Werner Mueller, Birgit Speh, Erez M. Lapid
Let G be the group GL(n) over a number field E and let A be the ring of adeles of E. In this paper we prove that the spectral side of the Arthur trace formula for G is absolutely convergent for all integrable rapidly decreasing functions on $G(A)^1$.
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In this paper we study the absolute convergence of the spectral side of the Arthur trace formula. We reduce the problem of the absolute convergence to a problem about local components of automorphic representations. The latter problem can be settled for GL(n).
We give upper bounds for the absolute value of the global coefficients appearing in the fine geometric expansion of Arthur's trace formula for GL(n).
We establish a fine expansion for the geometric part of the Arthur-Selberg trace formula (as it was conjectured by Werner Hoffmann). For the general linear group, we deduce an expression for the contributions of regular by blocks unipotent orbits (orbits with one Jordan block with multiplicity). As a consequence, we find formulas for Arthur's global coefficients attached to such orbits.
We show that the distributions occurring in the geometric and spectral side of the twisted Arthur-Selberg trace formula extend to non-compactly supported test functions. The geometric assertion is modulo a hypothesis on root systems proven when the group is split. The result extends the work of Finis-Lapid (and Müller, spectral side) to the twisted setting. We use the absolute convergence to give a geometric interpretation of sums of residues of certain Rankin-Selberg L-functions.
The theme of the article is the study of the unipotent part of Arthur's trace formula for general linear groups. The case of regular (or "regular by blocks") unipotent orbits has been essentially done in a previous paper. Here we are interested by the contribution of Richardson orbits that are induced by Levi subgroups with two-by-two distinct blocks. In this case, the contribution is remarkably given by a global unipotent weighted orbital integral. As a corollary, we get integral formulas for some of Arthur's global coefficients. We also present a new construction of Arthur's local unipotent …
We study the non-semisimple terms in the geometric side of the Arthur trace formula for the split symplectic similitude group or the split symplectic group of rank 2 over any algebraic number field. In particular, we show that the coefficients of unipotent orbital integrals are expressed by the Dedekind zeta function, Hecke L-functions, and the Shintani zeta function for the space of binary quadratic forms.
These are Notes prepared for nine lectures given at the Mathematical Sciences Research Institute, MSRI, Berkeley during the period January--March 1995. It is a pleasant duty to record here my gratitude to MSRI, and its staff, for making possible this 1994--95 Special Year in Automorphic Forms, and for providing such a setting for work. The purpose of these Notes is to describe the contents of Arthur's earlier, foundational papers on the trace formula. In keeping with the introductory nature of the lectures, we have sometimes illustrated the ideas of Arthur's general theory by applying them in …
Let F be a global function field with constant field $\mathbb{F}_q$. Let G be a reductive group over $\mathbb{F}_q$. We establish a variant of Arthur's truncated kernel for G and for its Lie algebra which generalizes Arthur's original construction. We establish a coarse geometric expansion for our variant truncation. As applications, we consider some existence and uniqueness problems of some cuspidal automorphic representations for the functions field of the projective line $\mathbb{P}^1_{\mathbb{F}_q}$ with two points of ramifications.
We pursue Arthur's invariant trace formula for certain coverings of connected reductive groups by deducing explicit formulas for its spectral side. This is based on some results in local harmonic analysis from an earlier preprint. The arguments are due to Arthur, and we will explain the necessary fine-tunings for coverings.
We show that, for a large class of test functions, the unipotent contributions in the trace formula for $GL(n)$ over a number field, can be obtained from zeta functions and integrals of Eisenstein series. The main innovation is a new truncation borrowed from a work of Schiffmann on Higgs bundles.
We show, using the trace formula, that any Newton stratum of a Shimura variety of PEL-type of types (A) and (C) is non-empty at the primes of good reduction. Furthermore we prove conditionally the non-emptiness for Shimura data associated to odd Spin groups. Our results are conditional on Rapoport-Langlands conjecture and Arthur's conjectures on the discrete spectrum. Both these results have been announced by Arthur and Kisin in significant cases.
