Local Trace Formula Research Articles

On the spectral theorem of Langlands

We show that the Hilbert subspace of L2(G(F)﹨G(A)) generated by wave packets of Eisenstein series built from discrete series is the whole space. Together with the work of Lapid [17], it achieves a proof of the spectral theorem of R.P. Langlands ([16], [19]) based on the work of J. Bernstein and E. Lapid [6] on the meromorphic continuation of Eisenstein series built from discrete data. I use truncation on compact sets as J. Arthur did for the local trace formula in [2].

A Local Trace Formula for p-adic Infinitesimal Symmetric Spaces: the Case of Guo–Jacquet

Abstract We establish an invariant local trace formula for the tangent space of some symmetric spaces over a non-archimedean local field of characteristic zero. These symmetric spaces are studied in Guo–Jacquet trace formulae, and our methods are inspired by works of Waldspurger and Arthur. Some other results are given during the proof including a noninvariant local trace formula, Howe’s finiteness for weighted orbital integrals, and the representability of the Fourier transform of weighted orbital integrals. These local results are preparations for the comparison of regular semi-simple terms, which are weighted orbital integrals, appearing in an infinitesimal variant of Guo–Jacquet trace formulae.

AST418 - A local trace formula for the Gan-Gross-Prasad conjecture for unitary groups: the archimedean case

ISBN 978-2-85629-919-7 This work would not have been possible without the support of several organizations. First, it was the open and stimulating atmosphere of the Max Planck Institute for Mathematics that made it so easy to begin our collaboration. Moreover, it is through the MPIM’s great generosity that we were able to continue work and finish the paper during several visits by VG and BW. Second, we benefited from the support and convivial setting of the Hausdorff Institute for Mathematics and its Trimester Program “Homotopy theory, manifolds, and field theories” during the summer of 2015. Third, the Oberwolfach Workshop “Factorization Algebras and Functorial Field Theories” in May 2016 allowed us all to gather in person and finish important discussions. In addition, OG enjoyed support from the National Science Foundation as a postdoctoral fellow under Award DMS-1204826, and BW enjoyed support as a graduate student research fellow under Award DGE-1324585. Finally, this research was carried out, in part, within the HSE University Basic Research Program and funded by the Russian Academic Excellence Project 5–100. For OG there is a large cast of mathematicians whose questions, conversation, and interest have kept these issues alive and provided myriad useful insights that are now hard to enumerate in detail. He thanks Kevin Costello for introducing him to the βγ system in graduate school—and for innumerable discussions since—as well as Dan Berwick-Evans, Ryan Grady, and Yuan Shen for grappling collaboratively with [15] throughout that period. Si Li’s many insights and questions have shaped this work substantially. Matt Szczesny’s guidance at the Northwestern CDO Workshop was crucial; his subsequent encouragement is much appreciated. OG would also like to thank Stephan Stolz and Peter Teichner for the still-running conversation about conformal field theory that influences strongly his approach to the subject. Finally, he thanks Andre Henriques, John Francis, and Scott Carnahan for letting him eavesdrop as they chatted about CDOs over a decade ago. BW feels fortunate to have stepped into this community early in his graduate work and has benefited from the support of many of the individuals mentioned above. First and foremost, he thanks his adviser Kevin Costello for guidance and Si Li for helping him to harness Feynman diagrams in the context of the BV formalism. He also thanks Ryan Grady, Matt Szczesny, and Stephan Stolz for invitations to talk about this project as well as valuable input on various aspects of it. In addition, numerous discussions with Dylan William Butson, Chris Elliott, and Philsang Yoo about perturbative QFT have informed his work. Finally, we would like to thank Matt Szczesny and James Ladouce for pointing out numerous typos and providing feedback on an earlier draft of this paper.

Local and global trace formulae for smooth hyperbolic diffeomorphisms

We define and study local and global trace formulae for discrete-time uniformly hyperbolic weighted dynamics. We explain first why dynamical determinants are particularly convenient tools to tackle this question. Then we construct counter-examples that highlight that the situation is much less well behaved for smooth dynamics than for real analytic ones. This suggests to study this question for Gevrey dynamics. We do so by constructing an anisotropic space of ultradistributions on which a transfer operator acts as a trace class operator. From this construction, we deduce trace formulae for Gevrey dynamics, as well as bounds on the growth of their dynamical determinants and the asymptotics of their Ruelle resonances.

A Local Relative Trace Formula for the Ginzburg-Rallis Model: The Geometric Side.

