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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.
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Distribution of signs of automorphic forms
Let's say we have an automorphic form $f$ on $GL(2)$ that is self-dual. In particular, the associated L-function $L(s,f)$ satisfies a functional equation with sign $\varepsilon_F = \pm 1$.
Is it know …
- 609
4
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Integrality of the support of matrix coefficients?
Consider a division quaternion algebra $D$ over a number field $F$. For an automorphic representation $\pi$ of $D$, I am interested in the associated matrix coefficients
$$f : \gamma \in G \longmapsto …
- 609
2
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Relations between spectral parameters of automorphic representations
Let $\pi$ be an automorphic representation (say, trivial central character) of $GL(2)$. Let $\alpha(p)$ and $\beta(p)$ denote its spectral parameters at the place $p$, that is to say the associated lo …
- 609
3
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Ramanujan conjecture in terms of representations
Given an automorphic representation, I would like to bound $\alpha_1^\nu(p) + \alpha_2^\nu(p)$ where the $\alpha_i$ are the Satake parameters of an automorphic form $f$ of, say, $GL_2$. So that $\alph …
- 609
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Local L-factors for automorphic representations
For Hecke L-functions associated to a holomorphic cusp form $f$ of level $N$, the local factors can be decomposed into
$$L_p(s, f) = (1-\lambda_f(p)p^{-s} + \chi_N(p)p^{-2s})^{-1}$$
where $\chi_N$ is …
- 609
2
votes
2
answers
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For what automorphic representations is Ramanujan-Petersson known?
I had in mind that Ramanujan-Petersson conjecture was essentially unknown in the case of number fields. I however recently heard that
If an automorphic representation on $GL(2)$ is ramified at a …
- 609
23
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2
answers
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What is the matter with Hecke operators?
This question is inspired by some others on MathOverflow. Hecke operators are standardly defined by double cosets acting on automorphic forms, in an explicit way.
However, what bother me is that Hec …
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