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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.
3
votes
Pullback of automorphic forms
In some interesting cases, decomposing an automorphic form/representation on a bigger group $G$ along a smaller group $H$ produces Euler products and similar helpful outcomes. Perhaps not "in general" …
- 23k
3
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Adelization of modular forms: What measure on $L^2(G_\mathbb{Q}Z_{\mathbb{R}}\backslash G_{\...
Yes, the two are the same, but I'd be inclined to separate the issues into first converting automorphic forms (with automorphy factors) on the domain $\mathfrak H$ into automorphic forms on the group …
- 23k
3
votes
a question about the proof of analytic continuation of Eisenstein series for GL(2)
I don't know whether this is what any of those authors were thinking, but if a function has a derivative in an $L^2$ sense, then it certainly has a derivative in a distributional sense. A distribution …
- 23k
11
votes
Meromorphic continuation of Eisenstein series
A belated response: so far as I can tell, Selberg's idea, taken literally, does not apply at all in rational rank above 1. One should note that Avakumovic and Roelcke had similar ideas, which also did …
- 23k
18
votes
Accepted
Automorphic forms on GL(3)
As in the comments and earlier answer: in short, there is nothing comparably elementary or accessible for GL(3), as holomorphic things for GL(2).
Even the explication of this apparent fact is not, an …
- 23k
4
votes
Does viewing an Eisenstein series as a sum over cusps explain the antagonism between Eisenst...
I'd say the approximate answer to the question in the title is "not exactly, but partly". But I'd also claim that clarity here is made considerably more difficult by the context in which the question …
- 23k
5
votes
Accepted
On the reductive group
It is not at all that automorphic representations can be defined only for reductive algebraic groups, but, rather, that the essential difficulties arise in that case... as opposed to (for example) abe …
- 23k
5
votes
Product of two cuspforms is not a cuspform
This sort of issue is significant when we're trying to get a grip on the analysis of automorphic forms, but/and, already non-automorphic situations provide much insight. I can't help but comment that …
- 23k
2
votes
Accepted
Left translation of automorphic form satisfies $K$-finiteness?
Presumable (in an automorphic forms context with contemporary left-right conventions) you mean that $\varphi$ is right $K$-finite. This could apply to any (complex-valued) function $\varphi$ on a topo …
- 23k
16
votes
Accepted
A stupid question about Automorphic forms
These conditions have technical/subtle interactions. It probably suffices to think about automorphic forms on a Lie group, and not think of the interaction with finite places.
For example, on each K …
- 23k
2
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Restriction map between spaces of automorphic forms
A belated data-point: in restricting a Siegel-type Eisenstein series from $Sp_2$ (meaning $4$-by-$4$ matrices) to the obvious imbedded copy of $SL_2\times SL_2$, any $f\otimes F$ for (strong-sense) cu …
- 23k
2
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Analytic continuation of intertwining operator
Another point is that (by Borel-Casselman-Matsumoto) in general subrepns and quotients of unramified principal series (for p-adic fields) are generated by their Iwahori-fixed vectors. For $GL_2$, ther …
- 23k
2
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Poles and residues of degenerate Eisenstein series on GL(n)
First, because the kind of induction that creates Eisenstein series can be done in stages, and commutes (after meromorphic continuation) with taking residues (even in the generalized sense required in …
- 23k
1
vote
Existence of a Hilbert modular form of parallel weight 6
You can construct some by (pluri-) harmonic theta series on even-dimensional quadratic spaces, imposing congruence conditions to assure not-identical-vanishing. For example,
$$
\theta(z_1,z_2) \;=\; …
- 23k
1
vote
Accepted
On the Weil representation of unitary groups.
At least as a place-holder answer: the issues may be more about just the geometric algebra rather than the Segal-Shale-Weil/oscillator repn. We can multiply hermitian and skew-hermitian forms by a non …
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