The endoscopic classification via the stable trace formula comparison provides certain character relations between irreducible cuspidal automorphic representations of classical groups and their global Arthur parameters, which are certain automorphic representations of general linear groups. It is a question of J. Arthur and W. Schmid that asks: How to construct concrete modules for irreducible cuspidal automorphic representations of classical groups in term of their global Arthur parameters? In this paper, we formulate a general construction of concrete modules, using Bessel periods, for cuspi…
In this paper we construct explicitly a square integrable residual automorphic representation of the special orthogonal group $SO_{2n}$, through Eisenstein series. We show that this representation comes from an elliptic Arthur parameter $ψ$ and appears in the space $L^2(SO_{2n}(\mathbb{Q})\backslash SO_{2n}(\mathbb{A}_{\mathbb{Q}}))$ with multiplicity one. Next, we consider parameters whose Hecke matrices, at the unramified places, have eigenvalues bigger (in absolute value), than those of the parameter constructed before. The main result is, that these parameters cannot be cuspidal. We establ…
The coarse trace formula of GL(4)
本文给出 GL(4) 的显式粗迹公式，证明其迹公式两侧发散项相等，并得到几何侧分支轨道的显式公式。
The author derives an expression for one side of the local relative trace formula, at the level of Lie algebras, by combining methods of Arthur and Harish-Chandra with the structure theory for reductive symmetric spaces.
Let $G=\mathbf{G}(\mathbb{R})$ be the group of real points of a quasi-split connected reductive algebraic group defined over $\mathbb{R}$. Assume furthermore that $G$ is a classical group (symplectic, special orthogonal or unitary). We show that the packets of irreducible unitary cohomological representations defined by Adams and Johnson in 1987 coincide with the ones defined recently by J. Arthur in his work on the classification of the discrete automorphic spectrum of classical groups (C.-P. Mok for unitary groups). For this, we compute the endoscopic transfer of the stable distributions on …
La formule des traces pour les revêtements de groupes réductifs connexes. II. Analyse harmonique locale
本文建立了连通约化群覆盖的局部调和分析结果，包括Plancherel公式、交织算子规范化、特征局部行为及不变局部迹公式，并由此推出温和特征的稠密性。
The trace formula of GL(3)
本文研究了GL(3)的迹公式，证明了几何侧和谱侧的发散项相等并抵消，推导了分支轨道积分的显式公式，并展示了谱侧展开式的归一化 intertwining 运算符的应用。
Arthur packets for metaplectic groups
本文定义了特征零局部域上亚纯群的 Arthur 包，并利用迹公式得出数域上真离散 $L^2$-自守谱的重数公式，证实了 Gan 的猜想，且与一阶和二阶现有结果相容。
Stabilization of the trace formula for metaplectic groups
本文通过内窥传递将亚辛群的 Arthur-Selberg 迹公式分解为分裂奇正交群乘积的稳定迹公式，并利用长归纳过程完成证明，同时稳定化了局部迹公式。
In this work, we study the intersection cohomology of Siegel modular varieties. The goal is to express the trace of a Hecke operator composed with a power of the Frobenius endomorphism (at a good place) on this cohomology in terms of the geometric side of Arthur's invariant trace formula for well-chosen test functions. Our main tools are the results of Kottwitz about the contribution of the cohomology with compact support and about the stabilization of the trace formula, Arthur's $L^2$ trace formula and the fixed point formula of the author. We "stabilize" this last formula, ie express it as a…
It is the last paper of a long series. We present the spectral side of the twisted trace formula and its stable version. Using the method of Arthur, we finish the proof of the stabilization of the two sides of this twisted trace formula.
In this paper an integral transform between spaces of nonstandard test functions on the affine space of dimension n is constructed. The integral transform satisfies a summation formula of Poisson type, which is derived from an analogue of the Arthur-Selberg trace formula for the Lie algebra of n by n matrices.
We begin the proof of the stabilization of the twisted trace formula. Here we prove that almost all "coefficients" appearing in this formula are equal to their endoscopic counterpart. It is the generalization to the twisted case of the result proved by Arthur in his second article on the stabilization of the trace formula.
