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Results tagged with automorphic-forms
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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.
49
votes
Accepted
Are there Maass forms where the expected Galois representation is $\ell$-adic?
Here's some piece of the bigger picture. Maass forms and holomorphic modular forms are both automorphic representations for $GL(2)$ over the rationals. An automorphic representation is a typically hug …
- 41.4k
30
votes
Accepted
What's the point of a Whittaker model?
This question is a bit like saying "what's the point of the theory of bases for vector spaces -- this just gives you an isomorphism of your space with $\mathbb{R}^n$. What is the point of defining thi …
- 41.4k
18
votes
Accepted
Is there a canonical notion of "mod-l automorphic representation"?
This is in my mind a central open problem.
Here is an explicit example which I believe is still wide open. Serre's conjecture (the Khare-Wintenberger theorem) says that if I have a continuous odd irre …
- 41.4k
16
votes
Accepted
Is this a subcase of the fundamental lemma?
I will offer some words on this, but only because no-one else has; I was holding out hoping that one of the more automorphic people would chip in. It might be worth taking much of the below with a pin …
- 41.4k
13
votes
Definition of L-function attached to automorphic representation
I think you're slightly misled. $\pi$ doesn't have an $L$-function "in abstracta". An unramified $\pi_v$ at a finite place $v$ gives rise, by Langlands' interpretation of the Satake isomorphism, to a …
- 41.4k
6
votes
Conductor of monomial forms with trivial nebentypus
Thinking about Idoneal's question to Emerton about what is being used about the Steinberg, it's not just standard facts about the Steinberg one needs via this approach, but also local-global compatibi …
- 41.4k
6
votes
Accepted
Finite dimensional automorphic representations of a definite quaternion with prime discrimin...
1) Yes, I think that's true. I guess it follows relatively easy from the statement that an automorphic representation of the algebraic group $D^\times$ is finite-dimensional iff it's 1-dimensional and …
- 41.4k
6
votes
Automorphic form encoding the orders of $N$ modulo $p$.
$\newcommand{\F}{\mathbf F}\newcommand{\Q}{\mathbf Q}$
I want to say two things about this question. First is that I don't understand some of it: I don't know what you mean by "$a_p=H^0(\F_p,M)$" (num …
- 41.4k
4
votes
Accepted
Base Change for Eigenvarieties
This is all a bit complicated -- the theory is still in its infancy and some arguments aren't quite as smooth as they should be.
If all you know is that "the eigenvarieties have been constructed" the …
- 41.4k
1
vote
How badly can strong multiplicity one fail in the theory of automorphic representations?
For what it's worth I can now answer Q0. I believe it is not true in general that $\pi_v$ and $\pi'_v$ will have to have the same central character. We can let $G$ be a torus $T$. If $S$ is a finite s …
- 41.4k