All Questions
Tagged with automorphic-forms analytic-number-theory
10 questions
3
votes
1
answer
1k
views
Are there infinitely many L-rigs?
$\DeclareMathOperator{\Q}{\mathbb{Q}}$Call "L-rig" any class $\mathcal{L}$ of L-functions of automorphic representations of $\operatorname{GL}_{n}(\mathbb{A}_{\Q})$ for some $n$ belonging to ...
- 6,974
15
votes
3
answers
1k
views
There is no lattice in PSL(2,R) which contains PSL(2,Z) properly?
How can I see that there is no lattice in $G=\mathrm{PSL}_2( \mathbb{R})$ which contains $\Gamma_1=\mathrm{PSL}_2( \mathbb{Z})$ properly, or equivalently, that $X_1 =\mathrm{PSL}_2(\mathbb{Z}) \...
- 11.2k
12
votes
1
answer
3k
views
Best record toward Selberg's eigenvalue conjecture?
What's the best record toward Selberg's eigenvalue conjecture:
a Maass form on $\Gamma_0(N)$ has eigenvalue greater than or equal to 1/4?
- 3,804
9
votes
3
answers
2k
views
Sato-Tate measure for GL(3) Automorphic forms
As we have known, the Sato-Tate measure for GL(2) turned out to be the half circle measure
$\frac{1}{2\pi} \sqrt{4-x^2}dx$ on [-2,2],
which appears in various versions of equi-distribution problems ...
- 3,804
8
votes
2
answers
839
views
On the consistency of the definition of the conductor for automorphic forms
Let $\pi$ be an irreducible admissible representation of $\mathrm{GL}_2(F)$, where $F$ is local non-archimedean. The local conductor associated to $\pi$ can be defined in two usual manners:
By its ...
- 5,424
8
votes
1
answer
530
views
How strong is the requirement of being a Gelbart-Jacquet lift?
Let $\pi$ be a cuspidal automorphic representation of $\mathrm{GL}(3)$ over a number field $F$. I am wondering how general are Gelbart-Jacquet lifts of automorphic representations of $\mathrm{GL}(2)$ ...
- 5,424
7
votes
1
answer
811
views
Alternative way to prove the functional equation for Eisenstein series?
Let $E(z,s):=\pi^{-s}\Gamma (s) \sum_{(m,n)=1}\frac{y^s}{|mz+n|^{2s}}$ be the real-analytic Eisenstein series.
It satisfies the functional equation $E(z,s)=E(z,1-s)$ with two poles at $s=0,1$.
The ...
- 3,804
5
votes
1
answer
616
views
Question on an application of Langlands' result on the constant term of Eisenstein series (Is this a typo?)
I would like to understand an argument in https://link.springer.com/content/pdf/10.1007/BF01393904.pdf, which uses Langlands' result on the constant term of Eisenstein series, but I'm not getting it ...
- 3,625
5
votes
2
answers
433
views
Asymptotic's for Fourier coefficients of $GL(3)$ Maass forms
Let $f$ be a $GL(3)$ Hecke-Maass cusp form and $A(m,n)$ denote its Fourier coefficients.
Are there any lower bounds known for $\sum_{p\leq x}|A(1,p)|^2$ or $\sum_{n\leq x}|A(1,n)|^2$ ? (we know the ...
- 153
4
votes
2
answers
1k
views
Real character modular forms: Fourier coefficient real?
Let $f$ be a modular form of level $N$ and real character $\chi$ of mod $N$ and weight $k$.
Does the Fourier coefficient or hecke-eigenvalue of $f$ have to be real?
What I knew is that if $N=1$ and $\...
- 3,804