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Results tagged with automorphic-forms
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user 128718
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.
6
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1
answer
253
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Divergence of integrals in the trace formula
I am trying to understand the following situation for $G=GL(2)$, when going from the compact trace formula to the non-compact case.
The integral over $G(\mathbb{A})^1_\gamma \backslash G(\mathbb{A})^1 …
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8
votes
1
answer
586
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Does the symmetric square L-function vanish at one?
Take a cuspidal automorphic representation $\pi$ of $GL(3)$ over a number field. My question is quite straightforward and can be related to this one :
Can $L(1, \pi, \mathrm{sym}^2)$ be zero? If y …
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5
votes
0
answers
104
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Archimedean L-factors for symplectic group
Let $\pi$ be an automorphic representation of $GSp(4)$. Provided a representation $r$ of the Langlands dual group of $GSp(4)$ (namely, the standard or the spinor one), it is possible to define a parti …
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6
votes
1
answer
540
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Analogues of Hecke relations for Maass forms
If a (suitably normalised) holomorphic cusp newform has q-expansion
$$f(z) = \sum_n \lambda_f(n) e(nz),$$
then we know the Hecke relations for $(mn,q)=1$,
$$(\star) \qquad \lambda_f(m)\lambda_f(n) = …
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8
votes
1
answer
565
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Functional equation for general number fields
When it comes to general number fields beyond $\mathbb{Q}$, the litterature is not so abundant in analytic number theory. For instance over $\mathbb{Q}$, for primitve Dirichlet characters modulo $q$, …
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4
votes
0
answers
133
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Plancherel measure and dimension
I would like to understand what the Plancherel measure is on the dual of a compact group, in more explicit terms than with an implicit definition. Indeed, to the extent of my knowledge it is defined a …
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7
votes
0
answers
127
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Bound for orbital integrals
Let $F$ be a number field, and $G$ be the group of units of a quaternion algebra $D$ over $F$. At a certain ramified place $v$, for $\gamma_v \in G(F_v)$, could we bound the orbital integral
$$\mathca …
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2
votes
0
answers
100
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Function equation over general number fields
Let $\chi$ be a Hecke character on a number field $k$, where could I find a precise reference for the function equation of the $GL(1)$ L-functions
$$L(s, \chi)?$$
I only find references for the case …
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1
vote
0
answers
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Mean value estimates for general number fields
Results are known in many different cases to bound powers of L-functions on average over a wide enough family. I am interested in results for general number fields, not only for the rationals, for pow …
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