Questions tagged [trace-formula]
Theoretical issues and applications of the Selberg, Arthur and relative trace formulas
7
votes
0answers
79 views
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
$$\...
0
votes
2answers
181 views
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
2answers
650 views
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 ...
9
votes
0answers
230 views
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 ...
5
votes
1answer
209 views
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 ...
22
votes
2answers
2k views
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 ...
7
votes
1answer
225 views
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 ...
6
votes
1answer
330 views
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
...
2
votes
1answer
229 views
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}\...
40
votes
1answer
2k views
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 ...
6
votes
1answer
285 views
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 ...
6
votes
2answers
308 views
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 ...
0
votes
0answers
59 views
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)?
1
vote
0answers
215 views
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 ...
8
votes
2answers
412 views
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 ...
3
votes
1answer
281 views
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 ...
15
votes
1answer
1k views
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
1answer
206 views
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 ...
10
votes
1answer
604 views
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
1answer
256 views
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 ...
4
votes
2answers
356 views
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:
$$
\int_{-\infty}^\infty ...
2
votes
0answers
292 views
on the fundamental lemma
I consider the fundamental lemma for the spherical Hecke algebra.
Let $G$ a connected reductive quasisplit group on $F$, a local field of equal characteristic $p$.
and $H$ an endoscopic group.
Can ...
11
votes
1answer
388 views
Trace formula for PSDOs
In Getzler's famous paper "Pseudodifferential Operators on Supermanifolds and the Atiyah-Singer Index Theorem", he states that for a (trace-class) pseudo-differential operator $P$ on a Riemannian ...
5
votes
0answers
217 views
On Langlands Pairing and transfer factors
In the paper "On the definition of transfer factors" Langlands and Shelstad define a certain number of factors $\Delta_{I}$, $\Delta_{II}$,$\Delta_{III,1}$,$\Delta_{III,2}$, which are roots of unity.
...
9
votes
1answer
987 views
Is this a subcase of the fundamental lemma?
Let $F$ be a local field and $G= GL(n,F)$.
Assume that $\gamma$ is an element of $G$ and $G_\gamma$ is its centralizer.
The orbital integral is defined as
$$ O_\gamma^G( \phi) = \int\limits_{G_\...
5
votes
0answers
651 views
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 ...
10
votes
1answer
1k views
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 ...
26
votes
0answers
1k views
Is there a Grothendieck-Riemann-Roch type of theorem generalizing Grothendieck's Lefschetz trace formula
Grothendieck deduced that the L-function of a (constructible) $\ell$-adic sheaf on a variety over $\mathbf{F}_p$ is rational from the generalized trace formula.
My first question is based on the ...
16
votes
0answers
843 views
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 ...