Following the method developed by Waldspurger and Beuzart-Plessis in their proofs of the local Gan-Gross-Prasad conjecture, we are able to prove the geometric side of a local relative trace formula for the Ginzburg-Rallis model. Then by applying such formula, we prove a multiplicity formula of the Ginzburg-Rallis model for the supercuspidal representations. Using that multiplicity formula, we prove the multiplicity one theorem for the Ginzburg-Rallis model over Vogan packets in the supercuspidal case.

The Spectral Side of Stable Local Trace Formula

Let G be a connected quasi-split reductive group over \mathbb{R} , and more generally, a quasi-split K -group over \mathbb{R} . Arthur had obtained the formal formula for the spectral side of the stable local trace formula, by using formal substitute of Langlands parameters. In this paper, we construct the spectral side of the stable local trace formula and endoscopic local trace formula directly for quasi-split K -groups over \mathbb{R} , by incorporating the works of Shelstad. In particular we give the explicit expression for the spectral side of the stable local trace formula, in terms of Langlands parameters.

Multiplicity Formula and Stable Trace Formula

Let $G$ be a connected reductive group over $\Bbb{Q}$. In this paper, we give the stabilization of the local trace formula. In particular, we construct the explicit form of the spectral side of the stable local trace formula in the Archimedean case, when one component of the test function is cuspidal. We also give the multiplicity formula for discrete series. At the same time, we obtain the stable version of $L^2$-Lefschetz number formula.

AST418 - A local trace formula for the Gan-Gross-Prasad conjecture for unitary groups: the archimedean case

In this paper, we prove, following earlier work of Waldspurger \cite{Wa1}, \cite{Wa4} a sort of local relative trace formula which is related to the local Gan-Gross-Prasad conjecture for unitary groups over a local field $F$ of characteristic zero. As a consequence, we obtain a geometric formula for certain multiplicities $m(\pi)$ appearing in this conjecture and deduce from it a weak form of the local Gan-Gross-Prasad conjecture (multiplicity one in tempered L-packets). These results were already known over $p$-adic fields \cite{Beu1} and thus are only new when $F=\mathbb{R}$.

A local trace formula for the generalized Shalika model

We study local multiplicities associated to the so-called generalized Shalika models. By establishing a local trace formula for these kind of models, we are able to prove a multiplicity formula for discrete series. As a result, we can show that these multiplicities are constant over every discrete Vogan $L$-packet and that they are related to local exterior square $L$-functions.

Geometric side of a local relative trace formula

Following a scheme suggested by B. Feigon, we investigate a local relative trace formula in the situation of a reductive $p$ -adic group $G$ relative to a symmetric subgroup $H= \underline{H}(F)$ where $\underline{H}$ is split over the local field $F$ of characteristic zero and $G = \underline{G} (F)$ is the restriction of scalars of $\underline{H} _{I E}$ relative to a quadratic unramified extension $E$ of $F$. We adapt techniques of the proof of the local trace formula by J. Arthur in order to get a geometric expansion of the integral over $H \times H$ of a truncated kernel associated to the regular representation of $G$.

A local relative trace formula for PGL(2)

Following a scheme inspired by B. Feigon, we describe the spectral side of a local relative trace formula for $G:= PGL(2,\rm E)$ relative to the symmetric subgroup $H:=PGL(2,\rm F)$ where $\rm E/\rm F$ is an unramified quadratic extension of local non archimedean fields of characteristic $0$. This spectral side is given in terms of regularized normalized periods and normalized $C$-functions of Harish-Chandra. Using the geometric side obtained in a more general setting by P. Delorme, P. Harinck and S. Souaifi , we deduce a local relative trace formula for $G$ relative to $H$. We apply our result to invert some orbital integrals.

Local trace formulae for commuting Hamiltonians in Toeplitz quantization

Local trace formulae for commuting Hamiltonians in Toeplitz quantization

A Local Trace Formula for Anosov Flows

We prove a local trace formula for Anosov flows. It relates Pollicott–Ruelle resonances to the periods of closed orbits. As an application, we show that the counting function for resonances in a sufficiently wide strip cannot have a sublinear growth. In particular, for any Anosov flow there exist strips with infinitely many resonances.

Stable local trace formula

Langlands program is the central concern of modern number theory. In general, we think that the main tool for studying this program is the trace formula. Arthur-Selberg’s trace formula is successful in solving the endoscopic case of Langlands program. Arthur stabilized the geometric side of the local trace formula in the general case, but did not stabilize the general spectral. This paper aims to solve the general case by different method in the Archimedean case, by directly stabilizing the spectral side of the local trace formula, and obtains the stable local trace formula.

