace formula - III: The …
We continue to work on \emph{Beyond Endoscopy} for $\mathsf{GL}_2$ over $\mathbb{Q}$ with ramification at $S = \{\infty, q_1, \dots, q_r\}$ (where $2 \in S$), generalizing the final step of Altu\u{g}'s work in the unramified setting. We…
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…
We continue the analysis of the elliptic part of the trace formula initiated in \cite{Altug:2015aa}. In that reference Poisson summation was applied to the elliptic part and the dominant term was analyzed. The main aim of this paper is to…
We continue our work on $\mathsf{GL}_2$ over $\mathbb{Q}$ in the ramified setting for \emph{Beyond Endoscopy}. We establish asymptotic formulas for each term of the trace formula when summing over $n<X$, using arbitrary smooth test…
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…
We generalize to $\mathrm{GL}(3,\mathbb{Q})$ the Poisson Summation method developed by Altu\u{g} for $\mathrm{GL}(2, \mathbb{Q})$ for the strategy of Beyond Endoscopy. Concretely, assuming Conjecture A, we isolate the contribution of the…
We continue generalizing Altu\u{g}'s work on $\mathsf{GL}_2$ over $\mathbb{Q}$ in the unramified setting for \emph{Beyond Endoscopy} to the ramified case where ramification occurs at $S=\{\infty,q_1,\dots,q_r\}$ with $2\in S$, after…
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…
In his paper "Beyond Endoscopy," Langlands tries to understand functoriality via poles of L-functions. The following paper further investigates the analytic continuation of a L-function associated to a $GL_2$ automorphic form through the…
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 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 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…
With analytic applications in mind, in particular Beyond Endoscopy ([13]), we initiate the study of the elliptic part of the trace formula. Incorporating the approximate functional equation to the elliptic part we control the analytic…
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 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…
We establish a relative trace formula on $\mathrm{GL}(n+1)$ weighted by cusp forms on $\mathrm{GL}(n)$ over number fields. The spectral side is a weighted average of Rankin-Selberg $L$-functions for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$…
For the group G = PGL(2) we prove nonstandard matching and the fundamental lemma between two relative trace formulas: on one hand, the relative trace formula of Jacquet for the quotient T\G/T, where T is a nontrivial torus; on the other,…
The continuous spectrum to the spectral side of the Arthur-Selberg trace formula is described in terms of intertwining operators, whose normalising factors involve quotients of $L$-functions. In this paper, we derive two expressions in 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.
Let f be a cusp form for SL(3, Z) associated with a generalized principal series representation of minimal weight d, spectral parameter r and associated L-function L(s, f). For $r \asymp d \asymp T$ the subconvexity bound $L(1/2, f) \ll…