Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with automorphic-forms
Search options answers only
not deleted
user 6518
An automorphic form is a well-behaved function from a topological group $G$ to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup $\Gamma \subset G$ of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups.
23
votes
Accepted
What is the status of Arthur's book?
Updated answer (Oct 2024):
While Arthur did not finish some preprints referred to in his book ([A24]-[A27]), [A24] was dealt with by Moeglin and Waldspurger, and this arXiv preprint which was just pos …
- 6,039
16
votes
What motivations for automorphic forms?
To give a brief answer, which I think applies to all audiences, and I hope is not too "elementary" for you (I'm not attempting to give details, which of course need to be specialized for the intended …
13
votes
Accepted
Why is the simple trace formula a weaker tool than the Arthur trace formula?
I must have missed this question a month ago, but hopefully you're still interested.
I haven't read that particular paper, but simple trace formulas have restricted test functions, which for one can …
- 6,039
12
votes
What motivations for automorphic forms?
Here's an answer for audiences (C), (D) about class field theory:
Class field theory provides a way classify abelian extensions of number fields $F$. In the case of $F=\mathbb Q$, this is answered b …
10
votes
Impact of the squarefreeness of the level for modular forms
There are several ways in which studying modular forms with squarefree level is "simpler" for general level. Here I assume trivial nebentypus. For instance:
You do not see CM forms.
You do not see …
- 6,039
9
votes
Accepted
Are all these representations supercuspidal
First of all, you will only get a non-trivial inner form at finitely many places (where $D$ ramifies). Even then, the group is not compact mod center, so not all representations will be supercuspidal …
- 6,039
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
9
votes
Accepted
Local component of cuspidal automorphic representation
Let me work in the category of $L^2$-automorphic representations. Assuming your global representation $\pi$ is irreducible, about the only thing you can say about an arbitrary local component $\pi_v$ …
- 6,039
8
votes
The space of Whittaker functionals is at most one-dimensional
This is proved in Shalika's multiplicity one paper:
Shalika, J. A.
The multiplicity one theorem for $GL_n$.
Ann. of Math. (2) 100 (1974), 171–193.
While Shalika starts off assuming $G=GL_n$ …
- 6,039
8
votes
Accepted
Distribution of signs of automorphic forms
For simplicity, let's consider the case of holomorphic modular forms over $\mathbb Q$ of squarefree level and trivial nebentypus. Then one knows from
Iwaniec, Henryk; Luo, Wenzhi; Sarnak, Peter. …
- 6,039
8
votes
Accepted
$GSp(4)$ vs $PSp(4)$
If we think about classical modular forms, one typically works on SL(2). One could also work on PSL(2), but one would like to write down congruence subgroups in terms of matrices, so one often phrase …
- 6,039
8
votes
Why is Langlands functoriality usually related with period integral in a third group?
A lot is known--too much to try to summarize--and I think this philosophy came about after seeing numerous examples, beginning with Harder-Langlands-Rapoport (base change for GL(2)), and thinking abou …
- 6,039
6
votes
Accepted
On Siegel mass formula
There are many, many references on quadratic forms. This is a huge area, depending on which way you want to go. One of the main approaches is to construct a theta series associated to your quadratic …
- 6,039
6
votes
Accepted
Relation between Hecke operators and coefficient of L-functions
A normalized version of your guess is right. First note that the $T_{p^n}$'s satisfy the relation
$$ T_{p^{n+1}} = T_p T_{p^n} - p T_{p^{n-1}} $$
(e.g., Bump Prop 4.6.4). This gives you a recursion …
- 6,039
5
votes
Do we care about multiple zeta functions?
To address the conceptual question, the $L$-function essentially characterizes the automorphic representation and can be studied locally (associating local $L$-functions to local components of the glo …
- 6,039