# Questions tagged [trace-formula]

Theoretical issues and applications of the Selberg, Arthur and relative trace formulas

49
questions

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### Divergence of integrals in the trace formula

I am trying to understand the following situation for $G=GL(2)$, when going from the compact trace formula to the non-compact case.
The integral over $G(\mathbb{A})^1_\gamma \backslash G(\mathbb{A})^1$...

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0
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### Grothendieck trace formula for schemes with étale fundamental groups that have no dense cyclic subgroup

This question may be more of a philosophical rather than mathematical nature.
Assume I have a scheme $X$ and an endomorphism $F:X\longrightarrow X$. For instance, $X$ might be of finite type over $\...

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1
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### Comparing Selberg and Eichler-Selberg trace formulas

The trace formula of Selberg gives an equality between trace of Hecke operators (a spectral sum) on spaces of Maass forms and sums over closed geodesics mostly. The Eichler-Selberg trace formula, ...

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### An application of Koike's Trace Formula

Koike's Trace Formula states that
\begin{equation}
\mbox{Tr}((U_p^{\kappa})^n) = - \sum_{0 \leq u < \sqrt{p^n}\\
(u,p)=1}H(u^2-4p^n)\frac{\gamma(u)^\kappa}{\gamma(u)^2 - p^n}-1,
\end{equation}
...

4
votes

1
answer

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### The meaning of $L_{\chi}^2(G(\mathbb Q) \backslash G(\mathbb A)^1)$

I'm reading James Arthur's notes on the trace formula and am confused on a point on pages 65 and 66. For $G$ a reductive group over $\mathbb Q$ we are going over the decomposition of the space $L^2(G(...

3
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0
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### Variants of Selberg trace formula

I am familiar with a basic case of Selberg's trace formula, in the case of quotients of upper half plane (for example, see Sections 5.1 - 5.3 of Bergeron's book). Section 5.1 describes a general setup ...

4
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1
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### $\DeclareMathOperator\SL{SL}$Multiplicities of irreducible representations in discrete part of $L^2(\SL(2,\mathbb{Z})\backslash{\SL(2,\mathbb R)})$

$\DeclareMathOperator\SL{SL}$It is well-known that the cuspidal (or discrete) part of $L^2(\SL(2,\mathbb{Z})\backslash{\SL(2,\mathbb{R})})$ decomposes into irreducible representations of $\SL(2,\...

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### Sum of Kloosterman sums with oscillating factor

Denote by $S(c;n,m)$ Kloosterman's sum. Take $X>0$ and take $n,m\in \mathbb Z$ smaller than a small power of $X$ in modulus. It is known that essentially
\[ \sum _{c\sim X}\frac {S(c;n,m)}{c}\ll ...

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0
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### An inequality regarding operator concave function

Crossposted from math.SE
Let $\mathbb P_n$ be the space of all $n \times n$ self-adjoint positive definite matrices. Consider the function $\varphi: \mathbb P_n \longrightarrow \mathbb R$ defined by $...

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### The specific connection between the Hecke operator and the t'Hooft Operator

As I was reading some articles concern about the Selberg trace formula and its general form, I have noticed that the Selberg trace formula and its general form can be understand as the energy spectrum ...

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### How to read the paper of Arthur on trace formula on general reductive groups

My question is about the correct order to read the papers by Arthur on trace formula. Arthur's papers are perfectly well-written, but maybe a little too hard for me to go through easily.
I would like ...

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0
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### Truncation and weighted orbital integrals in hyperbolic term of trace formula for $\mathrm{GL}(2)$

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PGL{PGL}$I am looking at Gelbart--Jacquet's article in the first Corvallis volume (the article entitled Forms of $\GL(2)$ from an analytic point of ...

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### The kernel $K(x,y)$ as an integral over Eisenstein series for $\operatorname{GL}_2$

Let $G = \operatorname{GL}_2$, $f \in C_c^{\infty}(G(\mathbb A)/Z(\mathbb A))$, and $V = L^2( G(\mathbb Q)Z(\mathbb A)\backslash G(\mathbb A))$ (trivial central character). Then the operator $R(f)$ ...

5
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1
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### Arthur's Simple Trace Formula

In Deligne–Kazhdan–Vigneras's "Représentations des groupes réductifs sur un corps local," they use the Simple Trace Formula to prove cases of the local Jacquet–Langlands correspondence ...

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### Why does norm map the $\sigma$-conjugacy classes to the conjugacy classes?

