Questions tagged [eisenstein-series]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
3 votes
1 answer
71 views

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 &...
  • 1,550
2 votes
0 answers
67 views

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 $SL(2,{\mathbb Z})$...
  • 21
1 vote
1 answer
92 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^...
9 votes
1 answer
212 views

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 ...
  • 5,758
3 votes
1 answer
100 views

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(...
  • 5,758
1 vote
0 answers
106 views

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 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) := \{\...
  • 83
3 votes
0 answers
62 views

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}_{...
  • 5,758
3 votes
1 answer
88 views

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 ...
  • 5,758
4 votes
0 answers
57 views

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 = ...
  • 5,758
9 votes
1 answer
183 views

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 ...
  • 5,758
10 votes
0 answers
125 views

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 ...
  • 1,550
2 votes
1 answer
138 views

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 $w=2\pi i z$, $...
  • 1,550
6 votes
0 answers
159 views

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 $...
  • 9,361
2 votes
0 answers
79 views

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 $R(f)$ ...
  • 5,758
2 votes
0 answers
166 views

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 ...
  • 821
5 votes
2 answers
396 views

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 votes
0 answers
147 views

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 vote
0 answers
176 views

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 views

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 ...
  • 1,525
1 vote
0 answers
139 views

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 ...
  • 1,525
1 vote
1 answer
133 views

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 ...
  • 141
6 votes
3 answers
254 views

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 ...
  • 61
1 vote
0 answers
80 views

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)...
  • 242
3 votes
0 answers
214 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 ...
  • 880
1 vote
0 answers
276 views

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 votes
0 answers
173 views

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 $2$, $...
  • 9,361
1 vote
0 answers
238 views

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, ...
  • 41
0 votes
0 answers
74 views

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 answers
57 views

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 answer
532 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 ...
6 votes
1 answer
338 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) = \...
5 votes
1 answer
308 views

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/...
  • 2,511
7 votes
2 answers
418 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) }{...
  • 71
2 votes
0 answers
214 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 ...
  • 5,758
10 votes
0 answers
264 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
1 answer
401 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\...
  • 543
6 votes
0 answers
357 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. ...
  • 123
11 votes
1 answer
522 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 ...
user avatar
3 votes
0 answers
55 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-...
user avatar
6 votes
1 answer
538 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 $\...
  • 598
21 votes
1 answer
1k 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}:=\...
  • 598
5 votes
0 answers
135 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
1 answer
240 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
0 answers
86 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
1 answer
383 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 ...
  • 123
11 votes
0 answers
211 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 ...
  • 238
7 votes
1 answer
420 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 ...
  • 7,062
9 votes
1 answer
437 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 + ...
  • 427
37 votes
2 answers
3k 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 \...