# Questions tagged [eisenstein-series]

The eisenstein-series tag has no usage guidance.

87
questions

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### Non-holomorphicity of Hecke regularization of weight-2 level-1 Eisenstein series

Originally asked on stackexchange since I figured it was a very elementary and silly error, but since I haven't had any response or interest in a couple days, I thought I'd post it here.
I have a &...

2
votes

0
answers

67
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### Extrema of real analytic Eisenstein series and more general modular functions

The real analytic Eisenstein series defined by the Poincare sum
$$E(s,z)=\sum_{(m,n)\neq (0,0)} {y^s\over |mz+n|^{2s}}$$
for $z\in{\mathbb H}$ and ${\rm Re}(s)>1$ is a manifestly $SL(2,{\mathbb Z})$...

1
vote

1
answer

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### 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^...

9
votes

1
answer

212
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### History of points of view on Eisenstein series

What is the history of Eisenstein series? Did the mathematician Eisenstein actually encounter them?
There are, as far as I know, two major perspectives on what Eisenstein series are. The first is in ...

3
votes

1
answer

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### The meaning of $L_{\chi}^2(G(\mathbb Q) \backslash G(\mathbb A)^1)$

I'm reading James Arthur's notes on the trace formula and am confused on a point on pages 65 and 66. For $G$ a reductive group over $\mathbb Q$ we are going over the decomposition of the space $L^2(G(...

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0
answers

106
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### Eisenstein series evaluated at $2i$

Consider the real analytic Eisenstein series defined by
$$
E(z,s) := \sum_{\gamma\in\Gamma_\infty\setminus\Gamma} Im(\gamma z)^s
$$
where as usual, $\Gamma=SL(2,\mathbb{Z})$ and $\Gamma_\infty$ is the ...

1
vote

0
answers

168
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### 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) := \{\...

3
votes

0
answers

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### Density of the Mellin transform inside the direct integral of induced representations

I'm trying to better understand the continuous spectrum of $G = \operatorname{GL}_2(\mathbb A_{\mathbb Q})$, which is the direct integral of induced representations $\mathbf H(s) = \operatorname{Ind}_{...

3
votes

1
answer

88
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### Calculating the residue of Eisenstein series from the residue of the intertwining operator

I've been reading the article Forms of $\operatorname{GL}(2)$ from the analytic point of view by Gelbart and Jacquet in Corvallis and am confused on a particular claim (equation 5.17 on page 232).
The ...

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votes

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### How to see the surjectivity of $L^2_{\text{cont}}$ onto the direct integral of Hilbert space representations?

I've been reading the article Forms of $\operatorname{GL}(2)$ from the analytic point of view by Gelbart and Jacquet in Corvallis and am confused on one point. Let $G = \operatorname{GL}_2$, and $V = ...

9
votes

1
answer

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### Why are characters orthogonal to cusp forms?

Let $G = \operatorname{GL}_2$, and let $V = L^2(Z(\mathbb A)G(\mathbb Q) \backslash G(\mathbb A),\omega)$, for $\omega$ a character of the ideles $\mathbb A^{\ast}$, identified with a central ...

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### Is there some precise way in which modular cocycle $c/(c\tau +d)$ "is" the generator of $H^1(\mathcal{M}_{ell}, \omega^2) \cong \mathbb{Z}/12$?

It is semiclassical that the extension class $\text{Ext}^1(\omega^{-1},\omega)\cong H^1(\mathcal{M}_{ell},\omega^2)$ over the modular stack $\mathcal{M}_{ell}/\mathbb{Z}$ is nonvanishing, being cyclic ...

2
votes

1
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138
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### Index and weight of Weierstrass $\sigma$-function as a Jacobi form, versus a statement in a note by Zagier

Let
$$\sigma_L(w; \tau):=\frac{w}{\exp\left(\sum_{k\ge 2} 2G_k(q)\frac{w^k}{k!}\right)}$$
be the version of the Weierstrass $\sigma$-function which is used to orient $\text{tmf}$; here $w=2\pi i z$, $...

6
votes

0
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159
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### Factorizing classical Eisenstein series

In the course of my research, I found some surprising (for me) factorizations
of Eisenstein series in levels $1$, $2$, $3$, and $4$. For instance, in level $1$
set with standard modular form notation
$...

2
votes

0
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79
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### The kernel $K(x,y)$ as an integral over Eisenstein series for $\operatorname{GL}_2$

Let $G = \operatorname{GL}_2$, $f \in C_c^{\infty}(G(\mathbb A)/Z(\mathbb A))$, and $V = L^2( G(\mathbb Q)Z(\mathbb A)\backslash G(\mathbb A))$ (trivial central character). Then the operator $R(f)$ ...

2
votes

0
answers

166
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### Eisenstein Series at CM points

Suppose that $L= \mathbb{Z}\tau + \mathbb{Z}$ is a lattice with CM. Consider the Eisenstein sum
$$ G_{2k}(L) = \sum_{(m,n)\neq (0,0)} \frac{1}{(m\tau+n)^{2k}}$$
where $k$ is a positive integer greater ...