Let $E/F$ be a cyclic extension of order $\ell$ (not assumed prime) of fields of characteristic $0$, and $\Sigma$ its Galois group; we denote by $\sigma$ a generator of $\Sigma$. We denote by $G(E), G(...

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1
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### Combinatorial Skeleton of a Riemannian manifold

In Chung and Yau's paper: "A combinatorial trace formula" (MSN), they proved
a combinatorial version of Selberg's trace formula for lattice graphs.
I learned also in the setup that it makes sense to ...

6
votes

2
answers

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### Reduction to Lie algebra version of fundamental lemma?

Ngo famously proved the Langlands-Shelstad fundamental lemma for Lie algebras using the geometry of the Hitchin fibration.
For the purposes of the trace formula, one actually needs the fundamental ...

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vote

1
answer

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### Small questions in studying Arthur 's book 'Introduction to the Trace formula'

I am reading Arthur's book "Introductionto the trace formula".
In reading the book, two small question has arised and so I would like to ask it.
Let $G$ be a connected reductive group over $\mathbb{...

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1
answer

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### Simple trace formula with different spectral footprint?

A standard idea when dealing with the Arthur-Selberg trace formula (or a relative trace formula, for that matter) is to impose local conditions on the test function $f=\prod_vf_v$ to obtain a simple ...

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### About height function in the Arthur trace formula

I am reading a book of Arthur's book "Introduction to the trace formula".
In page 24, Arthur defined height function $H_P:G(\mathbb{A}) \to \mathcal{a}_P$
by setting $H_p(nmk)=H_{M_p}(m)$ where $n\in ...

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votes

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answers

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### The number of rational semisimple conjugacy class/the Arthur-Selberg trace formula

I was trying to understand a statement in Theorem 1.5 of this where the author seems to imply that if $G$ is a reductive group over $\mathbb{Q}$ such that $G/Z(G)$ is anisotropic, then for any ...

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### Bound for orbital integrals

Let $F$ be a number field, and $G$ be the group of units of a quaternion algebra $D$ over $F$. At a certain ramified place $v$, for $\gamma_v \in G(F_v)$, could we bound the orbital integral
$$\...

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votes

2
answers

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### Some confusion about weights and roots in parabolic root systems

I was reading James Arthur's book An Introduction to the Trace Formula and had a couple of questions. Here $A_0$ is a maximal split torus of a reductive group $G$, $P_0 \supset A_0$ is a minimal ...

9
votes

2
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### Kuznetsov trace formula, orthogonality of Bessel functions

Sorry if this is a vague question. I remember from my younger days that
before proving his trace formula, Kuznetsov had a pretty result on
orthogonality of Bessel functions. The formulas that I am ...

11
votes

0
answers

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### The trace formula over function fields

There are many examples in number theory where an "arithmetic" problem (i.e. for number fields) has an easier analogue for function fields over finite fields. This is also true for questions ...

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votes

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answer

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### A trace formula for $\mathrm{GSp(4)}$

The Arthur trace formula and its variations provide general results for reductive groups, however to the extent of my knowledge only few specific instances of the formula have been really worked out ...

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2
answers

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### What is the matter with Hecke operators?

This question is inspired by some others on MathOverflow. Hecke operators are standardly defined by double cosets acting on automorphic forms, in an explicit way.
However, what bother me is that ...

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### Selberg trace formula, quadratic L-values, and generalization

It is known that the geometric side of the Selberg trace formula on GL(2) is related to values of quadratic L-functions (due to Sarnak, Zagier, etc).
Are there any conjectures or results about its ...

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### Reaching Hecke eigenvalues from a trace formula

I am interested in studying equidistribution of Hecke eigenvalues and proving statistical properties of arithmetical objects. On the road, I face the following problem: how to express sums of the form
...

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votes

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answer

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### Orbital integral for matrix coefficients

I am currently aiming at estimating orbital integrals. Maybe surprizingly, I hope for some help in the compact case (ramified places), in proving the usual formula
$$O_\gamma(f) = \int_G f(x^{-1}\...

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1
answer

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### What is the status of Arthur's book?

Arthur's long-awaited book project is now published (The endoscopic classification of representations: orthogonal and symplectic groups). However, in the book he takes some things for granted:
The ...

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votes

2
answers

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### Fundamental lemma: why is the transfer factor a power of q

Let $k$ be a finite field of sufficiently large characteristic, $F = k((t))$ and $\mathfrak{o} = k[[t]]$. Let $G$ be a reductive algebraic group defined over $\mathfrak{o}$. Roughly stated, for sake ...

