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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.
4
votes
Conductor of monomial forms with trivial nebentypus
Just to augment Kevin's series of comments: I think that the conductor of the induction of some
character $\chi$ over a quadratic field to $\mathbb Q$ would normally equal $D N(C)^2$, where $D$ is the …
- 57.6k
6
votes
Is there a canonical notion of "mod-l automorphic representation"?
If $G$ is a connected, linear, reductive group over $\mathbb Q$, let $q_0$ be the dimension so denoted in Borel--Wallach, i.e. $q_0 = (d - l_0)/2,$ where $d = \text{dim } G_{\infty} -
\text{ dim } A_ …
- 57.6k
3
votes
Accepted
extending cusp forms
I believe the answer should be yes, by some version of the following sketch of an argument:
(Note: by restriction of scalars, I regard all groups as being defined over $\mathbb Q$,
and I write ${\mat …
- 57.6k
11
votes
Accepted
Local to Global principle for reductive groups
If $\pi_v$ is supercuspidal, then (after making a twist if necessary) it should be possible to find an automorphic $\pi$ with $\pi_v$ as the local factor at $v$. This kind of result is usually proved …
- 57.6k
6
votes
Accepted
Automorphic forms and Galois representations over imaginary quadratic fields: generalizing T...
Various groups of people have thought about/are thinking about this. The natural source of the desired Galois reps. is a $U(2,2)$ Shimura variety. The problem is that the cohomology of this variety …
- 57.6k
11
votes
Accepted
strong approximation and one-dimensional automorphic representations
Let $G$ be the group of norm one elements in $D^{\times}$. An easy argument
shows that it suffices to prove the claim for $G$ in place of $D^{\times}$.
In other words, I will let $\pi$ be an automorp …
- 57.6k
8
votes
embedding of local tempered representation into cuspidal automorphic representation
As Joel points out, this is not possible. Even in the case of a discrete series rep'n, one can't control the value of the central character on the non-compact part of $F_v^*$. What one can do (in so …
- 57.6k
23
votes
Accepted
What is the current status of the function fields Langlands conjectures?
(1) Regarding the relationship between geometric Langlands and function field Langlands:
typically research in geometric Langlands takes place in the context of rather restricted ramification (everywh …
- 57.6k
16
votes
Accepted
Why isn't meromorphic continuation enough for converse theorems?
Dear Kevin,
My understanding of the current meromorphic continuation results is that they do something along the lines of expressing a given Galois representation as the induction of a virtual (i.e. …
- 57.6k
26
votes
What is the non-motivic motivation behind automorphic representations?
These are some comments that originally appeared on the OP's earlier question (linked above), gathered together here as an answer:
The notion of automorphic representation (as an irreducible represen …
- 57.6k
14
votes
Accepted
Constructing coherent sheaves on Shimura varieties.
If the Shimura variety is attached to the Shimura data $h:\mathbb S \to G_{/\mathbb R}$,
and if as usual $K$ denotes the centralizer in $G_{/\mathbb R}$ of $h$,
then the automorphic vector bundles are …
- 57.6k
12
votes
p-adic L-functions
You should distinguish between the method (of which there is not one, but several) and the result (namely, the existence of $p$-adic $L$-functions).
It is expected that $p$-adic $L$-function exist …
- 57.6k
11
votes
Unitary groups over number fields
Just to add to Jon Yard's answer: when one defines a unitary group as described in the OP, the group that one gets after extending scalars from $F$ to $F_v = \mathbb R$ for each archimedean place $v$ …
- 57.6k