Selberg Trace Formula Research Articles

Mean square in the prime geodesic theorem

We prove upper bounds for the mean square of the remainder in the prime geodesic theorem, for every cofinite Fuchsian group, which improve on average on the best known pointwise bounds. The proof relies on the Selberg trace formula. For the modular group we prove a refined upper bound by using the Kuznetsov trace formula.

Global rigid inner forms and multiplicities of discrete automorphic representations

We study the cohomology of certain Galois gerbes over number fields. This cohomology provides a bridge between refined local endoscopy, as introduced in Kaletha (Ann Math (2) 184(2):559–632, 2016), and classical global endoscopy. As particular applications, we express the canonical adelic transfer factor that governs the stabilization of the Arthur–Selberg trace formula as a product of normalized local transfer factors, we give an explicit constriction of the pairing between an adelic L-packet and the corresponding S-group (based on the conjectural pairings in the local setting) that is the essential ingredient in the description of the discrete automorphic spectrum of a reductive group, and we give a proof of some expectations of Arthur.

LOCAL AVERAGE OF THE HYPERBOLIC CIRCLE PROBLEM FOR FUCHSIAN GROUPS

Let be a finite-volume Fuchsian group. The hyperbolic circle problem is the estimation of the number of elements of the -orbit of in a hyperbolic circle around of radius , where and are given points of the upper half plane and is a large number. An estimate with error term is known, and this has not been improved for any group. Recently, Risager and Petridis proved that in the special case taking and averaging over in a certain way the error term can be improved to . Here we show such an improvement for a general ; our error term is (which is better than but weaker than the estimate of Risager and Petridis in the case ). Our main tool is our generalization of the Selberg trace formula proved earlier.

Selberg and Ruelle zeta functions for non-unitary twists

In this paper we study the dynamical zeta functions of Ruelle and Selberg associated with the geodesic flow of a compact hyperbolic odd-dimensional manifold. These are functions of a complex variable s in some right half-plane of \(\mathbb {C}\). Using the Selberg trace formula for arbitrary finite dimensional representations of the fundamental group of the manifold, we establish the meromorphic continuation of the dynamical zeta functions to the whole complex plane. We explicitly describe the singularities of the Selberg zeta function in terms of the spectrum of certain twisted Laplace and Dirac operators.

The Selberg trace formula as a Dirichlet series

Abstract We explore an idea of Conrey and Li of expressing the Selberg trace formula as a Dirichlet series. We describe two applications, including an interpretation of the Selberg eigenvalue conjecture in terms of quadratic twists of certain Dirichlet series, and a formula for an arithmetically weighted sum of the complete symmetric square L-functions associated to cuspidal Maass newforms of squarefree level N > 1 ${N>1}$ .

Hypoelliptic Laplacian and probability

The purpose of this paper is to describe the probabilistic aspects underlying the theory of the hypoelliptic Laplacian, as a deformation of the standard elliptic Laplacian. The corresponding diffusion on the total space of the tangent bundle of a Riemannian manifold is a geometric Langevin process, that interpolates between the geometric Brownian motion and the geodesic flow. Connections with the central limit theorem for the occupation measure by the geometric Brownian motion are emphasized. Spectral aspects of the hypoelliptic deformation are also provided on tori. The relevant hypoelliptic deformation of the Laplacian in the case of Riemann surfaces of constant negative curvature is briefly described, in connection with Selberg's trace formula.

Spectral correspondences for Maass waveforms on quaternion groups

We prove that in most cases the Jacquet–Langlands correspondence between newforms for Hecke congruence groups and newforms for quaternion orders is a bijection. Our proof covers almost all cases where the Hecke congruence group is of cocompact type, i.e. when a bijection is possible. The proof uses the Selberg trace formula.

