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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.
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
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
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
5
votes
Accepted
Relationship between motivic Galois groups and Langlands program
Take $K=\mathbb{Q}$ for simplicity, but this applies to any number field $K$, and let $L$ be the Langlands group and $\mathcal{G}$ the motivic Galois group of $\mathbb{Q}$.
Then the conjectural relat …
- 17.6k
3
votes
Poles of Rankin-Selberg $L(s,\pi\times\tilde \pi)$?
Warning. I was thinking on $\pi$ not neccesarily cuspidal. Jeremy Rouse's answer is the correct one.
$L(s,\pi_1\times \pi_2)$ has a pole at $s=1$ if $\pi_2=\tilde\pi_1$, but it is not neccesarily sim …
- 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
3
votes
Accepted
Non-vanishing of L-function of modular form
Yes, for primitive modular forms (both holomorphic and non-holomorphic) you can in fact something much stronger than the non-vanishing result with much (much!) less machinery than Langlands-Shalika.
…
- 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
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
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
4
votes
Lindelof Hypothesis implying Selberg Eigenvalue Conjecture?
To see that this is not the case, consider that it is well known that the Generalized Riemann Hypothesis implies the Lindelöf hypothesis.
On the other hand, not even the full GRH implies the Selberg …
- 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
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
8
votes
The status of automorphic induction
I'm quite interested in this myself. I'll try to answer, in hope that someone else can complete it if I'm missing something important.
The only two major cases of automorphic induction known still ar …
- 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