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

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Why is it so difficult to use geometric methods to construct weight one modular forms?

I am trying to understand why it is that geometric methods of constructing modular forms of weight one for $SL(2,\mathbb Z)$ fail. I have a relatively rudimentary understanding of it, but it would be ...
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Complex Galois Representaions

I'm trying to understand 1 and 2 dimensional complex representations, induced representations and associated Artin L-functions for a project. I'm finding it hard to find appropriate material to help ...
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An Efficient Route to Tate's Thesis

I want to learn Tate's thesis. My advisor suggested the Book "Algebraic Number Theory" By Lang. However, it seems to be a long read before I reach Tate's Thesis. I want to know what are other good ...
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The Fricke involution on arbitrary subgroups of $\operatorname{SL}_2(\textbf{Z})$

Let $\Gamma$ be a subgroup of $\operatorname{SL}_2(\textbf{Z})$. I want to understand how the Fricke involution $w_N : f \longrightarrow (\tau \longrightarrow N^{-k/2}\tau^{-k}f(-1/N\tau))$ acts on ...
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A sufficient condition for $\eta$-quotients to be modular forms

Let $f$ be the form : $$f(\tau)=\prod_{M\mid N}{\eta(M\tau)^{a_M}} \quad (\tau \in \mathcal{H})$$ Generally, we said that $f$ is an $\eta$-quotient when $(a_M)$ is a sequence of integers. One can ...
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Is there a natural interpretation of automorphic forms in terms of fractal geometry?

Disclaimer : This question is rather vague and thus might not be suitable for MathOverflow, so I prefer to ask it here. According to Wikipedia, an automorphic form is, roughly speaking, a ...
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On the partial zeta function

Let $F$ be a number field and $S$ be a finite set of places of $F$ including archimedian places. Let $\zeta^S(s)$ be the partial L-function, that is the meromorphic continuation of the product of ...
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Strong Approximation for adelic quaternionic groups

Let $H$ be a definite quaternion algebra defined over $\mathbb{Q}$, unramified at $p$, and ramified at $\infty$. Denote by $D$ the multiplicative group $H^\times$ divided by its center. In this ...
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phase portraits of automorphic functions (a la Indra's Pearls)

I would like to make movies of phase portraits of automorphic/kleinian functions with varying traces -- On page 375 of Indra's Pearls there is a phase portrait of an automorphic function/fonction ...
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What are the relations among Whittaker functions, Whittaker functionals, Whittaker models, and Whittaker vectors?

What are the relations among Whittaker functions, Whittaker functionals, Whittaker models, and Whittaker vectors? Are there some examples which describe the relations among Whittaker functions, ...
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30 views

How to show that a representation is supercuspidal?

Let $G$ be a reductive group. If we know a representation $\pi$ of $G$ explicitly, how could we determine that $\pi$ is supercuspidal or not? Are there some references about this? Thank you very much. ...
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Whittaker models and local L-functions.

If we have a Whittaker model, how to associate a local L-function to it? Are there some references? Thank you very much.
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How to prove that $\zeta*\zeta=\zeta$?

Let $F$ be a non-archimedean local field and $\mathcal{O}_F$ the ring of integers in $F$. Let $G_F=GL_2(F)$. Let $\pi_i$, $i=1,\ldots,n$,be non-equivalent finite dimensional irreducible ...
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1answer
36 views

Abstract Algebra: Automorphism and irreducible

I done how to prove this the question is A. Show that there is an automorphism of the ring of rational number $\mathbb{Q}(i)[x] $that has order of $4$. I suppose i can say that if $\varphi$ is an ...
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39 views

Abstract Algebra: Irreducible polynomials and automorphism

While I was studying and referring few examples I came up with few question. Hope you guys can help me solving my confusion. Irreducible polynomials problem 1 in this how to argue that why the ...
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1answer
69 views

Why a modular form is a highest weight vector of a discrete series summand of $L_2(SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R}))$?

It is said that a modular form is a highest weight vector of a discrete series summand of $L_2(SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R}))$. Why a modular form is a highest weight vector of a ...
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Invariance of measure on upper half plane

The upper half plane has the measure $|y|^{-2}dxdy$. Show that it is invariant under the action of $SL(2, \mathbb{R})$. I don't understand what any of this means. First, I don't understand what they ...
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1answer
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Equidistribution of lattice points on spheres in dimensions $d \ge 4$ (Pommerenke's theorem)

I am looking for either an English translation of: Pommerenke, C.: Über die Gleichverteilung von Gitterpunkten auf m-dimensionalen Ellipsoiden. Acta Arith. 5, 227-257 (1959) Or a reference in ...
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Is it true that $\theta_{1,1}^{4N} \in J_{2N,2N}(2N)$?

