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Results tagged with automorphic-forms
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user 9317
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.
10
votes
Any reference on Eisenstein Series for $\Gamma_0(N)$ in $\mathrm{GL}(2)$
Miyake, already recommended by GH, is a very complete reference, which is perhaps the only place to contain complete proofs about the subject. However, for that reason, and also because
he works with …
- 26k
7
votes
Most understandable notes on Jacquet-Langlands?
I think that you may like the notes by Ioan Badulescu: see
http://www-math.univ-poitiers.fr/~badulesc/pub/JLtata1.pdf
They are a 20 pages modern essentially complete exposition of the proof of Jacqu …
- 26k
5
votes
How badly can strong multiplicity one fail in the theory of automorphic representations?
I confirm what "someone once told you" about question 1 (so now "two people
once told you" or perhaps "someone twice told you"). This phenomenon
($\pi_\nu$ supercuspidal, and $\pi'_\nu$ principal ser …
- 26k
2
votes
Potential automorphy of abelian varieties
(i) is correct.
(ii) has nothing to do with Artin's conjecture I believe. It is known that (1) implies (2) in the case $K'/K$ solvable. This is the so-called solvable base change, solved by Arthur an …
- 26k
8
votes
Understanding the "idea" behind Langlands
To complement the answer from GH from MO:
Your first two bullet points, about Tate's thesis and Hecke characters are completely correct.
Your third point should be made more precise as follow. Yes w …
- 26k
10
votes
embedding of local tempered representation into cuspidal automorphic representation
This is not true, because the set of equivalence classes
of irreducible tempered representation of $Gl_n(F_v)$ has the power of the continuum
(think of the principal series for example) while the numb …
- 26k
17
votes
What is the non-motivic motivation behind automorphic representations?
Even without mentioning the relations with the arithmetic and algebraic geometry (motives, if you want), there are many reasons people have been interested in automorphic forms. One point of view is t …
- 26k
10
votes
Adelic methods for classical modular forms
A standard example of a theory that can be understood without the adelic setting but are better understood with it is the Atkin-Lehner theory of new forms. Atkin and Lehner did prove it without any re …
- 26k
4
votes
What can the theory of automorphic forms for $SL(n,\mathbb{Z})$ say about $SL(n,\mathbb{Z})$?
I don't exactly see how the second question is another version of the first, so let me answer them separately (and each very partially):
In what aspects, if any, automorphic forms for $SL(n,\ …
- 26k
5
votes
Accepted
What is an automorphic representation of CM type ?
1.-- in the $Gl_2$-case, $\pi$ is of CM type if it is the automorphic induction
of a Grossencharacter of a CM extension K of $F$. In terms of the Galois representation
of $Gal(\bar F/F)$ attached to $ …
- 26k
14
votes
A question on representation theory of p-adic groups
It is indeed well-known that the category of smooth admissible representations of $G$ (and other reductive $p$-adic groups)
is not semi-simple. The principal series, that is the representations induce …
- 26k
30
votes
Accepted
New Geometric Methods in Number Theory and Automorphic Forms
Knowing the organizers well and working in the field, I can try an answer, but this is nothing more than an educated guess.
First, the breakthroughs in question include
(i) The construction and stu …
- 26k
6
votes
Accepted
Holomorphic cusp forms and cohomology of GL(2,Z)
Let me try an answer. Instead of working with $H^1(SL(2,{\bf Z}),V_k)$, I'll work with a space which is naturally isomorphic to it, but more concrete, the space of modular symbols $Symb(V_k)$, defined …
- 26k
13
votes
central/critical/special values of L-functions terminology
I think everything you write is correct, and moreover very clear. For example, the value $\zeta(1/2)$ is not a special value of the Riemann Zeta function: $w=0$ in this case. On the contrary, if $E$ i …
- 26k