Operator-Valued Frames for the Heisenberg Group

A classical result of Duffin and Schaeffer gives conditions under which a discrete collection of characters on \({\mathbb {R}}\), restricted to \(E=(-\gamma ,\gamma )\subsetneq (-1/2,1/2)\), forms a Hilbert-space frame for \(L^2(E)\). For the case of characters with period one, this is just the Poisson Summation Formula. Duffin and Schaeffer show that perturbations preserve the frame condition in this case. This paper gives analogous results for the real Heisenberg group \(H_n\), where frames are replaced by operator-valued frames. The Selberg Trace Formula is used to show that perturbations of the orthogonal case continue to behave as operator-valued frames. This technique enables the construction of decompositions of elements of \(L^2(E)\) for suitable subsets \(E\) of \(H_n\) in terms of representations of \(H_n\).

A Selberg trace formula for hypercomplex analytic cusp forms

A breakthrough in developing a theory of hypercomplex analytic modular forms over Clifford algebras has been the proof of the existence of non-trivial cusp forms for important discrete arithmetic subgroups of the Ahlfors–Vahlen group. Hypercomplex analytic modular forms in turn also include Maaß forms associated to particular eigenvalues as special cases. In this paper we establish a Selberg trace formula for this new class of automorphic forms. In particular, we show that the dimension of the space of hypercomplex-analytic cusp forms is finite. Finally, we describe the space of Eisenstein series and give a dimension formula for the complete space of k-holomorphic Cliffordian modular forms.

Differences of the Selberg trace formula and Selberg type zeta functions for Hilbert modular surfaces

We present an example of the Selberg type zeta function for non-compact higher rank locally symmetric spaces. This is a generalization of Selberg's unpublished work [26] to non-compact cases. We study certain Selberg type zeta functions and Ruelle type zeta functions attached to the Hilbert modular group of a real quadratic field. We show that they have meromorphic extensions to the whole complex plane and satisfy functional equations. The method is based on considering the differences among several Selberg trace formulas with different weights for the Hilbert modular group. Besides as an application of the differences of the Selberg trace formula, we also obtain an asymptotic average of the class numbers of indefinite binary quadratic forms over the real quadratic integer ring.

On the Selberg trace formula for strictly hyperbolic groups

We show that, for the case of strictly hyperbolic groups, the right-hand side of the Selberg trace formula admits a representation in the form of a series in the eigenvalues of the Laplacian. The behavior of the Minakshisundaram function as t → 0 and t → ∞ is studied. Countably many conditions satisfied by the spectrum of the Laplacian are obtained in explicit form.

Periods of Jacobi forms and the Hecke operator

A Hecke action on the space of periods of cusp forms, that is compatible with that on the space of cusp forms, was first computed using continued fraction [20] and an explicit algebraic formula of Hecke operators acting on the space of period functions of modular forms was derived by studying the rational period functions [9]. As an application an elementary proof of the Eichler–Selberg trace formula was derived [27]. A similar modification has been applied to the space of period functions of Maass cusp forms with spectral parameter s[22,23,21]. In this paper we study the space of period functions of Jacobi forms by means of the Jacobi integral and give an explicit description of the action of Hecke operators on this space. A Jacobi–Eisenstein series E2,1(τ,z) of weight 2 and index 1 is discussed as an example. Periods of Jacobi integrals already appeared in a disguised form in the work of Zwegers in his study of the Mordell integral coming from Lerch sums [28], and mock Jacobi forms are typical examples of the Jacobi integral [10].

An Algorithm for the Computation of Eigenvalues, Spectral Zeta Functions and Zeta-Determinants on Hyperbolic Surfaces

We present a rigorous scheme that makes it possible to compute eigenvalues of the Laplace operator on hyperbolic surfaces within a given precision. The method is based on an adaptation of the method of particular solutions to the case of locally symmetric spaces and on explicit estimates for the approximation of eigenfunctions on hyperbolic surfaces by certain basis functions. It can be applied to check whether or not there is an eigenvalue in an ε-neighborhood of a given number λ > 0. This makes it possible to find all the eigenvalues in a specified interval, up to a given precision with rigorous error estimates. The method converges exponentially fast with the number of basis functions used. Combining the knowledge of the eigenvalues with the Selberg trace formula we are able to compute values and derivatives of the spectral zeta function again with error bounds. As an example we calculate the spectral determinant and the Casimir energy of the Bolza surface and other surfaces.

