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Results tagged with automorphic-forms
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user 2666
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
1
answer
207
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Local L-function $L(s,\pi_p\times \chi_p)=1$
Let $\pi_p$ be a ramified representation of $GL(n,\mathbb{Q}_p)$.
Let $\chi_p$ be a ramified representation of $GL(1,\mathbb{Q}_p)$.
Is it generally known that
$L(s,\pi_p\times \chi_p)=1$ if $\chi_ …
- 3,804
4
votes
1
answer
414
views
Fricke involution on GL(3)
Define $\Gamma_0(N)=\{\begin{pmatrix}
a&b&c\\
d&e&f\\
g&h&i
\end{pmatrix}
\in SL(3,\mathbb{Z})|g\equiv h\equiv 0(\mod N)\}$ be the $N$-level congruence subgroup on GL(3).
What should be a Fricke inv …
- 3,804
1
vote
0
answers
134
views
Is there an analysis theorem analogous to Kuznetsov/Petersson trace formula?
I am thinking about general differential operator acts on a compact manifold. Is there something similar to Kuznetsov trace formula?
For example, let $f_i $ be the eigenfunctions of an operator $D$, …
- 3,804
6
votes
2
answers
470
views
Symmetric powers of Ramanujan tau-function
Let $\Delta(z)$ be the modular form associated with Ramanujan $\tau$-function.
For any $k=2,3,...$, $Sym^k\Delta$ is conjectured to be an automorphic form on $\mathrm{GL}(k+1)$ and $L(s, Sym^k\Delta) …
- 3,804
2
votes
0
answers
335
views
Meaning of Ramanujan-Petersson conjecture? [closed]
I found it very hard to explain the Ramanujan-Petersson conjecture in a straightforward way.
I can only say now: think about automorphic forms as sound waves, and then the conjecture predicts that i …
- 3,804
7
votes
1
answer
774
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 m …
- 3,804
10
votes
0
answers
385
views
Residue of Eisenstein Series on GL(n)
Reference: Mœglin, C.; Waldspurger, J.-L.: Le spectre residuel de GL(n)
On GL($n$), the result of Waldspurger shows that if an automorphic representation $\pi$ is non-cuspidal and in the discrete spe …
- 3,804
13
votes
1
answer
852
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What kind of non-cuspidal automorphic representation are not isobaric sums?
Let's say $\pi$ is an automorphic representation on $GL_3(A_{\mathbb Q})$ (or $GL_n(A_{\mathbb Q})$).
If $\pi$ is not cuspidal, what $\pi$ can be other than isobaric sums?
If there is such a thing, …
- 3,804
7
votes
2
answers
474
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Rankin-Selberg integral for GL(3) form with Odd Maass form on GL(2)
Let $F$ be a Hecke-Maass cusp form for $SL_3(\mathbb Z)$.
Let $u$ be a Hecke-Maass cusp form for $SL_2(\mathbb Z)$.
The following integral
$$\mathcal L(s,F\times u)=\int_{{SL}(2,\mathbb{Z})\backslas …
- 3,804
4
votes
0
answers
106
views
Examples of conjectural functorial transfer which has $\times GL(1)$ functional equation?
I am look for some conjectural functorial transfer $X$ which
(A)for any $GL(1)$ automorphic representation $\pi$, we have
$L(s, X\times \pi)$ is holomorphic and satisfies certain functional equatio …
- 3,804
1
vote
0
answers
130
views
What is a common name for these automorphic objects?
I am looking for a name which includes these objects:
1. automorphic forms, cusp forms and non-cusp forms
2. Rankin-Selberg convolution between automorphic forms (which is conjectured to be automorphi …
- 3,804
9
votes
0
answers
417
views
Numerical Evidence for Grand Riemann Hypothesis?
Let $L(s)$ be an $L$-function coming from Hecke characters or automorphic forms (e.g. modular form on GL(2), Maass form on GL(2), and higher-rank analogues).
Is there any numerical evidence for Gran …
- 3,804
5
votes
0
answers
551
views
Why are Bessel function and Kloosterman sum similar?
It is a convention to say Kloosterman sums and Bessel functions are similar.
There are papers talking about Bessel functions on $p$-adic group (associated with a representation) such as Baruch's: htt …
- 3,804
7
votes
1
answer
310
views
Selberg trace formula, quadratic L-values, and generalization
It is known that the geometric side of the Selberg trace formula on GL(2) is related to values of quadratic L-functions (due to Sarnak, Zagier, etc).
Are there any conjectures or results about its ge …
- 3,804
3
votes
2
answers
333
views
Poles of Rankin-Selberg $L(s,\pi\times\tilde \pi)$?
Let $F$ be a number field and let $\pi$ be a cuspidal automorphic representation of $GL_n(\mathbb{A_F})$.
Do we know that $L(s,\pi\times \tilde\pi)$ has a simple pole at $s=1$?
Do we know that $\fra …
- 3,804