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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.
23
votes
Accepted
Explicit examples of algebraic Hecke characters with infinite image?
The "most obvious" algebraic Hecke characters of a field $K$ are the characters of the ideal class group of $K$, which have trivial infinity-type and trivial conductor. There might be no non-trivial e …
- 37k
21
votes
What computer program for automorphic forms
The only CAS's that have built-in support for modular and automorphic forms, as far as I know, are Sage and Magma. [Edit: I had forgotten Pari/GP, which will introduce substantial modular forms functi …
- 37k
18
votes
Accepted
What is the $p$-adic Langlands conjecture for $\mathbf{GL}_1$?
I am not sure it makes sense to ask "what is the p-adic local Langlands conjecture for $\mathrm{GL}_1$". Nobody has succeeded in even formulating a reasonable candidate for a p-adic LLC for $\mathrm{G …
- 37k
15
votes
Accepted
The correct determinant exponent of the weight $k$-operator for defining Hecke operators/ade...
This is a question which has no "right" answer.
A posh interpretation of the choice of exponent is that a Hecke eigenform $f$ determines an equivalence class of irreducible representations $\Pi = \big …
- 37k
14
votes
Accepted
Why the level of a half integral weight modular form must be a multiple of 4?
The problem isn't that $S_{k + 1/2}(\Gamma_0(N))$ is zero if $4 \nmid N$; it's that the space is not defined if $4 \nmid N$.
In order to make sense of what a "half-integer weight form of level $\Gam …
- 37k
13
votes
What kind of non-cuspidal automorphic representation are not isobaric sums?
EDIT. A colleague wrote to me to point out that my original answer to this question was actually completely wrong: I had confused "isobaric" representations with "pure" representations (which are not …
- 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
12
votes
Accepted
What is the relationship between (g,K)-module and Maass forms?
So you've seen that there are essentially three types of (g, K)-modules: finite-dimensional ones; principal series; and discrete series. The finite-dimensional ones don't interest us, since they are n …
- 37k
12
votes
Accepted
What's the status of Arthur's announced classification for GSp(4)?
This question is answered pretty definitively by the following recent paper:
Gee, Toby; Taïbi, Olivier,
Arthur’s multiplicity formula for $\mathrm{GSp}_4$ and restriction
to $\mathrm{Sp}_4$, J …
- 37k
11
votes
Accepted
modular form Fourier coefficients and associated automorphic representation
Jared Weinstein and I wrote a paper on how to compute $\pi_p$: see here.
As Olivier says, $a_p$ will often be zero, and in fact if the central character is trivial (or has conductor coprime to $p$) …
- 37k
11
votes
Accepted
"Classical" description of automorphic forms on unitary groups
These go by the name of "Hermitian modular forms". They occur very frequently in papers of Shimura (e.g. his monograph Arithmeticity on the theory of automorphic forms) and in other more recent works. …
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10
votes
Accepted
Is there a known construction of Cuspidal representations of GL(3) isomorphic to their own t...
Let $E / F$ be the cyclic cubic extension corresponding to $\chi$ by class field theory. Let $\sigma$ be a generator of $\operatorname{Gal}(E / F)$, and let $\psi$ be a character of $E^\times \backsla …
- 37k
10
votes
Symmetric powers of Ramanujan tau-function
It is indeed true that substantially more is known for holomorphic cusp forms than for general automorphic representations, as a consequence of modularity lifting theorems.
The strongest result so f …
- 37k
10
votes
Real character modular forms: Fourier coefficient real?
When acting on forms with trivial character, the Hecke operators $T_n$ for $n$ coprime to $N$ are Hermitian, so their eigenvalues are always real. This doesn't work for the Hecke operators of index no …
- 37k
10
votes
Accepted
conductor formula
The PhD thesis of Manami Roy (Univ. Oklahoma, 2019) gives a very detailed formula for the conductor of $\operatorname{Sym}^3(\pi_p)$, where $\pi_p$ is the representation of $\mathrm{GL}_2(\mathbf{Q}_p …
- 37k