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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.
7
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Computing explicit matrix coefficients
I would like to understand in a more explicit way the Fell topology on unitary duals, that is to say the convergence of matrix coefficients of local representations.
If I consider a local representati …
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votes
1
answer
375
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Relation between $\xi$-cohomological and discrete series
Sometimes, results on automorphic representations are available only under local assumptions. Typically, one could require the representation to be a $\xi$-cohomological cuspidal representation, and I …
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3
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2
answers
245
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Compactness of the automorphic quotient and genericity
Let $G$ be a reductive group defined over a field $F$. Let $\mathbf{A}$ denote the ring of adeles of $F$. My question is:
Assuming the automorphic quotient $[G]=G(F) \backslash G(\mathbf{A})$ is c …
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8
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1
answer
523
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How strong is the requirement of being a Gelbart-Jacquet lift?
Let $\pi$ be a cuspidal automorphic representation of $\mathrm{GL}(3)$ over a number field $F$. I am wondering how general are Gelbart-Jacquet lifts of automorphic representations of $\mathrm{GL}(2)$ …
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11
votes
2
answers
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Relation between Fourier coefficients and Satake parameters
Let $L(s)$ be an automorphic L-function (attached to a self contragredient automorphic representation on $GL(3)$), according to the following notations for $s$ of sufficiently large real part:
$$L(s) …
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5
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1
answer
277
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Archimedean Langlands classification
I am trying to clarify how the archimedean admissible dual is classified in the $G=GL(n, \mathbf{R})$ case.
Fix a semistandard Levi subgroup $M$ in $G$, $\delta$ a square-integrable representation of …
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8
votes
1
answer
461
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Equivalence between Ramanujan and Selberg conjectures
At first the Ramanujan conjecture for automorphic forms and the Selberg conjecture appear to be understood as totally independent. However, they are now known to be tighyly connected once viewed in th …
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7
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1
answer
574
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$GSp(4)$ vs $PSp(4)$
After some months wandering through examples of algebraic groups in the theory of automorphic forms and number theory, I wonder why so many efforts are spent in understanding $GSp(4)$ (local newforms, …
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6
votes
1
answer
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What is the conductor of an automorphic representation for $\Gamma_0(q)$ in $GSp(4)$?
Let $\pi$ be a generic cuspidal automorphic representation on $GSp(4)$, with level $\Gamma_0(q)$ (the group of symplectic matrices with lower left block divisible by $q$), i.e.
$$\Gamma_0(q) = \left\{ …
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6
votes
1
answer
185
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Fields of rationality as a notion of automorphic size
I want to interpret the degree of the field of rationality of an automorphic form as a notion of size, analogously to the conductor, and this question is about the possible obstructions to do so. The …
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5
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681
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Poles of $L$-functions associated to Maass forms
Let $\pi$ be an automorphic representation of $GL_2$ over a number field. What can I say concerning the order of the pole at $1$ of the $L$-function $L(s, \pi)$? Can we say more about $L(s, \mathrm{Sy …
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11
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332
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Fourier Transforms of Convolutions
Straightforward computations lead to the following standard property of Fourier transformation: it transforms convolutions into products, i.e. for functions $f$ and $g$ Schwartz class we have
$$\wideh …
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14
votes
1
answer
522
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Bound for $GL(3)$ symmetric square
Let $\pi$ be an automorphic representation of $GL(3)$ over a number field. Let $a_n$ be the coefficients of $L(s, \pi, \mathrm{sym}^2)$. Do we know if
$$\sum_{n>0} \frac{|a_n|}{n^s}$$
and
$$\sum_{n>0} …
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answer
223
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Are all these representations supercuspidal
Let $D$ a division quaternion algebra over a number field $F$, and consider $(V,q)$ be a $D$-hermitian space of $D$-dimension $2$, and introduce its group of isometries
\begin{align*}
\mathrm{GU}(V, q …
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Continuity of the conductor of automorphic representations
I am interested in properties of (semi-)continuity of the conductor of automorphic representation. Let $F$ be a number field.
The meta question is, given a function on the unitary dual of $PGL(2, F)$ …
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