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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.
5
votes
Reference Request: Test vectors for local Rankin-Selberg L-factors in ramified cases
Are you asking for a proof of existence, or an explicit construction? These are very different things!
It is immediate from the definition that there exists a finite family $(W_i, W_i')_{i \in I}$ wit …
- 37k
4
votes
What is the conductor of an automorphic representation for $\Gamma_0(q)$ in $GSp(4)$?
To make the question well-posed, I'm going to suppose that we fix $\pi$ and take $q$ to be the smallest integer such that $\pi$ has nonzero invariants under $\Gamma_0(q)$. Then $q$ gives you some info …
- 37k
13
votes
Accepted
Do people prefer working on $\mathrm{GSp}$ and $\mathrm{GU}$ rather than $\mathrm{Sp}$ and $...
Symplectic case: Here are two reasons (not necessarily the only ones) why $\operatorname{GSp}_{2n}$ is more convenient to work with than $\operatorname{Sp}_{2n}$.
Firstly: there is no Shimura datum w …
- 37k
2
votes
Accepted
Understand the $p$-adic local Langlands correspondence with examples
Let's look at the case of representations associated to modular forms. I'm going to switch the roles of $\ell$ and $p$, because I find $\ell$-adic Hodge theory disturbing; so I'm going to look at $\rh …
- 37k
2
votes
Accepted
$l$-adic sheaf associated to an algebraic representation of $\mathrm{GSp}_{4}(\mathbb{Q})$
This is a special case of Pink's "canonical construction" functor, which associates various kinds of coefficient sheaves on a Shimura variety (etale $\ell$-adic sheaves, vector bundles with connection …
- 37k
8
votes
Accepted
modularity lifting theorems for non-compact unitary groups
You might like to read the introduction of Harris' 2013 Crelle paper "The Taylor-Wiles method for coherent cohomology" (see link). Here is an excerpt:
In practice, all the higher-dimensional results, …
- 37k
6
votes
The Langlands parameters of the symmetric cube lifts of cusp forms
To understand this question better one should remember what Langlands parameters actually are. A Langlands parameter isn't just a list of numbers: these numbers are the components of a map from some a …
- 37k
3
votes
Accepted
Motive associated to a cuspidal representation of $GSp_{4}$
The formula you quote from Harris defines a Galois representation, not a motive. We expect that there is a motive whose etale realisation is Harris' space, but that is not immediate.
The problems are: …
- 37k
4
votes
Accepted
Global symplectic (orthogonal) type of automorphic representation compels its type to all it...
Here is a proof of the claim using results from Arthur's monograph The Endoscopic Classification of
Representations: Orthogonal and
Symplectic Groups.
Let $N = 2n$ be an even integer, and $\pi$ a cusp …
- 37k
2
votes
Accepted
Part of some generic representation is also generic?
Let $\pi$ be the irreducible generic unramified representation of $Sp(W) $ that is a subquotient of $Ind(\chi_1, \dots, \chi_n)$.
I think the key here is to realise that this does not exist for all …
- 37k
5
votes
Galois representations attached to a cusp form for different primes
At the most basic level, $\rho_p$ and $\rho_q$ are "nothing to do with each other". E.g. the kernels of $\rho_p \bmod p$ and of $\rho_q \bmod q$ cut out two finite Galois extensions of $\mathbf{Q}$ wh …
- 6,039
3
votes
Jacquet module and Frobenius reciprocity
In general, all we can say from "general abstract nonsense" is that if $\sigma$ is a subrepresentation of $Ind_P^G(\pi)$, then $\pi$ is a quotient of $J_N(\sigma)$; but you don't immediately get any f …
- 37k
8
votes
Accepted
Is the Sato-Tate conjecture known for Bianchi modular forms?
Does it follow from the work of [1] that the Sato-Tate conjecture is
known for some class of cuspidal automorphic forms for GL2 over CM
fields?
Tautologically: "yes, those which correspond to modula …
- 37k
6
votes
Watson's triple product for automorphic forms shifted by Maass rising operators
By Ichino's triple product formula, of which Watson's formula is a special case, this integral will be given by the expected ratio of L-values (depending only on the automorphic reps generated by the …
- 37k
6
votes
Accepted
Relation between $\xi$-cohomological and discrete series
This condition comes up because of $(\mathfrak{g}, K)$-cohomology, which is an extremely important invariant of automorphic representations.
If $\xi$ is an algebraic rep, then $\xi$ defines a locally- …
- 37k