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### Analytic continuation of intertwining operator

I was trying to understand the paper "Forms of GL(2) from the analytic point of view", by Gelbart and Jacquet.
On Page 226 in Remark (4.13) they mention that the kernel of the local intertwining ...

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0
answers

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### A source for integral operators in the context of Arthur-Selberg trace formula

Could you suggest a textbook for integral operators
(in the context of Arthur-Selberg trace formula)?

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0
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### About Theorem $3.1.3$ in Kubota's book: Elementary theory of eisenstein series

My question is about the proof of Theorem $3.1.3$ given in kubota's book, which shows how the function $\varphi(s)$ appearing in the Fourier expansion of eisenstein series can be continued ...

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votes

2
answers

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### Regularity assumption in the simple trace formula

In the simple trace formula of Deligne Kazhdan one assumes that the test function is supported at the elliptic regular elements at one place and is a supercusp form at another place. Why can't one ...

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votes

1
answer

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### What is the logarithmic derivative of an (intertwining) operator?

The constant term of the Eisenstein series (for an adele group $GL_2$, say) contains an intertwining operator, often written as $M(s)$. In the form given in Gelbart-Jacquet's Corvallis paper, for ...

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### What is the Twisted Trace Formula?

I am studying the trace formula using "An Introduction to the Trace Formula" by James Arthur. I would like to understand the twisted trace formula, but unfortunately I never came across a good ...

3
votes

1
answer

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### Spectral synthesis for central functions on locally compact groups

There is a large literature on harmonic analysis on locally compact group, that
I am just beginning to discover. However I have not seen so far anything that emphasizes the central functions on $G$. A ...

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votes

1
answer

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### Trace Class Functions on locally compact groups

Let $G$ be a locally compact subgroup, $\mu$ a Haar-measure.
For $f \in L^1(G)$, and for $\pi$ a unitary, topology irreducible, representation of $G$ on
an Hilbert space $H_\pi$, it is customary to ...

2
votes

1
answer

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### Arthur-Clozel Prop 3.1 for Function Fields?

The subject says it all. I would like to know if Proposition 3.1 in
Arthur-Clozel's book on the trace formula holds for local fields of positive
characteristic.
Thanks!
EDIT: Here is Prop 3.1 of ...

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votes

2
answers

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### vanishing of spectral term in Arthur-Selberg trace formula for GL(2)?

Hi,
In the Arthur-Selberg trace formula for $G = GL(2)/\mathbf Q$ (as seen for example in
Gelbart's "Lectures on the Trace Formula"), the spectral side includes terms
like:
$$
\int_{-\infty}^\infty ...

2
votes

0
answers

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### on the fundamental lemma

I consider the fundamental lemma for the spherical Hecke algebra.
Let $G$ a connected reductive quasisplit group on $F$, a local field of equal characteristic $p$.
and $H$ an endoscopic group.
Can ...

12
votes

1
answer

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### Trace formula for PSDOs

In Getzler's famous paper "Pseudodifferential Operators on Supermanifolds and the Atiyah-Singer Index Theorem", he states that for a (trace-class) pseudo-differential operator $P$ on a Riemannian ...

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votes

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### On Langlands Pairing and transfer factors

In the paper "On the definition of transfer factors" Langlands and Shelstad define a certain number of factors $\Delta_{I}$, $\Delta_{II}$,$\Delta_{III,1}$,$\Delta_{III,2}$, which are roots of unity.
...

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1
answer

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### Is this a subcase of the fundamental lemma?

Let $F$ be a local field and $G= GL(n,F)$.
Assume that $\gamma$ is an element of $G$ and $G_\gamma$ is its centralizer.
The orbital integral is defined as
$$ O_\gamma^G( \phi) = \int\limits_{G_\...

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votes

0
answers

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### Base change and Langlands' combinatorial exercise

Is it correct that Langlands' combinatorial exercise (as he terms it in his paper "Shimura varieties and the Selberg trace formula") is to establish base change identities between orbital integrals of ...

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### Carayol via the trace formula

Hi,
Is there a proof of the result that Carayol proves in "Sur les representations l-adiques..."
using the Langlands-Kottwitz method of comparing the Lefschetz trace formula and the Selberg trace ...

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votes

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### Computation of low weight Siegel modular forms

We have these huge tables of elliptic curves, which were generated by computing modular forms of weight $2$ and level $\Gamma_0(N)$ as N increased.
For abelian surfaces over $\mathbb{Q}$ we have very ...