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Questions tagged [automorphic-forms]

Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups.

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Why is a modular form an automorphic form?

According to Wikipedia a modular form is a holomorphic function $f:\mathbb H \to \mathbb C$ satisfying quasi-invariance under the action of $SL(2, \mathbb Z)$ on $\mathbb H$ and a growth condition. ...
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The Laplacian operator is invariant to $SL_2(\mathbb{R})$

I am reading Iwaniec's book on the spectral analysis of automorphic forms, where I bumped into the following statement in p.20 section 1.6. Given a function $f:\mathbb{H}\longrightarrow \mathbb{C}$, ...
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Classification of (finite dimensional) admissible representations of $F^{\times} = \mathrm{GL}(1, F)$.

Let $F$ be a non-archimedean local field. In Bump's textbook, there are two kinds of 2-dimensional such representations: $$ t\mapsto \begin{pmatrix} \xi(t) & \\ & \xi'(t)\end{pmatrix} $$ for ...
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Uniqueness of linear functional invariant under maximal torus action

This is an exercise 4.4.2 in Bump's automorphic forms textbook, which I can't solve. Let $T = T(F)$ be a maximal torus of $\mathrm{GL}(2, F)$, where $F$ is a nonarchimedean local field. ($T$ may ...
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Test vector for local zeta integral with ramified character

Suppose $\pi$ is an unramified principal series representation of ${\rm GL}_2(F)$, where $F$ is a non-archimedean local field with integers $\mathfrak{o}$. Let $W$ be a function in its Whittaker model....
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Application of Ngo's fundamental lemma

I'm studying Arthur-Selberg trace formula and trying to find various applications of it, from classical ones (Eichler-Selberg formular and the dimension of space of modular forms) to modern ones (...
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Asymptotic's for Fourier coefficients of $GL(3)$ Maass forms

Let $f$ be a $GL(3)$ Hecke-Maass cusp form and $A(m,n)$ denote its Fourier coefficients. (1) Are there any lower bounds known for $\sum_{p\leq x}|A(1,p)|^2$ or $\sum_{n\leq x}|A(1,n)|^2$ ? (we know ...
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Descent for admissible representations of algebraic groups over local fields

Let $G$ be a reductive group and $F$ be a local field which is a finite extension of $\mathbb{Q}_p$. Assume $\Pi$ is an irreducible smooth admissible representation of $G(F)$ over $\bar{\mathbb{Q}}_l$,...
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Whittaker model for $\mathrm{GL}(2, \mathbb{R})$

The complete list of irreducible admissible representations of $\mathrm{GL}(2, \mathbb{R})$ are known - principal series, discrete series, limit of discrete series, and finite dimensional ...
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Classification of $(\mathfrak{g},K)$-module of $\mathrm{GL}(2, \mathbb{R})$

This is a classification of $(\mathfrak{g},K)$-module of $\mathrm{GL}(2, \mathbb{R})$ in Bump: Here we assume that two complex numbers $s_{1}, s_{2}$ are given, and $\lambda = s(1-s), \mu = s_{1} + ...
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Steinberg representation for $\mathrm{GL}(2)$ is irreducible

Let $(\pi, V) = \mathcal{B}(\chi_{1}, \chi_{2})$ be a principal series representation of $\mathrm{GL}(2, K)$ ($K$ is a local field), where $\chi_{1}, \chi_{2}$ are characters of $K^{\times}$. If $(\...
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Supercuspidal representation of $\mathrm{GL}_{2}$ over finite field

I'm reading Bump's automorphic form, chapter 4.1, and this note written by Garrett. The later note said that there are $q(q-1)/2$ different supercuspidal representations of $\mathrm{GL}_{2}(\mathbb{F}...
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Definition of weight $k$ Laplacian

I just found that there are two different definitions of the weight $k$ Laplacian on the complex upper half plane. In Bump's book, he defines $\Delta_{k}$ as $$ \Delta_{k} = -y^{2} \left( \frac{\...
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Does Weil's converse theorem holds for weight 1?

I'm reading Iwaneic's "Topics in classical automorphic forms". Now, I'm reading the proof the theorem that for any Hecke character $\xi$ of a quadratic field $K/\mathbb{Q}$, there exists a $\mathrm{GL}...
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Whittaker model equation

This is from Cogdell and Piateski-Shapiro's paper Derivative and L-functions for $GL_n$. Let $\lambda$ be a non-trivial $\psi$-Whittaker functional. The Whittaker model is defined as$W_v(g)=\lambda(\...
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Functional equation for $GL(3)\times GL(2)\times GL(1)$ L-functions

For two Maass forms $$f(z)=\sum_{n\neq 0}a(n)\sqrt{2\pi y}K_{v_1-\frac{1}{2}}(2\pi|n|y)e^{2\pi inx}$$ and$$g(z)=\sum_{\gamma\in U_2(\mathbb{Z})\backslash SL(2,\mathbb{Z})} \,\,\,\,\,\sum_{m=1}^{\...
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Coefficient of Maass cusp forms are bounded

Let $$\phi(z)=\sum_{\gamma\in U_2(\mathbb{Z})\backslash SL(2,\mathbb{Z})} \,\,\,\,\,\sum_{m_1=1}^{\infty}\,\,\sum_{m_2\neq 0}a(m,n)W_{\text{Jacquet}}\left(\begin{pmatrix} |m_1m_2| & & \\ &...
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Is $ \pi\mapsto(s\mapsto L(s,\pi)) $ bijective?

