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Results tagged with automorphic-forms
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user 11919
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.
9
votes
Accepted
Generalizations of Hamburger's Theorem
Hamburger's theorem has been generalized in various ways to automorphic $L$-functions (of arbitrary degree). Such generalizations are called "converse theorems", and they play a central role in the La …
- 105k
6
votes
Accepted
Is "self-dual" equivalent to "dihedral" for Maass forms on $\mathrm{GL}(2)$?
Let $f$ be a Maass form on the upper half-plane with nebentypus $\chi$. It is known that
\begin{align*}
\text{$f$ is self-dual}&\qquad\Longleftrightarrow\qquad\text{$L(f\times f,s)$ has a pole at $s=1 …
- 105k
3
votes
The lower bound for the automorphic $L$-function $L(s,\pi)$ at the edge of the critical stri...
This is not known. The best known lower bound is due to Brumley (2013), to be found in an Appendix to a paper by Lapid, which gives that
$$|L(1+it,\pi)| \gg_{\pi_{\infty},\varepsilon} ( N (1+|t|) …
- 105k
7
votes
A question on hybrid subconvexity for individual L-functions
There are several confusions in your post.
1. Automorphic forms for the group $\mathrm{SL}_2(\mathbb{Z})$ have level $q=1$ by definition. You probably wanted to talk about newforms for the Hecke congr …
- 105k
5
votes
Accepted
'$\times$' or '$\otimes$' when writing $L$-functions?
The symbol $\times$ on the left-hand side is the Rankin-Selberg product. If $\pi$ and $\rho$ are automorphic representations of $\mathrm{GL}(m)$ and $\mathrm{GL}(n)$, respectively, then one can define …
- 105k
2
votes
Accepted
Question on automorphic $L$-functions
Let us restrict to automorphic representations of $\mathrm{GL}_n$ over $\mathbb{Q}$ with arbitrary $n$ and unitary central character.
If $\pi$ is an irreducible cuspidal representation, then $L(s,\pi) …
- 105k
8
votes
Accepted
On the notion of cuspidality
To every local admissible representation $\pi_v$ of $\mathrm{GL}_n(k_v)$, there is a local $L$-function $L(s,\pi_v)$. For a global admissible representation $\pi=\otimes_v \pi_v$ of $\mathrm{GL}_n(\ma …
- 105k
4
votes
On the precise form of $\mathrm{GL}(3)$ (and others) L-functions
Whether you use $A(1,n)$ or $A(n,1)$ is just a convention. If one sequence defines $L(s,\pi)$, then the other one defines the dual $L$-function $L(s,\tilde\pi)$. For example, we could define the Diric …
- 105k
4
votes
Accepted
$\DeclareMathOperator\SL{SL}$Multiplicities of irreducible representations in discrete part ...
You are asking what is known about the dimension of weight $k$ holomorphic cusp forms for $\mathrm{SL}_2(\mathbb{Z})$, and the multiplicities of Laplace eigenvalues of weight $0$ and weight $1$ Maass …
- 105k
4
votes
Is a reductive adelic group a Type I group?
Freitag and van Dijk proved that the adelic points of a reductive group over a global field is trace class (Theorem 2.3), while every trace class group is of type I (Theorem 1.7). So the answer is yes …
- 105k
5
votes
Accepted
A question related to newform and irreducible cuspidal representation of $\operatorname{GL}_n$
Newform theory for $\mathrm{GL}_n$ was originally developed over non-archimedean local fields (at least for $n\geq 3$). The local statements readily yield their global adelic counterparts, since a cus …
- 105k
3
votes
Accepted
Under Ramanujan conjecture, is primitivity equivalent to cuspidality and irreducibility?
Yes, and this is true without the Ramanujan conjecture (but see also Peter Humphries' comment below).
If $\pi$ is not irreducible, say $\pi=\pi_1\oplus\pi_2$, then $L(s,\pi)=L(s,\pi_1)L(s,\pi_2)$.
…
- 105k
4
votes
Accepted
Do we know absolute bounds for the norm of Satake parameters?
We don't know any upper bound for $|\alpha_{p,j}|$ that is independent of $p$. On the other hand, we do know that each $|\alpha_{p,j}|$ is bounded by $p^{1/2}$, hence if $\pi$ is an automorphic repres …
- 105k
1
vote
For what automorphic representations is Ramanujan-Petersson known?
From the way you phrase your question, I suspect that you misunderstand something. Being supercuspidal is a local condition. If $\pi$ is an automorphic representation of $\mathrm{GL}_2$ over $\mathbb{ …
- 105k
10
votes
Accepted
Are the L-functions of a normalized newform and the corresponding cuspidal representation eq...
$L(\pi,s)$ agrees with $L(f,s)$ if $f\in\pi$ is a newform, and this is even true for $\mathrm{GL}_n$. Of course, things are complicated by the fact that there are many ways to define $L(\pi,s)$ and $L …
- 105k