A relative trace formula for PGL(2) in the local setting

We develop the local Kuznetsov trace formula on a unitary group in two variables for an unramified quadratic extension of local, non-Archimedean fields E=F and compare it to a local relative trace formula on PGL.2; E/. To define the local distributions for the relative trace formula, we define a regularized local period integral and prove that it is a PGL.2; F/-invariant linear functional. By comparison of the two local trace formulas, we get an equality between a local PGL.2; F/-period and local Whittaker functionals.

Local trace formulae and scaling asymptotics for general quantized Hamiltonian flows

A class of trace formulae in the Toeplitz quantization of Hamiltonian flows on compact symplectic manifolds is given a local intepretation in terms of certain scaling asymptotics near a symplectic fixed locus; these are reminiscent of the near diagonal asymptotics of equivariant Szegö kernels. In spectral theory, the distributional trace of a positive elliptic operator encapsulates information on the asymptotic distribution of its eigenvalues; a trace formula relates the singularities of the trace to Poincaré data of an appropriate Hamiltonian flow. This leads to the asymptotics of certain smoothing kernels, related to the Fourier transform of the trace of a wave operator. In Toeplitz quantization, we consider analogues of these, with the wave operator replaced by a quantized Hamiltonian flow, possibly composed with a family of zeroth order Toeplitz operators. We study the concentration of these smoothing kernels near the fixed loci of the linearized dynamics, and specifically how the Poincaré map controls their scaling asymptotics. This generalizes previous results on holomorphic flows.

LOCAL TRACE FORMULAE AND SCALING ASYMPTOTICS IN TOEPLITZ QUANTIZATION

A trace formula for Toeplitz operators was proved by Boutet de Monvel and Guillemin in the setting of general Toeplitz structures. Here, we give a local version of this result for a class of Toeplitz operators related to continuous groups of symmetries on quantizable compact symplectic manifolds. The local trace formula involves certain scaling asymptotics along the clean fixed locus of the Hamiltonian flow of the symbol, reminiscent of the scaling asymptotics of the equivariant components of the Szegö kernel along the diagonal.

Sur les représentations elliptiques du groupe $\mathit{SL}(n)$ $p$-adique

The paper is devoted to classify all irreducible tempered representations of $\mathit{SL}(n)$ $p$-adic which appear in the spectral side of J. Arthur's local trace formula. We give description of such representations which are elliptic or irreducibly induced from elliptic representations of proper Levi subgroups with certain condition on the Weyl group.

Resonances and spectral shift function for a magnetic Schrödinger operator

We consider the three-dimensional Schrödinger operator H0 with a constant magnetic field and subject to an electric potential v0 depending only on the variable along the magnetic field x3. The operator H0 has infinitely many eigenvalues of infinite multiplicity embedded in its continuous spectrum. We perturb H0 by smooth scalar potentials V=O(⟨(x1,x2)⟩−δ⊥⟨x3⟩−δ∥), δ⊥>2, δ∥>1. We assume also that V and v0 have an analytic continuation, in the magnetic field direction, in a complex sector outside a compact set. We define the resonances of H=H0+V as the eigenvalues of the nonself-adjoint operator obtained from H by analytic distortions of Rx3. We study their distribution near any fixed real eigenvalue of H0, 2bq+λ for q∊N. In a ring centered at 2bq+λ with radii (r,2r), we establish an upper bound, as r tends to 0, of the number of resonances. This upper bound depends on the decay of V at infinity only in the directions (x1,x2). Finally, we deduce a representation of the derivative of the spectral shift function for the operator pair (H0,H) in terms of resonances. This representation justifies the Breit–Wigner approximation and implies a local trace formula.

RESONANCES AND SPECTRAL SHIFT FUNCTION FOR THE SEMI-CLASSICAL DIRAC OPERATOR

We consider the selfadjoint operator H = H0+ V, where H0is the free semi-classical Dirac operator on ℝ3. We suppose that the smooth matrix-valued potential V = O(〈x〉-δ), δ > 0, has an analytic continuation in a complex sector outside a compact. We define the resonances as the eigenvalues of the non-selfadjoint operator obtained from the Dirac operator H by complex distortions of ℝ3. We establish an upper bound O(h-3) for the number of resonances in any compact domain. For δ > 3, a representation of the derivative of the spectral shift function ξ(λ,h) related to the semi-classical resonances of H and a local trace formula are obtained. In particular, if V is an electro-magnetic potential, we deduce a Weyl-type asymptotics of the spectral shift function. As a by-product, we obtain an upper bound O(h-2) for the number of resonances close to non-critical energy levels in domains of width h and a Breit–Wigner approximation formula for the derivative of the spectral shift function.