I need examples of Jacobi forms for full congruence subgroups $\Gamma(N) $ of $SL(2,Z)$. As a particular case, take the theta function $\theta_{1,1}(t,z) := \sum_{n\in\mathbb{Z}} exp(\pi it(n + ...
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Definition of discrete and continuous spectrum

I am reading about automorphic forms for $GL(2)$ and I am having trouble understand the definition of "spectrum." For instance, in Goldfeld and Hundley's book on automorphic representations they ...
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Is the p-adic Schwartz function uniform continuous?

In p-adic case, Schwart function is the function which has compact support and locally constant. But can we say its uniform continuity from this? I think it would not be true, but I am not certain ...
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Why will we never be able write down all automorphic representations “explicitly”?

In a recent article (p. 7), J. Arthur writes In fact, it is pretty clear that we will never be able to write down all automorphic representations explicitly. I have often heard similar remarks ...
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Why can't you apply the right regular representation the “wrong” way?

Let $G = GL(2,\textbf{R})^+$ be the $2 \times 2$ invertible matrices with positive determinant and given some discontinuous subgroup $\Gamma$, let $V = L^2(\Gamma \backslash G, \chi)$, the space of ...
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Definition of Automorphic form

I am looking at how one maps the space of cusp forms $S_k(N,\psi)$ into the space of automorphic forms on $GL_2$ over the Adeles. And I have a question about the definition of $K$-finiteness. I am ...
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References request: Ramanujan's tau function.

References request: Ramanujan's tau function. Let $\Delta(z)=q\prod_{n=1}^{\infty} (1-q^n)^{24}=\sum_{n=1}^{\infty} \tau(n)q^n$, $q=e^{2\pi i z}$. How can one show the following using representation ...
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Definition of Hyperbolic n-space

I am a little bit confused about the definition of hyperbolic $n$-space. How do we see $\mathbb{H}^n$ as a homogeneous space model? We can think, the Poincare upper half plane $\mathfrak{H}$ as ...
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Help with an exercise in Shimura's book

Exercise 3.26(B) in Shimura's book Introduction to the Arithmetic Theory of Automorphic functions says the following. Let $\Gamma=SL_n(\mathbb{Z})$ and $G=GL_n(\mathbb{Q})$. Let $R^{(n)}_p$ denote ...
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A Question of Congruence Subgroup

Hi I have a question about congruence subgroups $Γ(N)$ of $SL(2,\mathbb{Z})$. How to compute the index $[SL(2,\mathbb{Z}): Γ(N)]$? BTW, can anyone tell me double cosets decomposition ...
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How important are automorphic representations among admissible ones?

I'm currently studying automorphic representations on Bump's book "Automorphic forms and representations" and on Gelbart's "Automorphic forms on Adele groups". And I have some problems in ...
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What is the correct definition of the cuspidal subspace of $L^2$?

I have a few (semi-)related questions regarding certain Hilbert space representations of locally compact groups that come up in the theory of automorphic forms. Let $G$ be a unimodular locally ...
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Routine question about derivatives of automorphic forms being L^2

I consider Automorphic forms on $G = SL_2 (\mathbb{R})$, which are $\Gamma$-invariant, $K$-finite, $Z(g)$ finite, and of moderate growth. If I have such an automorphic form, which happens to be in ...
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writing $M : \Gamma_{n,0} \backslash \Gamma_n$

Let $\Gamma_n = \operatorname{Sp}_n(\Bbb Z)$ and $\Gamma_{n,0} \subset \Gamma_n$ be a subgroup. We write $$ M = \begin{pmatrix}A & B\\ C & D \end{pmatrix} \in \operatorname{Sp}_n(\mathbb{Z})$$ ...
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Definition of Automorphic Representation

I would like to think of an automorphic representation as a representation weakly contained in $L^2(G_F\backslash G_A)$ where $G_A$ is the reductive group of rational points in the adeles over $F$, ...
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Undergraduate roadmap for Langlands program and its geometric counterpart

What are the topics which an undergraduate with knowledge of algebra, galois theory and analysis learn to understand Langlands program and its goemetric counterpart? I would also like to know what are ...
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Condition on infinite component for a cuspidal automorphic representation to be attached to a Hilbert newform

Let $F$ be a totally real field with real primes $\tau_1,\ldots,\tau_n$ and $\pi$ a cuspidal automorphic representation of $\mathrm{GL}_2(\mathbf{A}_F)$. My rough understanding is that in order for ...
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Must automorphic forms be square-integrable modulo the center?

