Relative Trace Formula Research Articles

On the cubic Shimura lift to $\operatorname{PGL}(3)$: The fundamental lemma

The classical Shimura correspondence lifts automorphic representations on the double cover of \mathrm{SL}_{2} to automorphic representations on \mathrm{PGL}_{2} . Here we take key steps towards establishing a relative trace formula that would give a new global Shimura lift, from the triple cover of \mathrm{SL}_{3} to \mathrm{PGL}_{3} , and also characterize the image of the lift. The characterization would be through the non-vanishing of a certain global period involving a function in the space of the automorphic minimal representation \Theta_{\mathrm{SO}_8} for split \mathrm{SO}_{8}(\mathbb{A}) , consistent with a conjecture of Bump, Friedberg and Ginzburg (2001). In this paper, we first analyze a global distribution on \mathrm{PGL}_{3}(\mathbb{A}) involving this period and show that it is a sum of factorizable orbital integrals. The same is true for the Kuznetsov distribution attached to the triple cover of \mathrm{SL}_{3}(\mathbb{A}) . We then match the corresponding local orbital integrals for the unit elements of the spherical Hecke algebras; that is, we establish the fundamental lemma.

Twisted Linear Periods and a New Relative Trace Formula

Twisted Linear Periods and a New Relative Trace Formula

Manin–Drinfeld cycles and derivatives of $L$-functions

We study algebraic cycles in the moduli space of \operatorname{PGL}_2 -shtukas, arising from the diagonal torus. Our main result shows that their intersection pairing with the Heegner–Drinfeld cycle is the product of the r -th central derivative of an automorphic L -function L(\pi,s) and Waldspurger's toric period integral. When L(\pi,1/2) \neq 0 , this gives a new geometric interpretation for the Taylor series expansion. When L(\pi,1/2) = 0 , the pairing vanishes, suggesting higher order analogues of the vanishing of cusps in the modular Jacobian, as well as other new phenomena. Our proof sheds new light on the algebraic correspondence introduced by Yun and Zhang, which is the geometric incarnation of “differentiating the L -function”. We realize it as the Lie algebra action of e+f \in \mathfrak{sl}_2 on (\mathbb{Q}_\ell^2)^{\otimes 2d} . The comparison of relative trace formulas needed to prove our formula is then a consequence of Schur–Weyl duality.

Density theorems for {textrm{GL}}(n)

Strong bounds—going beyond Sarnak’s density hypothesis—are obtained for the number of automorphic forms for the group Gamma _0(q) subseteq textrm{SL}(n, {mathbb {Z}}) violating the Ramanujan conjecture at any given unramified place. The proof is based on a relative trace formula of Kuznetsov type and best-possible bounds for certain Kloosterman sums for textrm{GL}(n). Further applications are given.

A relative trace formula for obstacle scattering

We consider the case of scattering by several obstacles in Rd for d≥2. In this setting, the absolutely continuous part of the Laplace operator Δ with Dirichlet boundary conditions and the free Laplace operator Δ0 are unitarily equivalent. For suitable functions that decay sufficiently fast, we have that the difference g(Δ)−g(Δ0) is a trace-class operator and its trace is described by the Krein spectral shift function. In this article, we study the contribution to the trace (and hence the Krein spectral shift function) that arises from assembling several obstacles relative to a setting where the obstacles are completely separated. In the case of two obstacles, we consider the Laplace operators Δ1 and Δ2 obtained by imposing Dirichlet boundary conditions only on one of the objects. Our main result in this case states that then g(Δ)−g(Δ1)−g(Δ2)+g(Δ0) is a trace-class operator for a much larger class of functions (including functions of polynomial growth) and that this trace may still be computed by a modification of the Birman–Krein formula. In case g(x)=x1 2, the relative trace has a physical meaning as the vacuum energy of the massless scalar field and is expressible as an integral involving boundary layer operators. Such integrals have been derived in the physics literature using nonrigorous path integral derivations and our formula provides both a rigorous justification as well as a generalization.