The local trace formula gives strong relations between two types of invariant distributions on a reductive group defined over a local field: orbital integrals and characters of representations. For connected reductive groups, the formula was proved by J. Arthur. We extend the formula to twisted spaces, that is, roughly speaking, to non-connected reductive groups. The formula has the same consequences as in the connected case. In particular, we compute the elliptic product of tempered elliptic twisted representations.
This paper begins a new approach to the $r$-trace formula, without removing the nontempered contribution to the spectral side. We first establish an invariant trace formula whose discrete spectral terms are weighted by automorphic $L$-functions. This involves extending the results of Finis, Lapid, and Müller on the continuity of the coarse expansion of Arthur's noninvariant trace formula to the refined expansion, and then to the invariant trace formula, while incorporating the use of basic functions at unramified places.
This paper is concerned with two problems. One is to count Hitchin bundles on a projective curve and the other is to get an explicit formula for the nilpotent part of the Arthur-Selberg trace formula for a simple test function. The fact that the two problems are in fact related has been noticed in a previous paper. We expand the nilpotent part of the Arthur-Selberg trace formula in a sum of adelic integrals indexed by nilpotent orbits. For "regular by blocks" orbits, we get an explicit formula for these integrals in terms of the zeta function of the curve.
This paper is one of a series whose goal is to stabilize the twisted Arthur-Selberg's trace formula. Here we define the objects appearing in the geometric side of the twisted trace formula. We define also the similar stable and endoscopic objects. We state the principal theorems concerning the stabilization of this geometric side. The proofs will be given in forthcoming papers.
We study the genuine part of the Arthur-Selberg trace formula for some nonlinear covers of connected reductive groups. As a first step towards the invariant trace formula, we express the geometric side in terms of weighted orbital integrals. In particular, our results apply to the covers constructed by Brylinksi and Deligne.
Beyond endoscopy for $\mathsf{GL}_2$ over $\mathbb{Q}$ with ramification 5: cancellation theory
该研究完成了有理数域上分歧为5的GL₂的超越内窥镜相关工作，证明了迹公式项的渐近公式为o(X)及迹公式极限形式的恒等式，为相关领域提供了关键理论结果。
In the preceeding article, we have generalized to the twisted case the proposition 3.1 of the third Arthur's article on the stabilization of the trace formula. The base-field was p-adic. Here, we prove the same result over the real base-field. We prove also auxiliary results on the differential equations and the jump's conditions satisfied by the weighted orbital integrals.
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 trace formula and endoscopy 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 Langlan…
We establish the invariant trace formula (à la Arthur) for the adélic covers of connected reductive groups over a number field, under the hypothesis that the trace Paley-Wiener theorem is verified for all Levi subgroups at the real archimedean places. For instance, this hypothesis can be verified for the metaplectic covers of GL(n), or the twofold metaplectic cover of Sp(2n). We also give simple trace formulae when the test function satisfies certain cuspidality properties.
We establish endoscopic and stable trace formulas whose discrete spectral terms are weighted by automorphic $L$-functions, by the use of basic functions that are incorporated into the global spectral and geometric coefficients. This is a continuation of a previous work of the author which established the corresponding weighted invariant trace formula for noncompactly supported test functions. The meromorphic continuation of these weighted trace formulas would yield $r$-trace formulas, and can therefore be seen as precursors to them. Along the way, we formulate a weighted form of the Langlands-…
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…
On the absolute convergence of the spectral side of the twisted trace formula
研究约化代数群扭曲迹公式谱侧的绝对收敛性，核心方法是将Finis - Lapid - Müller的定理扩展到扭曲情形并给出详细证明，主要贡献是为相关研究及应用提供了理论基础。
This paper proves the existence of cuspidal automorphic forms for a reductive group, invariant under an automorphism of finite order. The techniques used are a local analysis of orbital integrals and the Arthur-Selberg trace formula.