I've recently needed to learn the basics of automorphic forms and automorphic representations. I've seen two apparently different definitions of automorphic forms, and I'm wondering which is more ...
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Base-Change for $GL_1$

I try to understand the base-change in the theory of automorphic forms for the simplest case: $GL_1$. Let $L/K$ be an abelian extension of number fields of degree $n$ and let $\Gamma$ be the set of ...
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Residue of Rankin-Selberg L-function for non-trivial nebentypus

Let $f\in S_k(\Gamma_0(N),\chi)$ be a normalized holomorphic newform (i.e. weight $k$, level $N$, nebentypus $\chi$) and write its Fourier expansion as $$ f(z)=\sum_{n\ge 1} ...
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Questions about the L-function for Eisenstein Series

$E_a(z,s)$ denotes the Eisenstein series expanded at the cusp $a$. For each cusp $a=\frac{u}{w}$ of $\Gamma_0(N)$, we define the Eisenstein series $$ ...
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Notation confusion in “On Some Results of Atkin and Lehner” article

I'm currently reading an article on Automorphic Forms, and I'm a bit confused about some of the notation used. The article is "On Some Results of Atkin and Lehner" by William Casselman, published in ...
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Extending Galois automorphism to group automorphism

Let $F \subset K$ be a field extension of degree $n$, then $F^n \cong K$ as $F$-vectorspaces. Now $K^\times$ acts on $F^n$, by multiplication on $K$, and so $K^\times$ embeds into $GL_n(F)$, and every ...
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1answer
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Can characters occur in automorphic representation

Let $\pi$ be an irreducible cuspidal automorphic representation of $GL(2)$ over a global field with factorisation $\pi = \otimes \pi_v$. Then at most finitely many $\pi_v$ are not spherical. ...
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141 views

Right Regular Representation and Automorphic Forms

Let $R(g)$ denote the right regular representation of $SL_{2}(\mathbb{R})$ in $L^{2}(\Gamma\backslash SL_{2}(\mathbb{R}))$ with $\Gamma$ a congruence subgroup. Why is decomposing $R(g)$ equivalent to ...
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Haar Measure and Automorphic Forms

Let $G = SL_{2}(\mathbb{R})$ and $\Gamma = \Gamma_{0}(N)$. Every element $g =\begin{pmatrix}a & b\\ c& d\end{pmatrix}\in G$ can be written as $$\begin{pmatrix} y^{1/2} & xy^{-1/2} \\ 0 ...
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Matrix Groups and the Upper Half Plane

Why is it that the upper half plane $\mathbb{H}$ can be thought of as equivalent to $SL_{2}(\mathbb{R})/SO(2)$?
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bounds for Fourier coefficients of non-holomorphic automorphic forms of weight 2

Are there any results about the bounds for Fourier coefficients of non-holomorphic automorphic forms of weight 2? More precisely, let $k$ be a positive integer and $m=4/k$. Write \begin{equation*} ...
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Do general automorphic factors arise in some canonical way?

Disclaimer: none of what I'm about to say I understand particularly well; corrections and clarifications are not only welcome but will be accepted with great gratitude. The wikipedia article on ...
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The Langlands program for beginners

Assuming that a person has taken standard undergraduate math courses (algebra, analysis, point-set topology), what other things he must know before he can understand the Langlands program and its ...
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1answer
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Where can I find rigorous statements about the spectral decomposition of reductive groups?

Given a global field $F$ and a reductive group $G$, where can I find the spectral decomposition of $$ L^2( Z(\mathbb{A}) G(F) \backslash G( \mathbb{A})).$$ I will need the result in this generality, ...
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Eisenstein spectrumfor $GL(n)$

Fix a global field $F$. Does every automorphic representation of $GL(n)$ appear as an arbitrary twists in the continuous spectrum of $GL(m)$, $m>n$? What happens for the automorphic ...