A density theorem for Sp(4)$\operatorname{Sp}(4)$

Strong bounds are obtained for the number of automorphic forms for the group $\Gamma_0(q) \subseteq \operatorname{Sp}(4,\mathbb{Z})$ violating the Ramanujan conjecture at any given unramified place, which go beyond Sarnak's density hypothesis. The proof is based on a relative trace formula of Kuznetsov type, and best-possible bounds for certain Kloosterman sums for $\operatorname{Sp}(4)$.

Trace singularities in obstacle scattering and the Poisson relation for the relative trace

We consider the case of scattering by several obstacles in {mathbb {R}}^d, d ge 2 for the Laplace operator Delta with Dirichlet boundary conditions imposed on the obstacles. In the case of two obstacles, we have the Laplace operators Delta _1 and Delta _2 obtained by imposing Dirichlet boundary conditions only on one of the objects. The relative operator g(Delta ) - g(Delta _1) - g(Delta _2) + g(Delta _0) was introduced in Hanisch, Waters and one of the authors in (A relative trace formula for obstacle scattering. arXiv:2002.07291, 2020) and shown to be trace-class for a large class of functions g, including certain functions of polynomial growth. When g is sufficiently regular at zero and fast decaying at infinity then, by the Birman–Krein formula, this trace can be computed from the relative spectral shift function xi _mathrm {rel}(lambda ) = -frac{1}{pi } {text {Im}}(Xi (lambda )), where Xi (lambda ) is holomorphic in the upper half-plane and fast decaying. In this paper we study the wave-trace contributions to the singularities of the Fourier transform of xi _mathrm {rel}. In particular we prove that {hat{xi }}_mathrm {rel} is real-analytic near zero and we relate the decay of Xi (lambda ) along the imaginary axis to the first wave-trace invariant of the shortest bouncing ball orbit between the obstacles. The function Xi (lambda ) is important in the physics of quantum fields as it determines the Casimir interactions between the objects.

On the stabilization of relative trace formulae: Descent and the fundamental lemma

Motivated by the study of periods of automorphic forms and relative trace formulae, we develop the theory of descent necessary to study orbital integrals arising in the fundamental lemma for a general class of symmetric spaces over a p-adic field F. More precisely, we prove that a connected symmetric space over F enjoys a notion of topological Jordan decomposition, which may be of independent interest, and establish a relative version of a lemma of Kazhdan that played a crucial role in the proof of the Langlands–Shelstad fundamental lemma.As our main application, we use these results to prove the endoscopic fundamental lemma for the unit element of the Hecke algebra for the symmetric space associated to unitary Friedberg–Jacquet periods.

A symplectic restriction problem

We investigate the norm of a degree 2 Siegel modular form of asymptotically large weight whose argument is restricted to the 3-dimensional subspace of its imaginary part. On average over Saito–Kurokawa lifts an asymptotic formula is established that is consistent with the mass equidistribution conjecture on the Siegel upper half space as well as the Lindelöf hypothesis for the corresponding Koecher–Maaß series. The ingredients include a new relative trace formula for pairs of Heegner periods.

Transfer operators and Hankel transforms between relative trace formulas, II: Rankin–Selberg theory

