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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.
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 …
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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 …
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4
votes
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Counting local representations for $\mathrm{GL}_2$
Yes, you can count automorphic forms with given local representations. If you want to count discrete series, you can just fix the $\pi_p$. For principal series, you can fix a parameter in a small in …
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4
votes
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Automorphic classification of different types of abelian surfaces
Yes, knowing the endomorphism algebra of $A$ (conjecturally) translates to certain properties of an associated automorphic representation $\pi$.
First, you should look at the Galois type, which is lab …
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3
votes
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Modular forms on central division algebra of degree $\ge 3$
For the first question, it is only true that if $D$ is a (totally) definite quaternion algebra over a number field $K$, then the weight 0 automorphic forms factor through a finite set (1-sided ideal c …
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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
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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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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 …
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5
votes
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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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8
votes
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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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8
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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 …
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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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4
votes
Langlands Reciprocity and Fermat's Last Theorem
I think the confusion here lies in what is being called reciprocity (and perhaps the interpretation of "simple"). If by Langlands reciprocity, you mean a correspondence between classical Artin repres …
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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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4
votes
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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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