All Questions
18 questions
39
votes
2
answers
4k
views
How can one understand the Eisenstein series E2 in terms of automorphic representation?
The weight 2, level 1 Eisenstein series $E_2(z)$ is a non-holomorphic automorphic form. It is defined as the analytic continuation to $s = 0$ of the series
$$ E_2(z, s) = \sum_{\substack{m, n \in \...
- 497
17
votes
4
answers
2k
views
Where do the real analytic Eisenstein series live?
In obtaining the spectral decomposition of $L^2(\Gamma \backslash G)$ where $G=SL_2(\mathbb{R})$, and $\Gamma$ is an arithmetic subgroup (I am satisfied with $\Gamma = SL (2,\mathbb{Z})$) we have a ...
11
votes
0
answers
231
views
Eisenstein series for non congruence subgoups
What is the present status of the Eisenstein series for noncongruence subgroups?
I am aware of work of A. Scholl and Rohrlich work on the subject.
Is there any specific examples that has been ...
- 248
8
votes
1
answer
666
views
Atkin-Lehner theory for nonholomorphic Eisenstein series
I am currently reading something about nonholomorphic Eisenstein series $E_\mathfrak{a}(z,1/2+it)$ for $\Gamma_0(q)$, where $\mathfrak{a}$ is a cusp (cf. Iwaniec, H. Spectral Methods of Automorphic ...
- 135
8
votes
1
answer
459
views
Simplest case of Langlands-Shahidi method
I would like to read the simplest examples of Langlands-Shahidi method carried out to prove the functional equation of $L$-function.
Could the constant term of $\mathrm{GL}(2)$-Eisenstein series be ...
- 3,804
8
votes
1
answer
369
views
Eisenstein series over a definite division algebra
Let $D$ be the definite quaternion division algebra over $\mathbb{Q}$. $\mathcal{O}$ is a maximal order inside $D$, let's fix $\mathcal{O}$ to be the Hurwitz quaternion. Let $\Gamma=PGL_2(\mathcal{O})$...
- 1,692
7
votes
1
answer
639
views
History of spectral methods to the study of real analytic $GL_2$-Eisenstein series
I'm trying to sort out the history of spectral methods in the study of real analytic $GL_2$-Eisenstein series. From what I read so far, I would say that the subject was really kicked off by the ...
- 7,609
7
votes
2
answers
708
views
Eisenstein Series on Siegel Space
I am looking for any reference dealing explicitly with Eisenstein series on Siegel space (the simplest case of $\rm{SP}_4$ is fine). Anything would be welcome, but in particular I'm interested in the ...
- 2,824
6
votes
1
answer
557
views
Analogues of Hecke relations for Maass forms
If a (suitably normalised) holomorphic cusp newform has q-expansion
$$f(z) = \sum_n \lambda_f(n) e(nz),$$
then we know the Hecke relations for $(mn,q)=1$,
$$(\star) \qquad \lambda_f(m)\lambda_f(n) = \...
- 927
5
votes
1
answer
784
views
Special values of real analytic Eisenstein series
Given $\tau$ in the upper half plane, define the normalized real-analytic Eisenstein series by
$$
E(\tau, s) = \frac{1}{2} \sum_{(m,n)}' \frac{y^s}{|m\tau + n|^{2s}}
$$
It is initially defined for $\...
- 1,821
5
votes
0
answers
147
views
For Hida theory on $GU(2,2)$ can $p$ be inert in the imaginary quadratic field $K$?
I am familiar with the theory of Hida families of modular forms, so Hida theory on $GL_2$, but I am not familiar with Hida theory on any other algebraic groups. My question concerns Hida families of ...
- 323
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
4
votes
0
answers
189
views
About the structure of smooth automorphic forms
Recently I read Prof. Cogdell's notes: Lectures on L-functions, Converse Theorems, and Functoriality for $GL_n$. (Co)
In chap.2.3, the conception of smooth automorphic forms is introdued. Explicitly, ...
- 111
3
votes
0
answers
242
views
Explicit expression of automorphic representations as automorphic forms
Let‘s take $G=GL_n$ over a number field $F$ for example.
It's already known that every irreducible automorphic representation $\pi$ is a irreducible component of a induced representation $I(G,P;\...
- 111
3
votes
0
answers
291
views
Derivatives of Eisenstein series
A book of Moeglin-Waldspurger says that there was a conjecture that every automorphic form arises as the derivative of an Eisenstein series which is proved there for function field case and proved by ...
- 1,024
2
votes
1
answer
429
views
A computation about Whittaker functions and Eisenstein series
I have some questions about the computation of Eisenstein series and Whittaker functions in the book. The question is on page 29, Theorem 4.3.
My questions are in the following.
(1) I think that $B(...
- 6,201
1
vote
1
answer
189
views
Analytic continuation of the Eisenstein series defined over Hecke and Fricke subgroups
It is well known that the (real analytic) Eisenstein series is defined, in the slash notation, as follows
$$E_{s}(\tau) = \sum\limits_{\gamma\in\Gamma_{\infty}\backslash\text{SL}(2,\mathbb{Z})}\left.y^...
- 449
1
vote
0
answers
242
views
Constant coefficient of Eisenstein series
Let $\chi$ be a character of $\mathbb{Q}^{\times}\setminus\mathbb{A}^{\times}$. We define an induced representation of $Mp_2(\mathbb{A})\simeq SL_2(\mathbb{A})\times \mathbb{C}^1$,
$$I(s,\chi) := \{\...
- 103