The Langlands functoriality conjecture, as reformulated in the “beyond endoscopy” program, predicts comparisons between the (stable) trace formulas of different groups G1,G2 for every morphism G1L→LG2 between their L-groups. This conjecture can be seen as a special case of a more general conjecture, which replaces reductive groups by spherical varieties and the trace formula by its generalization, the relative trace formula.The goal of this article and its precursor [11] is to demonstrate, by example, the existence of “transfer operators” between relative trace formulas, which generalize the scalar transfer factors of endoscopy. These transfer operators have all properties that one could expect from a trace formula comparison: matching, fundamental lemma for the Hecke algebra, transfer of (relative) characters. Most importantly, and quite surprisingly, they appear to be of abelian nature (at least, in the low-rank examples considered in this paper), even though they encompass functoriality relations of non-abelian harmonic analysis. Thus, they are amenable to application of the Poisson summation formula in order to perform the global comparison. Moreover, we show that these abelian transforms have some structure — which presently escapes our understanding in its entirety — as deformations of well-understood operators when the spaces under consideration are replaced by their “asymptotic cones”.In this second paper we use Rankin–Selberg theory to prove the local transfer behind Rudnick's 1990 thesis (comparing the stable trace formula for SL2 with the Kuznetsov formula) and Venkatesh's 2002 thesis (providing a “beyond endoscopy” proof of functorial transfer from tori to GL2). As it turns out, the latter is not completely disjoint from endoscopic transfer — in fact, our proof “factors” through endoscopic transfer. We also study the functional equation of the symmetric-square L-function for GL2, and show that it is governed by an explicit “Hankel operator” at the level of the Kuznetsov formula, which is also of abelian nature. A similar theory for the standard L-function was previously developed (in a different language) by Jacquet.

Transfer operators and Hankel transforms between relative trace formulas, I: Character theory

The Langlands functoriality conjecture, as reformulated in the “beyond endoscopy” program, predicts comparisons between the (stable) trace formulas of different groups G1,G2 for every morphism G1L→LG2 between their L-groups. This conjecture can be seen as a special case of a more general conjecture, which replaces reductive groups by spherical varieties and the trace formula by its generalization, the relative trace formula.The goal of this article and its continuation [21] is to demonstrate, by example, the existence of “transfer operators” between relative trace formulas, which generalize the scalar transfer factors of endoscopy. These transfer operators have all properties that one could expect from a trace formula comparison: matching, fundamental lemma for the Hecke algebra, transfer of (relative) characters. Most importantly, and quite surprisingly, they appear to be of abelian nature (at least, in the low-rank examples considered in this paper), even though they encompass functoriality relations of non-abelian harmonic analysis. Thus, they are amenable to application of the Poisson summation formula in order to perform the global comparison. Moreover, we show that these abelian transforms have some structure — which presently escapes our understanding in its entirety — as deformations of well-understood operators when the spaces under consideration are replaced by their “asymptotic cones”.In this first paper we study (relative) characters for the Kuznetsov formula and the stable trace formula for SL2 and their degenerations (as well as for the relative trace formula for torus periods in PGL2), and we show how they correspond to each other under explicit transfer operators.

The relative trace formula approach for spherical varieties

We present a general framework for the relative trace formula approach to the conjecture of Sakellaridis and Venkatesh on period integrals for spherical varieties over global fields.This approach is somehow well-known for experts but it seems lack of literature.Our approach is based on the spectrum isolation technique developed by Beuzart-Plessis et al. (2019) for number fields,which avoids the difficulty from the refined spectral expansion. This technique is also available in the function field case.

Isolation of cuspidal spectrum, with application to the Gan–Gross–Prasad conjecture

We introduce a new technique for isolating components on the spectral side of the trace formula. By applying it to the Jacquet--Rallis relative trace formula, we complete the proof of the global Gan--Gross--Prasad conjecture and its refinement Ichino--Ikeda conjecture for $\mathrm{U}(n) \times \mathrm{U}(n+1)$ in the stablecase.

Multiplicities and Plancherel formula for the space of nondegenerate Hermitian matrices

This paper contains two results concerning the spectral decomposition, in a broad sense, of the space of nondegenerate Hermitian matrices over a local field of characteristic zero. The first is an explicit Plancherel decomposition of the associated L2 space thus confirming a conjecture of Sakellaridis-Venkatesh in this particular case. The second is a formula for the multiplicities of generic representations in the p-adic case that extends previous work of Feigon-Lapid-Offen. Both results are stated in terms of Arthur-Clozel's quadratic local base-change and the proofs are based on local analogs of two relative trace formulas previously studied by Jacquet and Ye and known as (relative) Kuznetsov trace formulas.

