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Results tagged with automorphic-forms
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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.
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 …
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1
vote
On tetrahedral Artin representation
Say $\rho$ is a tetrahedral representation of $G_F$, i.e., it is an irreducible 2-dimensional representation and the projection $\bar \rho$ to $PGL_2(\mathbb C)$ has image $A_4$. Then you can take $E …
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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. …
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5
votes
Accepted
Global Waldspurger packet is finite or infinite?
Revised. The global Waldspurger packet of $\pi$ is indeed finite, as you say in your comment. It's elements are metaplectic representations which are in bijection with the Vogan packet of $\pi$, i.e. …
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5
votes
A question on twisted L-function
I see this was posted awhile ago, but I would like to offer a counterpoint to Marc's answer: namely, yes there is a relation, at least in certain situations if you twist by quadratic characters, but i …
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4
votes
Accepted
Waldspurger's formula and toric periods — classical and adelic versions
These are two separate theorems, proved in different papers of Waldspurger (I think in 1980/1981 and 1985, respectively), so you shouldn't conflate them. The first theorem can be viewed as an "$L$-va …
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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 $\pi_v$ …
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1
vote
Regularity assumption in the simple trace formula
Probably the OP has figured this out by now, but for posterity let me explain (in the general setting of simple trace formulas, rather than the specific Deligne-Kazhdan case):
A trace formula $I(f) = …
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0
votes
Accepted
Factorizability of Subquotients of Principal Series Representations
It is not even true for finite direct products. Take $G = G_1 \times G_2$,
$\chi_i, \chi_i'$ to be characters of $G_i$. Then let $\rho = (\chi_1 \otimes \chi_2) \oplus (\chi_1' \otimes \chi_2')$. T …
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3
votes
On the local automorphic components of classical Siegel modular forms
Look at the tables of Ralf Schmidt and Brooks Roberts:
Tables for Representations of GSp(4)
Section 2 also tells you which representations you should find in Saito-Kurokawa versus non-Saito-Kurokawa …
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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 $G=GL_n$ …
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2
votes
Accepted
Integrality of the support of matrix coefficients?
According to the comments, I understand you to mean the following local question: Say $D$ is the quaternion division algebra over an $p$-adic field $F$,
and $\pi$ is a smooth representation of $D^\tim …
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4
votes
Accepted
Conductor of quaternionic representation
I consider a Casselman type of local newform theory on quaternion algebras in my paper on the basis problem (sections 2 and 3), which gives you a positive answer to your question half of the time (Cas …
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5
votes
Compactness of the automorphic quotient and genericity
I'm not sure what you mean by tempered or generic, or if you even know what you mean (defined locally or globally? in terms of representations or parameters?). But basically the answer is no. For in …
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3
votes
Fundamental lemma and transfer of characteristic functions of congruent subgroups
There are a couple of reasons one typically works with just fundamental lemmas for spherical functions:
1) the unramified comparison suffices for a trace formula comparison (well, a fundamental lemma …
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