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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.
6
votes
L and epsilon factors of Gelbart-Jacquet lifts
The function $s \mapsto \epsilon(s, \pi)$ has the form $s \mapsto A e^{Bs}$ for some constants $A, B$, so it vanishes nowhere on $\mathbb{C}$. That has nothing to do with being a GJ lift, it's a gener …
- 37k
4
votes
Accepted
Global symplectic (orthogonal) type of automorphic representation compels its type to all it...
Here is a proof of the claim using results from Arthur's monograph The Endoscopic Classification of
Representations: Orthogonal and
Symplectic Groups.
Let $N = 2n$ be an even integer, and $\pi$ a cusp …
- 37k
1
vote
Antiholomorphic cusp forms of negative weight
I am not entirely sure what Clozel is trying to do here. He doesn't make it terribly precise what he means, and as you yourself have realised, if you read the article literally it seems that one case …
- 37k
1
vote
Existence of a Hilbert modular form of parallel weight 6
There is functionality for computing Hilbert modular forms in MAGMA, based on the algorithms of Dembele, Donnelly, Greenberg, and Voight. If you don't have access to a copy, try the online MAGMA calcu …
- 37k
4
votes
Accepted
On the Cartan decomposition of unitary group
Theorem: Let $G$ be a reductive algebraic group over a local field $F$, let $K$ be any maximal compact subgroup of $G(F)$, and let $Z = Z(G)$. Then $K \cap Z(F)$ is the unique maximal compact subgroup …
- 37k
10
votes
Accepted
Is there a known construction of Cuspidal representations of GL(3) isomorphic to their own t...
Let $E / F$ be the cyclic cubic extension corresponding to $\chi$ by class field theory. Let $\sigma$ be a generator of $\operatorname{Gal}(E / F)$, and let $\psi$ be a character of $E^\times \backsla …
- 37k
5
votes
What is the meaning of the $L$-group?
I find this question somewhat strange; you ask "what is the meaning of the L-group?", but the survey article of Casselman which you link to is largely devoted to explaining the historical and conceptu …
- 37k
12
votes
Accepted
What is the relationship between (g,K)-module and Maass forms?
So you've seen that there are essentially three types of (g, K)-modules: finite-dimensional ones; principal series; and discrete series. The finite-dimensional ones don't interest us, since they are n …
- 37k
11
votes
Accepted
modular form Fourier coefficients and associated automorphic representation
Jared Weinstein and I wrote a paper on how to compute $\pi_p$: see here.
As Olivier says, $a_p$ will often be zero, and in fact if the central character is trivial (or has conductor coprime to $p$) …
- 37k
9
votes
Accepted
Exceptional primes
No, this does not work: for any modular form of weight $k \ge 2$, the image of the projective representation $\tilde\rho_{f, \lambda}$ is infinite for every prime $\lambda$.
Proof: if $\tilde{\rho}_ …
- 37k
2
votes
Fourier expansion at inequivalent cusps
A chunk of the Princeton PhD thesis of Dan Collins is devoted to this problem. See Collins' preprint here: https://arxiv.org/abs/1802.09740, and the accompanying Sage code.
- 37k
3
votes
Jacquet module and Frobenius reciprocity
In general, all we can say from "general abstract nonsense" is that if $\sigma$ is a subrepresentation of $Ind_P^G(\pi)$, then $\pi$ is a quotient of $J_N(\sigma)$; but you don't immediately get any f …
- 37k
5
votes
Galois representations attached to a cusp form for different primes
At the most basic level, $\rho_p$ and $\rho_q$ are "nothing to do with each other". E.g. the kernels of $\rho_p \bmod p$ and of $\rho_q \bmod q$ cut out two finite Galois extensions of $\mathbf{Q}$ wh …
- 37k
2
votes
Accepted
Part of some generic representation is also generic?
Let $\pi$ be the irreducible generic unramified representation of $Sp(W) $ that is a subquotient of $Ind(\chi_1, \dots, \chi_n)$.
I think the key here is to realise that this does not exist for all …
- 37k
6
votes
Accepted
Relation between $\xi$-cohomological and discrete series
This condition comes up because of $(\mathfrak{g}, K)$-cohomology, which is an extremely important invariant of automorphic representations.
If $\xi$ is an algebraic rep, then $\xi$ defines a locally- …
- 37k