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Results tagged with automorphic-forms
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user 43108
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
votes
Accepted
Corvallis 1979 proceedings
Given the importance of the proceedings, I leave a link to the copy at Library Genesis:
A. Borel and W. Casselman (Editors) Automorphic Forms, Representations and L-functions (1979)
- 17.6k
3
votes
Existence of a Hilbert modular form of parallel weight 6
In this paper, van der Geer and Zagier come across a concrete Hilbert cusp form of weight $6$ on $\mathrm{SL}_2(\mathcal{O})$, namely
$$f=-16(x+x^{-1})^3 (q+(27x^3-39x-39x^{-1}+27x^{-3})q^2+(285x^6-7 …
- 17.6k
8
votes
Accepted
Fourier expansion of automorphic forms
I don't see the relation between the first sentence and the question, so perhaps I'm missing something, but the answer to the question is yes, regardless.
Fourier expansion is a very important tool i …
- 17.6k
1
vote
Is the twisted symmetric fifth power $L$-function holomorphic?
This is definitely an open problem.
For a very recent reference of $Sym^m\pi$ being known only for $m \leq 4$ ($\pi$ arbitrary cuspidal), see:
Huixue Lao, Mark McKee, Yangbo Ye, Asymptotics for cus …
- 17.6k
5
votes
Accepted
Known degrees of L-functions F and G whose Rankin-Selberg convolution is an L-function
An automorphic L-function $L(\pi,s)$ belongs to the Selberg class if and only if the generalized Ramanujan conjecture for $\pi$ is known.
This includes two important cases: Hecke characters and holom …
- 17.6k
7
votes
Reference for the proof of Langlands conjecture for $GL_n$ over function fields
It depends on what you mean by reference, but this might work
Gérard Laumon, "The Langlands Correspondence for Function Fields following Laurent Lafforgue" (1999)
It is a brief sketch of the ever …
- 17.6k
6
votes
Accepted
Special values of real analytic Eisenstein series
Combining paul garret's comments (see also his wonderful notes, Standard compact periods for Eisenstein series) with the class number formula, the functional equation of the Dedekind zeta function and …
- 17.6k
6
votes
1
answer
609
views
Converse to Modularity II: Maass cusp forms
(This comes from this other question. You can find more details there)
The following bijection is now a theorem:
Odd irreducible 2-dim Galois repn $\longleftrightarrow$ weight 1
newforms
note …
- 17.6k
12
votes
1
answer
567
views
Non-algebraic Hecke characters
Algebraic Hecke characters are ubiquitous in modern number theory. They are in 1-1 correspondence with one dimensional complex Galois representations, and in some precise sense they are the building b …
- 17.6k
13
votes
0
answers
620
views
No Siegel-Landau zeros for $\mathrm{GL}(n)$
The problem of non-existance of Siegel-Landau zeros seems to be uncharacteristically easier for cuspidal automorphic representations $\pi$ on $\mathrm{GL}(n)$ if $n\geq2$. We have in fact:
There a …
- 17.6k
5
votes
Simplest case of Langlands-Shahidi method
I think that the best introduction to the Langlands-Shahidi method still is the following book by Shahidi and Gelbart:
Gelbart & Shahidi, "Analytic Properties of Automorphic L-Functions" (1988)
It …
- 17.6k
16
votes
1
answer
2k
views
Automorphic factorization of Dedekind zeta functions
It is well known that for abelian number fields, the factorization of its Dedekind zeta function goes like this:
$$\zeta_K(s)=\zeta(s)\prod_{\chi \neq 1} L(s,\chi)$$
with the Dirichlet characters di …
- 17.6k
7
votes
Philosophy behind cohomological representations
This is just a long comment, and I'm not sure if it is the kind of thing you're looking for. Hopefully someone can give a proper answer to your question.
In an old question, Paul Garrett made the fol …
- 17.6k
23
votes
2
answers
2k
views
Even Galois representations "mod p"
Consider an irreducible $\mathrm{mod}$ $p$ representation:
$$\rho: \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_2(\bar{\mathbb{F}}_p)$$
If $\rho$ is odd, it was conjectured by Serre in th …
- 17.6k
19
votes
Underlying idea for (automorphic) L-function?
This is not exactly an answer, but should illustrate how tentative the opinion of experts on L-functions is, when it comes to explain what L-function really are. Two quotes from the first volume on nu …
- 17.6k