The trace formula for a point scatterer on a compact hyperbolic surface

An exact trace formula for the perturbation of the Laplacian by a Dirac delta potential on a compact hyperbolic Riemann surface is derived. The formula can be considered an analogue of the Selberg trace formula. The difference of perturbed and unperturbed trace is expressed as an identity term plus a sum over combinations of diffractive orbits which visit the position of the potential.

Dimension formulas for spaces of vector-valued Siegel cusp forms of degree two

We give a general arithmetic dimension formula for spaces of vector-valued Siegel cusp forms of degree two. Then, using this formula, we derive explicit dimension formulas for arithmetic subgroups of any level for each Q -form of Sp ( 2 ; R ) . Tsushima has already given the dimension formulas for some congruence subgroups of the split Q -form in Tsushima (1983, 1997) [32,33]. We obtain an alternative proof for his results by using the Selberg trace formula and the theory of prehomogeneous vector spaces. As for the non-split Q -forms, our results are new. We generalize the results and proofs given in Arakawa (1981) [1], Christian (1969, 1975, 1977) [5,6], Hashimoto (1983, 1984) [12,13], Morita (1974) [25] for the scalar-valued case to the vector-valued case using the Selberg trace formula.

On the Arthur–Selberg trace formula for $\mathrm{GL}(2)$

We extend the terms on the geometric side of the trace formula for \mathrm{GL}(2) over \mathbb{Q} continuously to a natural Fréchet algebra of non-compactly supported test functions. For the spectral side the analogous result had been obtained previously (in much greater generality) in collaboration with W. Müller.

Transfert d’intégrales orbitales pour le groupe métaplectique

AbstractWe set up a formalism of endoscopy for metaplectic groups. By defining a suitable transfer factor, we prove an analogue of the Langlands–Shelstad transfer conjecture for orbital integrals over any local field of characteristic zero, as well as the fundamental lemma for units of the Hecke algebra in the unramified case. This generalizes prior work of Adams and Renard in the real case and serves as a first step in studying the Arthur–Selberg trace formula for metaplectic groups.

A Selberg Trace Formula for Non-unitary Twists

A Selberg Trace Formula for Non-unitary Twists

Le lemme fondamental pondéré. I. Constructions géométriques

AbstractThis work is the geometric part of our proof of the weighted fundamental lemma, which is an extension of Ngô Bao Châu’s proof of the Langlands–Shelstad fundamental lemma. Ngô’s approach is based on a study of the elliptic part of the Hichin fibration. The total space of this fibration is the algebraic stack of Hitchin bundles and its base space is the affine space of ‘characteristic polynomials’. Over the elliptic set, the Hitchin fibration is proper and the number of points of its fibers over a finite field can be expressed in terms of orbital integrals. In this paper, we study the Hitchin fibration over an open set larger than the elliptic set, namely the ‘generically regular semi-simple set’. The fibers are in general neither of finite type nor separated. By analogy with Arthur’s truncation, we introduce the substack of ξ-stable Hitchin bundles. We show that it is a Deligne–Mumford stack, smooth over the base field and proper over the base space of ‘characteristic polynomials’. Moreover, the number of points of the ξ-stable fibers over a finite field can be expressed as a sum of weighted orbital integrals, which appear in the Arthur–Selberg trace formula.

The zeta functions of Ruelle and Selberg for hyperbolic manifolds with cusps

In this paper, we study the Ruelle zeta function and the Selberg zeta functions attached to the fundamental representations for real hyperbolic manifolds with cusps. In particular, we show that they have meromorphic extensions to \({\mathbb{C}}\) and satisfy functional equations. We also derive the order of the singularity of the Ruelle zeta function at the origin. To prove these results, we completely analyze the weighted unipotent orbital integrals on the geometric side of the Selberg trace formula when test functions are defined for the fundamental representations.