5
votes

2
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396
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### Is there a similar formula like Ramanunjan's Eisenstein series identity for $\sum_{k=1}^{n-1}k^2 \sigma(k)\sigma(n-k)$?

This question is related to the last question about van der Pol's identity for the sum of divisors.
In Touchard (1953) it is mentioned that the sum of divisors $\sigma(n)$ satisfies the following ...

3
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147
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### Maass--Selberg for any Eisenstein series on higher rank

Does there exist a Maass--Selberg relation for any Langlands Eisensein series on $\mathrm{GL}(n)$? By any I mean an Eisenstein series which is induced from any standard parabolic with any discrete ...

1
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0
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176
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### Multidimensional series: an application of quantum field theory

While computing the quantum vacuum energy of a real scalar field defined on $\mathbb{R}\times \mathbb{T}^3$, I encountered the following sum:
$$ \sum_{n_1^2+n_2^2+n_3^2\geq 1}^{\infty} \frac{1}{(n_1^2+...

2
votes

1
answer

167
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### Question on the residual representation

Let $G=SO_n$ and fix a borel subgroup $P_0$ of $G$. Let $P=MN$ be a standard maximal parabolic subgroup $G$ and $\sigma$ a cuspidal representation of $M$
Consider the normalized parabolic induced ...

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0
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139
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### Intertwining operator is not an isomorphism?

Let $F$ be a number field and $G$ a symplectic group over $F$.
Let $P=MN$ is a maximal parabolic subgroup of $G$ and $W_M=N_G(M)/M$. Since $P$ is maximal, $W_M \simeq S_2$. Let $w$ be a non-trivial ...

1
vote

1
answer

133
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### Quantum ergodicity of Eisenstein series on arithmetic quotients of hyperbolic space

Let $E(z,1/2+it)$ be the Eisenstein series furnishing the continuous spectrum of the Laplace operator $\Delta$ on $X=PSL_2(\mathbb{Z})\setminus H^2$ and $dV(z)=y^{-2} \,dx \,dy$ be the volume element ...

6
votes

3
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254
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### An explicit formula for a cuspidal form of weight $2$ and arbitrarily large prime level

In Miyake's book on modular forms explicit formulas for the $q$-expansions of a basis of the space Eisenstein series of arbitrary level and weight were given. I guess similar formulas for a basis of ...

1
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0
answers

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### Decomposing functions on the fundamental domain of the torus into cusp forms, eisenstein series

I am trying to do some elementary calculations to understand the properties of the following spectral resolution on $H/SL(2,\mathbf{Z})$. (Half plane mod modular group; fundamental domain of the torus)...

3
votes

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214
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### 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 ...

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0
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276
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### What is definition of Cohen–Eisenstein series?

I only find Cohen–Eisenstein series of weight 3/2 (in the paper A note on the Fourier coefficients of a Cohen-Eisenstein series). I founded some general definition in "Modular Forms with Integral and ...

13
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### Hypergeometric representation of Eisenstein series

It is well known (Fricke ?) that $E_4^{1/4}$ and $E_6^{1/6}$ can be represented as Gauss hypergeometric functions of $1728/j$ and $1728/(1728-j)$ respectively.
The same result is true in levels $2$, $...

1
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0
answers

238
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### Why can there be holomorphic modular forms of negative half integral weight?

In Shimura's paper "ON THE HOLOMORPHY OF CERTAIN DIRICHLET SERIES", he constructed a family of Eisenstein series $E(z,s)$ by summing factor of automorphy. $E(z,s)$ is of negative half-integral weight, ...

0
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0
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### What is meant by "roots in $Lie(N)$" in root space decomposition of Lie algebras?

Let $G = GL_n$ and $T$ the invertible diagonal matrices and $N$ the upper triangular matrices with only $1$'s on the diagonal. Then the Lie algebra $\mathcal{G}$ has the roots space decomposition
$$
\...

3
votes

0
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57
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### What is meant by singular hyperplane of $c(w, \cdot)$? (global intertwining operator related to Eisenstein series)

Let $P_0$ be a minimal $\mathbb{Q}$-parabolic subgroup of $G$, a semisimple linear algebraic group over $\mathbb{Q}$. Then $P_0 = M_0 N_0$ where $M_0$ is a Levi subgroup of $P_0$. Let $E^G_{P_0}$ be ...

5
votes

1
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532
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### 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 ...

6
votes

1
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338
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### 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) = \...

5
votes

1
answer

308
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### An easier reference than "On the Functional Equations Satisfied by Eisenstein Series"?

I'd like to learn about Eisenstein series so I started reading "On the Functional Equations Satisfied by Eisenstein Series"by Langlands.
http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/...

7
votes

2
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418
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### 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

0
answers

214
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### 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 ...

10
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### 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
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1
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401
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### 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\...

6
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### 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.
...

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### 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
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### 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-...

6
votes

1
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538
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### 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 $\...

21
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1
answer

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### 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}:=\...

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### 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

1
answer

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### 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

0
answers

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### 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

1
answer

383
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### 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
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0
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### 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 ...

7
votes

1
answer

420
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### 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

1
answer

437
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### 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 + ...

37
votes

2
answers

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### 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 \...