Questions tagged [eisenstein-series]
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5
votes
1answer
168 views
Bound on an expression involving J-function coefficients
I would like to show that
$$(m+1)\,c_m- \sum_{n=1}^{m-1} c_n \,\sigma_{m-n} > 0$$
for all $m$, where $c_i$ is the $i$th coefficient of Klein's $J$-function
$$J(q)= \frac{1728 \;E_4^3(q) }{...
2
votes
0answers
130 views
Question about the Fourier expansion of adelic Eisenstein series for $\operatorname{GL}_2$
My reference is Daniel Bump's book, Automorphic Forms and Representations, Chapter 3.7. Let $k$ be a number field, $G = \operatorname{GL}_2$, $B$ and $T$ the usual Borel subgroup and maximal torus ...
5
votes
0answers
132 views
Generalized Jacobians and modular units
Let $X$ be a proper algebraic smooth curve over a characteristic zero field $k$ and let $J$ be the Jacobian variety of $X$. Let $K$ be the function field of $X$. Assume that we are given $n$ distinct ...
7
votes
1answer
266 views
Algebraic independence of $P,Q,R$ or $E_2,E_4,E_6$ over $\mathbb C(z)$
Let $P,Q,R$ be the Fourier series of the Eisenstein series $E_2,E_4,E_6$, that is,
$$
P(q)=1-24\sum_{n=1}^{\infty}\sigma_1(n)q^n,
$$
$$
Q(q)=1+240\sum_{n=1}^{\infty}\sigma_3(n)q^n,
$$
$$
R(q)=1-504\...
5
votes
0answers
170 views
Eisenstein series of Hilbert modular forms
I am reading Shimura's paper "The Special Values of the Zeta Functions Associated With Hilbert Modular Forms" and I do not exactly understand his definition of the Eisenstein series in section 3.
...
11
votes
1answer
360 views
Decategorification of Gaitsgory's strange functional equation?
Let $G$ a complex reductive algebraic group, $X$ be a smooth compact complex curve. The moduli stack $Bun_G$ of principal $G$-bundles on $X$ is generally speaking not quasi-compact (so we do not have ...
3
votes
0answers
35 views
Spectral theory for Fuchsian groups of the first kind
There are tons of material on the spectral theory of $L^2(\Gamma\backslash G)$ for a lattice $\Gamma$ in $G=PSL_2({\mathbb R})$. There are also many papers on the case of $\Gamma$ being convex-...
5
votes
1answer
343 views
How to compute Coefficients in Chudnovsky's Formula?
My aim is to understand all three coefficients arising in the Chudnovsky-Formula (see also Question 300385). Two of them are easily computed, but I failed with the third:
It is known that for all $\...
20
votes
1answer
868 views
Why does this quasi-modular function have integral values?
It is a well-known result that the modular function $1728J(\tau) := \frac{1728E_4(\tau)^3}{E_4(\tau)^3-E_6(\tau)^2}$ has integral values if $\tau$ has class number 1 - for example at $\tau_{163}:=\...
3
votes
0answers
96 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 ...
4
votes
1answer
180 views
Main result of Shimura's On Eisenstein Series
The paper I'm referring to is https://projecteuclid.org/euclid.dmj/1077303203. Here Shimura constructs an Hermitian Eisenstein series $E_m(z,k,s,\psi,\mathfrak{b})$ for, in the case I am interested in,...
2
votes
0answers
78 views
Growth of a modified Zeta function appearing in the non-holomorphic Siegel Eisenstein series
In a paper (Eisenstein series for Siegel modular groups, https://link.springer.com/content/pdf/10.1007/BF01459520.pdf) Mizumoto obtains an explicit Fourier expansion for the non-holomorphic Siegel ...
6
votes
1answer
230 views
Eisenstein series for quadratic number fields
I am familiar with the theory of modular forms and weight k Eisenstein series, and I am wondering if such a theory exists when the base field is not $\mathbb{Z}$.
Is there a theory of modular forms ...
11
votes
0answers
187 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 ...
6
votes
1answer
227 views
Critical values of L-functions and weights of Eisenstein Series
I have been reading Serre's paper on p-adic modular forms and there seems to be a connection between critical values of L-functions and weights of Eisenstein series in the following sense:
For the ...
9
votes
1answer
358 views
Properties of coefficients in expansion of $E_6/E_4$ and $E_8/E_6$
Let $a(n)$ and $b(n)$ be define by the following;
$E_6/E_4 = 1 - 744q + 159768q^2 - 36866976q^3 + 8507424792q^4 - 1963211493744q^5 + \cdots = \Sigma a(n)q^n,$
$E_8/E_6 = 1 + 984q + 574488q^2 + ...
