24
votes
What is the matter with Hecke operators?
To keep things simple, let $G$ be a finite group and $K$ a subgroup of it. The simplest definition of the Hecke algebra associated to this pair $(G, K)$ is that it is the algebra of $G$-endomorphisms $...
- 118k
17
votes
Accepted
Using the Eichler-Selberg Trace formula to compute class numbers?
Generally speaking, formulas like the trace formula admit an uncertainty principle: To obtain an identity where one side is highly concentrated (e.g. a sum over a small number of class numbers, or ...
- 148k
16
votes
Accepted
What is the matter with Hecke operators?
If $K$ is a local field and $\mathcal O$ is its ring of integers, we say an irreducible representation of $G(K)$ is unramified if it contains a vector invariant under $G(\mathcal O)$. It is known that ...
- 148k
13
votes
Accepted
Kuznetsov trace formula, orthogonality of Bessel functions
The Bessel functions $J_\ell$ for $\ell\geq 1$ odd are pairwise orthogonal on the positive axis with respect to the measure $dx/x$. They correspond to the holomorphic spectrum (of various even weights ...
- 105k
10
votes
Using the Eichler-Selberg Trace formula to compute class numbers?
A similar approach, using the closely related Selberg Trace Formula, is used by Ce Bian, Andrew R. Booker, Austin Docherty, Michael J. Jacobson, Jr.,
and Andrei Seymour-Howell in their paper ...
- 1,570
9
votes
Accepted
Reaching Hecke eigenvalues from a trace formula
Yes, this is a standard thing to do. If you want to look at traces of Hecke operators on a definite quaternion algebra, this is the same as what are known as "traces of Brandt matrices." These have ...
- 6,039
7
votes
Accepted
How to read the paper of Arthur on trace formula on general reductive groups
One of the best places to learn about trace formula, other than David Whitehouse's wonderful notes, are the notes by Erez Lapid Introductory notes on the trace formula.
Arthur's trace formula relies ...
7
votes
Kuznetsov trace formula, orthogonality of Bessel functions
I was acquainted with Nikolay Vasil'evich Kuznetsov while worked in Vladivostok, 1990s. And he was very kind to mee, too. He tought me that many asymptotics for Bessel functions are not valid, many ...
- 1,560
6
votes
Combinatorial Skeleton of a Riemannian manifold
For construction of a sequence of discrete Laplacians whose spectrum converges to that of a given Riemannian manifold, see
Dodziuk, Jozef, Finite-difference approach to the Hodge theory of harmonic ...
- 6,337
5
votes
Arthur's Simple Trace Formula
Yes, I have not known Deligne, Kazhdan and Vigneras to lie. A sketch of the proof, at least with the key details for GL(2), is given in Lecture V of
Steve Gelbart, Lectures on the Arthur--Selberg ...
- 6,039
5
votes
Comparing Selberg and Eichler-Selberg trace formulas
Trace formulas, and in particular the Selberg trace formula, is an identity $I(f) = J(f)$ of spectral and global distributions where $f$ is a test function. There are different ways to use the trace ...
- 6,039
4
votes
Accepted
Divergence of integrals in the trace formula
I take it that the question is "Why does this iterated integral diverge?," but correct me if that is not the question.
The issue here is the outer integral: when $\gamma$ is as you describe, ...
- 2,516
4
votes
Accepted
$\DeclareMathOperator\SL{SL}$Multiplicities of irreducible representations in discrete part of $L^2(\SL(2,\mathbb{Z})\backslash{\SL(2,\mathbb R)})$
You are asking what is known about the dimension of weight $k$ holomorphic cusp forms for $\mathrm{SL}_2(\mathbb{Z})$, and the multiplicities of Laplace eigenvalues of weight $0$ and weight $1$ Maass ...
- 105k
4
votes
Accepted
Reduction to Lie algebra version of fundamental lemma?
For the purpose of this answer let us say that "fundamental lemma" means "fundamental lemma for the unit of the unramified Hecke algebra". I do not think that "FL for Lie algebras => FL for groups" ...
- 56
4
votes
Accepted
A trace formula for $\mathrm{GSp(4)}$
There are many kinds of trace formulas on a given group $G$, and different things you could mean by decomposition of the spectrum. As mentioned in the comments, there's Arthur's article in the ...
- 6,039
3
votes
Selberg trace formula, quadratic L-values, and generalization
If I understand correctly what you are looking for, then yes, a fair amount of work has been done. Deitmar and Hoffman use a simple trace formula on SL(3) to get asymptotics of class number of cubic ...
- 6,039
3
votes
Reduction to Lie algebra version of fundamental lemma?
You can find it in Waldspurger's paper titled Le lemme fondamental implique le transfert.
DOI: https://doi.org/10.1023/A:1000103112268
- 899
3
votes
How to read the paper of Arthur on trace formula on general reductive groups
I agree with @kimball's suggestions. The first thing to do is to get a handle on what it looks like for GL(2), then upgrade to general $G$ only as needed. For a basic introductory references, starting ...
- 3,799
2
votes
The meaning of $L_{\chi}^2(G(\mathbb Q) \backslash G(\mathbb A)^1)$
Arthur doesn't work mod centre or $A_G(\mathbb R)^0$ because he works with $G(\mathbb A)^1$ instead. For a nice explanation of the subtle difference between these choices, take a look at Section 6 of ...
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2
votes
Some confusion about weights and roots in parabolic root systems
I only find one paper (a book chapter, not a book itself) with the indicated title, Arthur - An introduction to the trace formula, and I can't find in it the sentences you quote, so it's hard to speak ...
- 12.9k
1
vote
Accepted
Small questions in studying Arthur 's book 'Introduction to the Trace formula'
I will try to answer both your questions in the context of Arthur's notes.
There does not exist a canonical action of a general connected reductive group $G$ on $N_P$ where $P = N_P M_P$ is a ...
- 899
1
vote
Simple trace formula with different spectral footprint?
You should have a look at the article
Bernstein, J.; Deligne, P.; Kazhdan, D., Trace Paley-Wiener theorem for reductive p-adic groups, J. Anal. Math. 47, 180-192 (1986). ZBL0634.22011.
which ...
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