Questions tagged [eisenstein-series]

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79 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
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1answer
101 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 ...
6
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3answers
205 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 ...
1
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0answers
48 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)...
3
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0answers
147 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 ...
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0answers
206 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 ...
12
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0answers
125 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 $2$, $...
1
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0answers
134 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, ...
0
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0answers
67 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
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0answers
48 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
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1answer
465 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
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1answer
186 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
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1answer
189 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/...
6
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1answer
266 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
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0answers
186 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 ...
10
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0answers
214 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
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1answer
321 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
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0answers
257 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
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1answer
429 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
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0answers
43 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-...
6
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1answer
483 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
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1answer
985 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
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0answers
112 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
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1answer
208 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
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0answers
84 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
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1answer
292 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
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0answers
196 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
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1answer
306 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
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1answer
381 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 + ...
30
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2answers
2k 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
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1answer
170 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
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1answer
363 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
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0answers
96 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
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1answer
149 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
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0answers
406 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
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1answer
488 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
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1answer
130 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
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2answers
536 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
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1answer
772 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
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1answer
347 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
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1answer
359 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
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1answer
879 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
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1answer
358 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(...
13
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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
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1answer
857 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
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1answer
504 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
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1answer
142 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
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0answers
246 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
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0answers
135 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
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0answers
381 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. ...