Let $ \pi $ be an automorphic representation of $ \operatorname{GL_{n}}(\mathbb{A}_{\mathbb{Q}}) $ and $ L(s,\pi) $ the associated L-function. Is the map $ \pi\mapsto L(s,\pi) $ bijective ? In ...
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The Dual Maass form for $SL(3,\mathbb{Z})$

Let $\phi(z)$ be a Maass form of type $(v_1,v_2)\in \mathbb{C}^2$ for $SL(3,\mathbb{Z})$. Then the dual Maass form $$ \tilde{\phi}(z):= \phi(w.(z^{-1})^{\intercal}.w)\,,\,\,\,\,\,\,\,\,\,\,w=\begin{...
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self-intertwining operators of an automorphic representation

I stumbled a few days ago on the definition of an intertwining operator in https://www.encyclopediaofmath.org/index.php/Intertwining_operator Considering the case where $ \pi_{1}=\pi_{2}=\pi$, $ ...
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Bounds on Hecke eigenvalues

Let $\pi$ be an automorphic representation of a certain general linear group $GL(k)$. Write its L-function as $$L(s,\pi) = \sum_{n>0} \frac{\lambda(n)}{n^s}$$ for $Re(s) \gg 1$. What is the best ...
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About the Hasse-Weil zeta function of modular curves

It is an important philosophy in the Langlands program that (roughly) the zeta function of a Shimura variety can be written as a product of L-functions of automorphic representations. The first ...
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What does the following distribution look like?

I want to pick $x+\sqrt{-1}y$ at random with respect to the hyperbolic measure from the standard domain $\{z:|z|\geq1,|\mathcal R(z)|\leq\frac12\}$ for $SL_2(\mathbb Z)\backslash\mathbb H$. What does ...
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Points of Scholze's Anticanonical Tower

In his paper "On torsion of locally symmetric spaces", Schole introduces a perfectoid space, which is the anticanonical tower. He claims that points coming from $\text{Spa}(C,C^+)$, where $C$ is a ...
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on Selberg trace formula

The selberg trace formula has two forms: one is for a setting of a semi simple group $G$ and its cocompact subgroup $\Gamma$, and relates the geometric and spectral side of the canonical automorphic ...
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Square integrable Maass forms are cusp forms, local version near a cusp

Let $Y$ be a finite volume quotient of the upper half plane. I can show that a weight $0$ nonharmonic Maass form is cusp iff it is square integrable. One direction uses only local information near a ...
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Need explanation of a statement from Gelbert book of automorphic forms on Adele groups.

I was going through the book Automorphic forms on Adele groups by Stephen S. Gelbert and in the second page of first chapter I got a statement like all Fuchsian groups have a finite numbers of Γ-...
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Are automorphic forms eigenfunctions?

The Eisenstein series are eigenfunctions of the non-Euclidean Laplacian operator. Are automorphic forms in general by definition eigenfunctions of certain operators?
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Meromorphic functions on $X(1)$ are rational functions of $j$

I'm reading the proof given by Diamond and Shurman that the field of meromorphic functions on $X(1)$ is the set of rational functions of $j$ and it seems I'm missing something. Starting with $f$ ...
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Is there any known explicit value of dimension of space of Maass forms?

It is known that there exists a simple formula of the dimension of space of (holomorphic) modular forms on $\mathrm{SL}_{2}(\mathbb{Z})$ in terms of its weight. Also, we have similar but rather ...
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Non-vanishing of K-Bessel function

I don't know much about the spectral theory of $\text{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$ nor do I know much about Bessel functions, hence the following question. Suppose $f$ is Maass form of ...
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Sub representation generated by a Laplace eigenfunction

Let $G = \text{PGL}_2(\mathbb{R})$ and let $\Gamma = \text{PGL}_2(\mathbb{Z})$. Let $\varphi : \Gamma \backslash G \rightarrow \mathbb{C}$ be smooth and in $L^2(\Gamma\backslash G)$. Let $\mathfrak{g} ...
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Fuchsian Groups of the First Kind and Lattices

I am trying to compare various definitions and theorems I have seen recently concerning Fuchsian groups. Some of these seem to contradict each other, so I was hoping to get some clarification. First,...
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Evaluating $\displaystyle\int_{-\infty+iy}^{\infty+iy}(cv)^{-k}e\left(\frac{-m}{c^2v}-nv\right)dv$.

