Related papers: Beyond Endoscopy via the Trace Formula-I: Poisson …
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 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…
Recent work of Altu\u{g} continues the preliminary analysis of Langlands' Beyond Endoscopy proposal for $GL(2)$ by removing the contribution of the trivial representation to the trace formula using a Poisson summation formula. We show that…
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 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…
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 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…
We derive special forms of the Poisson summation formula for even and odd functions, which are applied to obtain representations for Euler-type numbers and to sum various series related to elliptic functions.
In this paper we analyze the endoscopy for $SU(2,1)$. The new results are a precise realization of the discrete series representations (in Section 2), a computation of their traces (Section 3) and an exact formula for the Poisson summation…
The group is interesting as the first example of split rank 2 semisimple group, all the irreducible unitary representations of which are known. We make a precise realization of the discrete series representations (in Section 2) by using the…
In this paper we analyze the endoscopy for $\Sp(4,\mathbb R)$. The new results are a precise realization of the discrete series representations (in Section 2), a computation of their traces (Section 3) and an exact formula for the…
Multiperforated plates exhibit high gradients and a loss of regularity concentrated in a boundary layer for which a direct numerical simulation becomes very expensive. For elliptic equations the solution at some distance of the boundary is…
Elliptic functions are largely studied and standardized mathematical objects. The two usual approaches are due to Jacobi and Weierstrass. From a contour integral which allowed us to unify many summation formulae (Euler-MacLaurin, Poisson,…
In the first part of this paper, we present a direct proof of a Poisson summation formula on the Whittaker space of $\mathrm{GL}_2$, which underlies the local Hankel transform computed by H. Jacquet. In the second part, we derive a…
We investigate convolution semigroups of probability measures with continuous densities on locally compact abelian groups, which have a discrete subgroup such that the factor group is compact. Two interesting examples of the quotient…
Semiclassical approximations often involve the use of stationary phase approximations. This method can be applied when $\hbar$ is small in comparison to relevant actions or action differences in the corresponding classical system. In many…
This note presents a rigorous introduction to a selection of distributions along with their Fourier transforms, which are commonly encountered in signal processing and, in particular, magnetic resonance imaging (MRI). In contrast to many…
We prove that a Poisson-Newton formula, in a broad sense, is associated to each Dirichlet series with a meromorphic extension to the whole complex plane of finite order. These formulas simultaneously generalize the classical Poisson formula…
For two distinguished prime $\ell$ and $p$, we prove a $\ell$-adic version of the Poisson formula for reductive $p$-adic groups. In order to do this we write an identity for the trace of regular representation and orbital integrals. Next we…
Summability has been a central object of study in difference algebra over the past half-century. It serves as a cornerstone of algebraic methods to study linear recurrences over various fields of coefficients and with respect to various…