28
votes
2answers
1k 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
$$ E_2(z, s) = \sum_{\substack{m, n \in \...
3
votes
1answer
142 views
Rate of convergence of Siegel Eisenstein series
Let $\Gamma_n=Sp_{2n}(\mathbb{Z})$ and write $\gamma=\left(\matrix{A & B \\ C & D}\right)\in\Gamma_n$ for its block decomposition. Further let $\Gamma_n^0$ be the subset consisting of ...
7
votes
1answer
341 views
Analytic properties of Eisenstein series
Let $\Gamma$ be a discrete subgroup of $SL_2(\mathbb{R})$ which has a cusp at $\infty.$ suppose that $\mu(\Gamma\setminus\mathbb{H})<\infty,$ consider the Eisenstein series :$$E(z,s,\Gamma)=\sum_{\...
3
votes
0answers
77 views
Reference request: Eisenstein series built from principal series representation
One can build an Eisenstein series $E_s(v,g)$ from a vector $v\in D_s$, the principal series representation of $G_{\mathbb{A}_\mathbb{Q}}$. The space $D_s$ has a restricted tensor product structure
$$ ...
7
votes
1answer
140 views
Region of convergence of Eisenstein series is a union of Weyl chambers when groups have discrete series?
Let $G$ be a reductive algebraic group, and let $M$ be a Levi subgroup of $G$. In Urban's paper Eigenvarieties for Reductive Groups, it seems to be assumed that if $G(\mathbb{R})$ and $M(\mathbb{R})$ ...
4
votes
0answers
292 views
Eisenstein series of weight one
Let $\psi$ be an odd Dirichlet character of $G_{\mathbb{Q}}$ with conductor equal to $N$ and $p \nmid N$ be a prime number. Assume that $\psi(Frob_p)=1$.
Denote by $E_{\psi,1} \in S_1(\Gamma_1(N))$ ...
5
votes
1answer
389 views
Special values of real analytic Eisenstein series
Given $\tau$ in the upper half plane, define the normalized real-analytic Eisenstein series by
$$
E(\tau, s) = \frac{1}{2} \sum_{(m,n)}' \frac{y^s}{|m\tau + n|^{2s}}
$$
It is initially defined for $\...
1
vote
1answer
112 views
Factorizability of Subquotients of Principal Series Representations
Fix number field $F$, its ring of adeles $\mathbb{A}$, a "nice" algebraic group defined over $F$ (at least reductive but for my purposes I can assume simple and simply connected) and a parabolic ...
7
votes
2answers
453 views
Eisenstein Series on Siegel Space
I am looking for any reference dealing explicitly with Eisenstein series on Siegel space (the simplest case of $\rm{SP}_4$ is fine). Anything would be welcome, but in particular I'm interested in the ...
10
votes
1answer
678 views
How much can an Eisenstein series be truncated?
For ease of exposition, I will stick to the simplest case: consider the Eisenstein series for $SL_2(\bf R)$
$$E(z,s)=\sum_{\gamma\in P_{\bf Z}\backslash SL_2(\bf Z)}\text{Im}(\gamma z)^s=\sum_{(c,d)\...
6
votes
1answer
304 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 ...
6
votes
1answer
310 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 ...
6
votes
1answer
655 views
Fourier expansion of Eisenstein Series
I have been reading a bit about the Fourier expansion of Eisenstein series (weight 1/2). I came across the fact that the coefficients contain Modified Bessel functions.
Further reading I found ...
2
votes
1answer
325 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(...
12
votes
1answer
1k views
Primer on Eisenstein series
My apologies if this question is a duplicate. I seached, and the closest I could locate is this question, which has very intriguing and intractable (for me) responses.
In my continuing journey of ...
1
vote
1answer
696 views
Eisenstein series of weight $2$ for $\Gamma_0(N)$ : where am I wrong?
Let $A_{N,2}$ be the set of triples $(\psi,\varphi,t)$ such that $\psi$ and $\varphi$ are primitive Dirichlet characters modulo $u$ and $v$ with $(\psi\varphi)(-1)=1$, and $t$ is an integer such that $...
7
votes
1answer
436 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 ...
4
votes
1answer
130 views
Intertwining Operators Associated to Simple Reflections
Let $G$ be a quasi-split reductive group, over a local field, with a Borel subgroup $B=T\cdot N$ and the associated Weyl group $W$. Given a family of induced representations $\pi_s = Ind_B^G \chi\cdot ...