We consider the problem of explicitly computing dimensions of spaces of automorphic or modular forms in level one, for a split classical group $\mathbf{G}$ over $\mathbb{Q}$ such that $\mathbf{G}(\R)$ has discrete series. Our main contribution is an algorithm calculating orbital integrals for the characteristic function of $\mathbf{G}(\mathbb{Z}_p)$ at torsion elements of $\mathbf{G}(\mathbb{Q}_p)$. We apply it to compute the geometric side in Arthur's specialisation of his invariant trace formula involving stable discrete series pseudo-coefficients for $\mathbf{G}(\mathbb{R})$. Therefore we e…
We finalize the analysis of the trace formula initiated in \cite{Altug:2015aa} and developed in \cite{Altug:2015ab}, and calculate the asymptotic expansion of the beyond endoscopic averages for the standard $L$-functions attached to weight $k\geq3$ cusp forms on $GL(2)$ (cf. Theorem \ref{mainthm}). This, in particular, constitutes the first example of beyond endoscopy executed via the Arthur-Selberg trace formula, as originally proposed in \cite{Langlands:2004aa}. As an application we also give a new proof of the analytic continuation of the $L$-function attached to Ramanujan's $Δ$-function.
We describe an approach to express the geometric side of the Arthur-Selberg trace formula in terms of zeta integrals attached to prehomogeneous vector spaces. This will provide explicit formulas for weighted orbital integrals and for the coefficients by which they are multiplied in the trace formula. We implement this programme for the principal unipotent conjugacy class. The method relies on certain convergence results and uses the notions of induced conjugacy classes and canonical parabolic subgroups. So far, it works for certain types of conjugacy classes, which covers all classes appearing…
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 prepared for the comparison of regular semi-simple terms, which are weig…
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 study the remaining terms after Poisson summation. We analyze the the Fourier transforms of (smoothed) orbital integrals and obtain exact asymptotic expansions. As an application we recover, using the Arthur-Selberg trace formula, Kuznetsov's result (cf. \cite{Kuznetsov:1980aa}) that the trace of the $p$th Hecke operator on cuspidal automorphic representation…
It is one of a series of papers whose goal is to stabilize the twisted trace formula. We consider here a "twisted space" over the real field. We prove in this twisted situation the results obtained by Arthur in his Selecta's paper. That is the existence of transfer of tempered representations. We use the Paley-Wiener's theorems of Renard and Delorme-Mezo, the work of Shelstad about transfer of functions and a recent result of Moeglin about super-tempered representations. As corollaries, we obtain a stable version of the Paley-Wiener theorem and the fact that transfer of functions preserves K-f…
We establish an infinitesimal variant of Guo-Jacquet trace formula for the case of $(GL_{2n,D}, GL_{n,D}\times GL_{n,D})$. It is a kind of Poisson summation formula obtained by an analogue of Arthur's truncation process. It consists in the equality of the sums of two types of distributions which are non-equivariant in general: one type is associated to rational points in the categorical quotient, while the other type is the Fourier transform of the first type. For regular semi-simple points in the categorical quotient, we obtain weighted orbital integrals.
Limit Multiplicity For Unitary Groups and The Stable Trace Formula
该研究利用Arthur的迹公式稳定化及相关表示内窥分类，给出酉群$U(a,b)$部分非tempered表示的极限重数上界，并应用于酉群余紧算术子群的上同调增长研究。
We state a variant of Arthur's weighted fundamental lemma for the metaplectic group of Weil, which will be an essential ingredient of the stable trace formula. Over a local field of large enough residual characteristic, we give a proof using the method of descent, which is conditional upon the weighted nonstandard fundamental lemma on Lie algebras. In view of the works of Chaudouard and Laumon, this condition is expected to hold.