So the book I'm reading tells me to derive \begin{align*} \mathcal{J}_c(m,n)&=\displaystyle\int_{-\infty+iy}^{\infty+iy}(cv)^{-k}e\left(\frac{-m}{c^2v}-nv\right)dv\\ &=\displaystyle\frac{2\pi}...
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orbit representatives for the group of unipotent matrix acting on the set of skew-symmetric matrices

Suppose $F$ is a p-adic field, I was trying to compute orbit representatives for the group of n by n upper-triangular unipotent matrces $U_n(F)$ acting on the set of n by n skew-symmetric matrices. I ...
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Solution to algebraic equations over $\mathbb{C}$ and $\mathbb{C}[x]$

$t^n=a$, we get one solution to the equation: $$t=e^{\frac{1}{n}\int^a_1 \frac{1}{x}}$$, generalizing this result by replacing the exponential with an elliptic modular function and the integral with ...
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1answer
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Coset representatives of $\mathbb{A}/\mathbb{Q}$ and cusp forms

I'm reading Deitmar's "Automorphic form" and it said that a function $\varphi\in L^{2}(\mathrm{GL}_{2}(\mathbb{Q})\backslash \mathrm{GL}_{2}(\mathbb{A})^{1})$ is a cusp form if $$ \int_{N(\mathbb{Q})\...
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Basic question about automorphic representations

I'm reading Gelfand/Graev/Pyatetskii-Shapiro's book Representation Theory and Automorphic Functions. On page 20, the setting is: Let $G$ be a locally compact topological group, $\Gamma\le G$ a ...
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Transformations between the fundamental domains of Eisenstein series

I am self-learning elliptic functions and modular form but struggling with the very basics. I have programmed the Eisenstein series $$E_6 = \sum_{m,n \in \mathbb{Z}} \frac{1}{(m +n \tau)^6} $$ and ...
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How to write Poincaré Series for $\Gamma_0(4)$?

The space of cusp forms on $\mathbb{H}/\Gamma_0(4)$ is finite dimensional and spanned by the Poincare series for $m \in \mathbb{Z}$: $$ P_m \big(z; \Gamma_0(4) \big) = \sum_{\tau \in \Gamma_\infty \...
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Two definitions of the Jacquet functor: why are they the same?

Let $G$ be (the rational points of) a connected, reductive group over a local field $F$. Let $P$ be a parabolic subgroup of $G$ with unipotent radical $N$ and Levi subgroup $M$. The inclusion $M \...
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An irreducible representation of $G$ is a $G$-subspace of $\textrm{Ind}_B^G(\chi)$

Let $F$ be a local field, and let $G = \textrm{GL}_2(F)$. Let $B \subseteq G$ be the subgroup of upper triangular matrices, $T$ the diagonal matrices, and $N$ the upper triangular matrices with $1$s ...
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Non-vanishing of left- vs. right-averages over lattices in $SL(2,\mathbb{R})$

EDIT: I have now asked the same question on MO. Background. Let $G=SL(2,\mathbb{R})$, let $K=SO(2)$, and let $\Gamma$ be a lattice in $G$, e.g. $SL(2,\mathbb{Z})$. Let $\phi \in L^1(G)$ be $\...
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$H(G') \otimes_{H(G)} V$ and $f^{-1} \mathcal G \otimes_{f^{-1}\mathcal O_Y} \mathcal O_X$, the connection

1 . Let $f: (X,\mathcal O_X) \rightarrow (Y,\mathcal O_Y)$ be a morphism of ringed spaces, and let $\mathcal G$ be a sheaf of $\mathcal O_Y$-modules. Define the inverse image $f^{\ast} \mathcal G$ ...
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Isotypic component of class $\mathfrak d$

I am reading through Representations of $\mathfrak p$-adic groups (P. Cartier, Corvallis proceedings Vol I) and had a question about the following section. $G$ is a topological group of totally ...
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Directed system of irreducible modules

I have a question on a paper in the Corvallis proceedings on automorphic forms. Background: Let $G$ be a topological group of td type. This means that $G$ is Hausdorff, and every neighborhood of the ...
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Diophantine equation related to coset representatives of $\Gamma_0(N)\backslash SL_2(\mathbb Z)$

I am trying to verify the coset representatives of $\Gamma_0(N)\backslash SL_2(\mathbb Z)$ used in the proof of Proposition 1.43 in Introduction to the Arithmetic Theory of Automorphic Functions by ...
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1answer
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Congruence Relations between Modular Forms

Recently I read papers of Ahlgren and Boylan's paper: Ahlgren, S. and Boylan, M., 2003. Arithmetic properties of the partition function. Inventiones mathematicae, 153(3), pp.487-502. In their ...
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What is the spectral decomposition of $L^2(G(\mathbb{Q}) \backslash G(\mathbb{A}))$ for compact quotient?

I'm trying to work out explicitly the spectral decomposition of $L^2(G(\mathbb{Q}) \backslash G(\mathbb{A}))$ when $G$ is anisotropic -- it has no split tori defined over $\mathbb{Q}$. This should ...
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How does the definition of Eisenstein series relate to the general definition?

In a first course in modular forms, the Eisenstein series of weight 2k are often introduced: $E_{2k}(z)= \Sigma_{(c,d)=1} 1/2 (cz+d)^{-2k}$. There is a general definition of Eisenstein series ...