1
vote
0answers
214 views
About Theorem $3.1.3$ in Kubota's book: Elementary theory of eisenstein series
My question is about the proof of Theorem $3.1.3$ given in kubota's book, which shows how the function $\varphi(s)$ appearing in the Fourier expansion of eisenstein series can be continued ...
4
votes
0answers
96 views
Counting cosets of matrices of determinant > 1 under the action of a congruence subgroup
I tried asking this on math exchange, but no luck, so thought I'd try here.
Let $M_2(m,\mathbb{Z}) $ be the $2\times 2$ matrices with integer entries and determinant $m$. Let $\Gamma^0(N)$ be the ...
8
votes
0answers
366 views
Moduli interpretation of Eisenstein series
Let $N \geq 11$ be an integer and consider the basis of Eisenstein series for $M_2(\Gamma_0(N))$ described in Theorem $4.6.2$ of Diamond--Shurman's book. Pick and Eisenstein series $F$ in this basis. ...
5
votes
1answer
488 views
Ternary quadratic form theta series as Hecke eigenforms and class number one
At
Simple comparison of positive ternary quadratic form representation counts
Jeremy answered:
"The reason is that the theta series for the sums of three squares form is an eigenfunction for all the ...
4
votes
1answer
148 views
Simple comparison of positive ternary quadratic form representation counts
Something came up yesterday in a referee request and I was surprised to find that I did not know the facts in full generality. This is about positive quadratic forms in three variables with integer ...
8
votes
1answer
266 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 $\Gamma=PGL_2(\mathcal{O})$...
3
votes
1answer
350 views
about lemma 5.9 of Mazur's famous Eisenstein ideal paper
In Lemma 5.9 of Chapter II of his famous Eisenstein ideal paper, Mazur proved that
when $1/N$ is invertible in the ring $R$, if $\phi$ is a holomorphic modular form in $\omega^k$ over $\Gamma_0(N)$ ...
1
vote
1answer
202 views
Real cusp forms
Most literature on modular functions (invariant or covariant with weight k under the full modular PSL_2(Z) group) treats holomorphic functions and introduce the notion of cusp forms (modular functions ...
1
vote
0answers
148 views
Twists in “Eisenstein property” in Geometric Langlands
I am trying to read and understand (parts of) Gaitsgory's “Outline of the proof of the Geometric Langlands conjecture for GL(2)” [arXiv link]. In Section 6.4.8 he states "Property Ei", which basically ...
3
votes
0answers
379 views
On nonholomorphic Eisenstein series
Could you suggest me a reference where the following non-holomorphic generalization of the Eisenstein series is discussed?
$$ G_{k,l}(\tau,z) = \sum_{m,n} (z+m+n\tau)^{-k}(\bar z+m+n\bar \tau)^{-l}
$...
2
votes
0answers
505 views
Numerical methods for Eisenstein series
Are there any existing numerical libraries for Eisenstein series? In particular I am interested in calculating values of parabolic Eisenstein series on $ SL(n,\mathbb Z) \setminus GL(n,\mathbb R) / (...
4
votes
2answers
1k views
Fourier expansion of Eisenstein series at various cusps
Here is my setting: Let $E\in\mathcal{M}_k(\Gamma_0(N))$ be an Eisenstein series (of trivial Nebentypus) that is a normalized eigenform for all the Hecke operators at level $\Gamma_0(N)$. Assume that ...
8
votes
1answer
792 views
Eisenstein series and 163?
Given $q = e^{2\pi i \tau}$ and the Eisenstein series $E_{2k}(\tau)$, i.e.,
$$E_2(\tau) = 1-24\sum_{n=1}^\infty \frac{n q^n}{1-q^n}$$
$$E_4(\tau) = 1+240\sum_{n=1}^\infty \frac{n^3 q^n}{1-q^n}$$
...
4
votes
0answers
542 views
What is different about the Residual Spectrum
In the context of spectral decomposition of functions in $L^2(\Gamma \backslash \mathfrak{h})$, or Selberg trace formula, we come across three different types of spectrum.
First off there is the ...
16
votes
4answers
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 ...
7
votes
2answers
1k views
Relation between Theta series and Eisensteinseries
In "Mackey - Unitary Group Representation in Physics, Probability and Number Theory" on page 326, George Mackey mentions a result of Ludwig Siegel, which was later generalized to semi-simple Lie ...
