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

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Regarding the weight of the given modular form

I am following this course. A question regarding the modular form of weight two that is constructed in lecture 33 emerged. Let me briefly tell you what's going on there. Let $\mathbb{H}$ denote the ...
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Automorphic forms and admissible representations

In the book Automorphic Forms representations and L-functions(Corvallis), Borel and Jacquet wrote a document with the title "Automorphic Forms and automorphic Representations". Proposition 4.5 reads ...
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Does the supposed to exist functor considered in Langlands program bear a peculiar name?

I'm trying to figure out what a very rough sketch of the Langlands program could be. From what I (think I) understand, objects called reductive algebraic groups together with related so-called ...
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28 views

Motivation of the definition of principal series.

I am reading the book representation theory of semisimple groups. On page 33, the principal series representation $\mathcal{P}^{k,iv}$ is defined as follows. What are motivations of the above ...
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Why unitary principal series is unitary?

I am reading the book representation theory of semisimple groups. On page 33, I tried to verify that $\mathcal{P}^{k,iv}$ is unitary. We need to verify that $$ \left|\left| ...
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Arithmetically equivalent number fields and Langlands Program

Two (number) fields are arithmetically equivalent if their Dedekind zeta functions are the same. It is known that any two arithmetically equivalent fields are not necessarily isomorphic; Prasad ...
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How to show that $v \mapsto \pi(f)v$ is differentiable?

Let $G$ be a compact group. Let $(\pi, V)$ be a representation of $G$ and $f$ a smooth function on $G$. Define \begin{align} \pi(f)v = \int_G f(x)\pi(x) v dx. \end{align} We have \begin{align} & ...
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1answer
48 views

Example of function smooth on upper half plane that is not holomorphic

I understand that Maass forms are the eigenfunctions of the Laplacian that are smooth on $\mathbb{H}$ but not holomorphic. But I can't write down an explicit formula for such a function. How might I ...
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Proving any automorphism of the group $\Bbb R^\times$ [closed]

Let$\Bbb R^\times$ denote the non-zero real numbers. Prove that any automorphism of the group $\Bbb R^\times$ under multiplication maps positive to positive numbers and negative numbers to negative ...
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Automorphic forms on $GL(3)$

It is well known and mentioned/proved in most introductory texts that the only automorphic forms on lower general groups are (constant functions and other details aside): $GL(1)$: Hecke characters ...
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41 views

Holomorphicity and modularity of Jacobi forms

I am reading this paper Suerconformal algebras and mock theta functions by T. Eguchi and K. Hikami. In section 3.2, the authors define a function $$J(z;w;\tau) = ...
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Multiplicity one theorem for GL(n) and SL(n) [closed]

I am reading Dorian Goldfeld's book Automorphic forms and L functions for the groups GL(n,R) ...
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32 views

Why $\int_{U} d(m n m^{-1}) = \int_{mUm^{-1}} dn = \omega^2_E(t_1) \int_U dn$?

I am reading the lecture notes. On page 5, formula (1.24) is $$ \int_{U} d(m n m^{-1}) = \int_{mUm^{-1}} dn = \omega^2_E(t_1) \int_U dn, $$ where $dn$ is the Haar measure on $N$, $U \subset N$ is ...
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51 views

Decomposition of standard Borel subgroup.

I am reading the lecture notes. On page 3, formulas (1.7), (1.8), (1.9), let $P$ be the standard Borel subgroup, we have $P=MN$. Why $M, N$ must be (1.8), (1.9)? I know that $P$ should be a upper ...
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28 views

Where was it originally proved that $K$-finite automorphic forms are of uniform moderate growth?

I'm providing an overview of some classical results about automorphic forms in one section of a paper I'm working on, and I've realized I don't have a good reference for the following. Let $k$ denote ...
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1answer
53 views

About algebraic groups defined over Q

I'm studying automorphic forms and there's something I don't understand, when we talk about a connected reductive algebraic group $G$ defined over $\mathbb{Q}$, connected means connected as an ...
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46 views

$\overline\pi$ vs. $\check\pi$

Let $\pi$ be the automorphic representation of ${\rm GL}_2$ or $B^\times$ ($B$ an indefinite quaternion algebra over $\Bbb Q$) of central character $\varepsilon$ attached to some holomorphic newform ...
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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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53 views

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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127 views

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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1answer
87 views

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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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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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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42 views

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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132 views

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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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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1answer
138 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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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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1answer
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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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1answer
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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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1answer
71 views

Computing the index $[\text{SL}(2,\mathbb{Z}) : \Gamma(N)]$

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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1answer
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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 ...
4
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1answer
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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 ...