We establish an infinitesimal variant of Guo-Jacquet trace formula for the case of a central simple algebra over a number field $F$ containing a quadratic field extension $E/F$. It is an equality between a sum of geometric distributions on the tangent space of some symmetric space and its Fourier transform. To prove this, we need to define an analogue of Arthur's truncation and then use the Poisson summation formula. We describe the terms attached to regular semi-simple orbits as explicit weighted orbital integrals. To compare them to those for another case studied in our previous work, we sta…
Let G be a reductive algebraic group over Q, and suppose that Gamma is an arithmetic subgroup of G(R) defined by congruence conditions. A basic problem in arithmetic is to determine the multiplicities of discrete series representations in L^2(Gamma \ G(R)), and in general to determine the traces of Hecke operators on these spaces. In this paper we give a conjectural formula for the traces of Hecke operators, in terms of stable distributions. It is based on a stable version of Arthur's formula for L^2-Lefschetz numbers, which is due to Kottwitz. We reduce this formula to the computation of elli…
Approche non-invariante de la correspondance de Jacquet-Langlands : analyse spectrale
本文利用非不变迹公式的谱分析完成整体Jacquet-Langlands对应证明，定义并证明非不变谱转移与几何转移一致，从而肯定Arthur猜想并推广Kazhdan定理。
This thesis provides an explicit, general trace formula for the Hecke and Casimir eigenvalues of GL(2)-automorphic representations over a global field. In special cases, we obtain Selberg's original trace formula. Computations for the determinant of the scattering matrices, the residues of the Eisenstein series, etc. are provided. The first instance of a mixed, uniform Weyl law for every algebraic number field is given as standard application. "Mixed" means that automorphic forms with preassigned discrete series representation at a set of real places are counted. "Uniform" indicates that the e…
A note on Poisson summation for GL(2)
本文从adelic视角重新审视GL(2)上通过修改泊松求和公式抵消平凡表示贡献的方法。
The stabilization of the Frobenius--Hecke traces on the intersection cohomology of orthogonal Shimura varieties
本文研究有理数域上特殊正交群的 Shimura 簇，证明并稳定化 Baily–Borel 紧化交上同调上 Frobenius–Hecke 迹的 Morel 公式，并借助自守表示内窥镜分类计算特殊情形的 Hasse–Weil ζ 函数。
For any reductive group G over a global function field, we use the cohomology of G-shtukas with multiple modifications and the geometric Satake equivalence to prove the global Langlands correspondence for G in the "automorphic to Galois" direction. Moreover we obtain a canonical decomposition of the spaces of cuspidal automorphic forms indexed by global Langlands parameters. The proof does not rely at all on the Arthur-Selberg trace formula.
In this paper we establish the endoscopic classification of tempered representations of quasi-split unitary groups over local fields, and the endoscopic classification of the discrete automorphic spectrum of quasi-split unitary groups over global number fields. The method is analogous to the work of Arthur on orthogonal and symplectic groups, based on the theory of endoscopy and the comparison of trace formulas on unitary groups and general linear groups.
This paper gives combinatorial formulas for discrete series constants, both stable and unstable, on real reductive groups. It also carries out one step of the comparison of the topological trace formula for Hecke operators with Arthur's trace formula for Hecke operators, the step in which the topological terms are combined in such a way that stable discrete series characters appear.
Développement fin de la contribution unipotente à la formule des traces sur un corps global de caractéristique p>0, I
本文发展了连通约化群上的单幂 $F$-层理论，并据此确立了特征 $p>0$ 整体域上迹公式中单幂元素贡献的精细展开。
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 $L^2$ 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 …
In arXiv:1208.1945, Shin and Templier proved certain equidistribution bounds on local components of certain families of automorphic representations. We extend their weight-aspect results to families of automorphic representations where the Archimedean component is restricted to a single discrete-series representation instead of an entire $L$-packet. We do this by using a so-called "hyperendoscopy" version of the stable trace formula developed by Ferrari. The main technical difficulties are defining a version of hyperendoscopy that works for groups without simply connected derived subgroup and …
Asymptotic behavior for twisted traces of self-dual and conjugate self-dual representations of $\mathrm{GL}_n$
本文研究了在某些条件下 $\mathrm{GL}_n$ 的自对偶或共轭自对偶离散自守表示扭曲迹之和的渐近行为，基于 Arthur 扭曲迹公式推导出主项及 Fourier 变换显式形式，并应用于局部表示的整体化及推导自守密度定理。
We establish an infinitesimal version of the Jacquet-Rallis trace formula for general linear groups. Our formula is obtained by integrating a kernel truncated a la Arthur multiplied by the absolute value of the determinant to the power $s \in \mathbb{C}$. It has a geometric side which is a sum of distributions $I_{\mathfrak{o}}(s, \cdot)$ indexed by the invariants of the adjoint action of $\mathrm{GL}_n(\mathrm{F})$ on $\mathfrak{gl}_{n+1}(\mathrm{F})$ as well as a "spectral side" consisting of the Fourier transforms of the aforementioned distributions. We prove that the distributions $I_{\mat…
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