Weil representation and Arithmetic Fundamental Lemma

We study a partially linearized version of the relative trace formula for the arithmetic Gan--Gross--Prasad conjecture for the unitary group $\mathrm{U}(V)$. The linear factor in this relative trace formula provides an $\mathrm{SL}_2$-symmetry which allows us to prove by induction the arithmetic fundamental lemma over $\mathbb{Q}_p$ when $p$ is odd and $p\geq \mathrm{dim}\ V$.

An analogue of the Grothendieck–Springer resolution for symmetric spaces

Motivated by questions in the study of relative trace formulae, we construct a generalization of Grothendieck's simultaneous resolution over the regular locus of certain symmetric pairs. We use this space to prove a relative version of results of Donagi and Gaitsgory about the automorphism sheaf of regular stabilizers. We also obtain partial results toward applications in Springer theory for symmetric spaces.

Fourier–Jacobi cycles and arithmetic relative trace formula (with an appendix by Chao Li and Yihang Zhu)

In this article, we develop an arithmetic analogue of Fourier--Jacobi period integrals for a pair of unitary groups of equal rank. We construct the so-called Fourier--Jacobi cycles, which are algebraic cycles on the product of unitary Shimura varieties and abelian varieties. We propose the arithmetic Gan--Gross--Prasad conjecture for these cycles, which is related to central derivatives of certain Rankin--Selberg $L$-functions, and develop a relative trace formula approach toward this conjecture. As a necessary ingredient, we propose the conjecture of the corresponding arithmetic fundamental lemma, and confirm it for unitary groups of rank at most two and for the minuscule case.

Relative Trace Formula for Compact Quotient and Pseudo-Coefficients for Relative Discrete Series

Abstract We introduce the notion of relative pseudo-coefficient for relative discrete series representations of real spherical homogeneous spaces of reductive groups. We prove that $K$-finite relative pseudo-coefficient does not exist for semisimple symmetric spaces of type $G_{\mathbb{C}}/G_{\mathbb{R}}$, where $K$ is a maximal compact subgroup of $G_{\mathbb{C}}$, and construct strong relative pseudo-coefficients for some hyperbolic spaces. We establish a toy model for the relative trace formula of H. Jacquet for compact discrete quotient $\Gamma \backslash G$. This allows us to prove that a relative discrete series representation, which admits strong pseudo-coefficients with sufficiently small support, occurs in the spectral decomposition of $L^2(\Gamma \backslash G)$ with a nonzero period.

Functorial transfer between relative trace formulas in rank 1

According to the Langlands functoriality conjecture, broadened to the setting of spherical varieties (of which reductive groups are special cases), a map between L -groups of spherical varieties should give rise to a functorial transfer of their local and automorphic spectra. The “beyond endoscopy” proposal predicts that this transfer will be realized as a comparison between limiting forms of the (relative) trace formulas of these spaces. In this paper, we establish the local transfer for the identity map between L -groups, for spherical affine homogeneous spaces X = H \ G whose dual group is SL 2 or PGL 2 (with G and H split). More precisely, we construct a transfer operator between orbital integrals for the ( X × X ) / G -relative trace formula, and orbital integrals for the Kuznetsov formula of PGL 2 or SL 2 . Besides the L -group, another invariant attached to X is a certain L -value, and the space of test measures for the Kuznetsov formula is enlarged to accommodate the given L -value. The transfer operator is given explicitly in terms of Fourier convolutions, making it suitable for a global comparison of trace formulas by the Poisson summation formula, hence for a uniform proof, in rank 1 , of the relations between periods of automorphic forms and special values of L -functions.

On the fine expansion of the unipotent contribution of the Guo–Jacquet trace formula

For a useful class of functions (containing functions whose one finite component is essentially a matrix coefficient of a supercuspidal representation), we establish three results about the unipotent contribution of the Guo-Jacquet relative trace formula for the pair $(GL_n(D),GL_n(E))$. First we get a fine expansion in terms of global nilpotent integrals. Second we express these nilpotent integrals in terms of zeta integrals. Finally we prove that they satisfy certain homogeneity properties. The proof is based on a new kind of truncation introduced in a previous article.