This lecture is intended as a general introduction to the trace formula. We shall describe a formula that is a natural generalization of the Selberg trace formula for compact quotient. Selberg also established trace formulas for noncompact quotients of rank 1, and our formula can be regarded as an analogue for general rank of these. As an application, we shall look at the "finite case" of the t...

La formule des traces sur les corps de nombres a été introduite par Selberg, puis développée par Arthur. Parmi les applications importantes de la formule des traces d’Arthur-Selberg, on trouve plusieurs cas du principe de fonctorialité de Langlands et l’expression de la cohomologie des variétés de Shimura en termes automorphes. Cependant, pour beaucoup de ces applications, il est nécessaire de ...

An elliptic BCn generalization of the classical two parameter Bailey Lemma is proved, and a basic one parameter BCn Bailey Lemma is obtained as a limiting case. Several summation and transformation formulas associated with the root system BCn are proved as applications, including a 6φ5 summation formula, a generalized Watson transformation and an unspecialized Rogers–Selberg identity. The last ...

For a random sample of size n obtained from a p-variate normal population, the likelihood ratio test (LRT) for the covariance matrix equal to a given matrix is considered. By using the Selberg integral, we prove that the LRT statistic converges to a normal distribution under the assumption p/n → y ∈ (0, 1]. The result for y = 1 is much different from the case for y ∈ (0, 1). Another test is stu...

The values of L-functions at special points have been the subject of intensive studies. For example, a good positive lower bound for the central value of Hecke L-functions would rule out the existence of the Landau-Siegel zero, see the notable paper [IS]; the nonvanishing of certain Rankin-Selberg L-functions is a crucial ingredient in the current development of the generalized Ramanujan conjec...

Revue internationale du TravailEarly View Livres Race, Gender and Contemporary International Labor Migration Regimes. 21st-Century Coolies?, ouvrage publié sous la direction de Leticia Saucedo et Robyn Magalit Rodriguez Niklas Selberg, Selberg Professeur associé, Faculté droit, Université Lund, SuèdeSearch for more papers by this author First published: 20 September 2023 https://doi.org/10.1111...

In this paper we study one-parameter families of signed transfer operators P± q associated to the Farey map, as well as two-parameter families of operators for the Gauss map, obtained from the Farey map by inducing. We characterise all the analytic eigenfunctions of P± q with eigenvalue λ not embedded in the continuous spectrum, which for λ = 1 are the period functions for Maass forms studied b...

This bears upon the construction of non-trivial residual square-integrable automorphic forms coming from cuspforms on smaller groups, anticipating that such automorphic forms occur as residues of Eisenstein series. For example, we can see why there is no interesting (i.e., non-constant) non-cuspidal discrete spectrum for GL(2) nor for GL(3), but only for GL(4) and larger groups. Namely, the Eis...

Abstract. We investigate the simplest class of hyperdeterminants defined by Cayley in the case of Hankel hypermatrices (tensors of the form Ai1i2...ik = f(i1+i2+· · ·+ik)). It is found that many classical properties of Hankel determinants can be generalized, and a connection with Selberg type integrals is established. In particular, Selberg’s original formula amounts to the evaluation of all Ha...

The connection between Lefschetz formulae and zeta function is explained. As a particular example the theory of the generalized Selberg zeta function is presented. Applications are given to the theory of Anosov flows and prime geodesic theorems.