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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.
1
vote
1
answer
116
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Maass-Hecke construction
I heard this name that it can construct GL(2) automorphic forms or L-functions from GL(1)?
I did not find it anywhere.
Or does it have another name which we are familiar with?
- 3,804
4
votes
2
answers
1k
views
Any reference on Eisenstein Series for $\Gamma_0(N)$ in $\mathrm{GL}(2)$
What is the best reference on Eisenstein Series for $\Gamma_0(N)$ in $\mathrm{GL}(2,\mathbb{R})$?
For fixed $\Gamma_0(N)$, should there be several Eisenstein series (corresponding to each cusp)?
- 3,804
1
vote
0
answers
130
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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
5
votes
0
answers
551
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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 …
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2
votes
2
answers
820
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What is the relationship between (g,K)-module and Maass forms?
What is the relationship between (g,K)-module and Maass forms for GL(2)?
(g,K)-modules are defined in chapter 2 of Bump, Automorphic forms and representations.
There is a classification of (g,K)-mod …
- 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
1
vote
1
answer
333
views
Does FE of Selberg Zeta function imply Trace formula?
Does the functional equation of the Selberg Zeta function imply the Selberg trace formula?
BTW, the trace formula implies the functional equation.
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9
votes
1
answer
727
views
Self-dual automorphic forms on $GL(4)$
As is known among experts, all self-dual automorphic forms on $GL(3)$ come from symmetric square lifts from $GL(2)$. You can find this in Ramakrishnan (http://www.math.caltech.edu/~dinakar/papers/exer …
- 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
11
votes
2
answers
1k
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Using Eichler-Selberg trace formula to compute dimension of modular forms?
Is it possible to use Eichler-Selberg trace formula to compute the dimension of modular forms of weight $k$ for $SL(2,\mathbb Z)$? This was computed by classical methods such as Riemann-Roch.
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3
votes
1
answer
203
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Lower bound of first moment of $L$-function on $\mathrm{GL}(3)$
Let $\pi$ be an automorphic form on $\mathrm{GL}(3,\mathbb{A}_{\mathbb{Q}})$.
Do we know any case that
\[\int_0^{T} \left|L(\frac{1}{2} + it, \pi)\right| dt \gg T\]
holds unconditionally?
I know the …
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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 …
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1
vote
0
answers
134
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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 $D$, …
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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
6
votes
1
answer
1k
views
classification of irreducible admissible (g,K)-module for GL(3,R)
classification of irreducible admissible (g,K)-module for GL(3,R)
Is there a classification of irreducible admissible (g,K)-module for GL(3,R)?
For GL(2,R) we have principal series, discrete series …
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