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

learn more… | top users | synonyms

-1
votes
0answers
37 views

problem in meaning of symbol in commutator subgroup

i was reading paper "OUTER AUTOMORPHISMS IN NILPOTENT p-GROUPS OF CLASS 2, H. LlEBECK" in page 2 there is a symbol i dont get. if G is generated by a basis $a_λ, λ∈Λ$ and z∈Z⋂Φ(G) for σ(z,μ) be ...
1
vote
0answers
22 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) = ...
1
vote
0answers
70 views

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) ...
1
vote
0answers
29 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 ...
0
votes
1answer
35 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 ...
0
votes
0answers
42 views

What's the meaning of $d^{\times } a$?

In the lecture notes, the last line of on page 51, what is the meaning of $d^{\times } a$ in the integral? Thank you very much.
4
votes
0answers
21 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 ...
0
votes
0answers
22 views

Arithmetic zeta functions

I understand that all motivic zeta functions are expected to be automorphic. How do arithmetic zeta funcions, this is, the zeta functions of an arithmetic scheme, fits in this picture? Are they ...
1
vote
1answer
40 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 ...
1
vote
0answers
38 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 ...
0
votes
2answers
74 views

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 ...
1
vote
1answer
47 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 ...
1
vote
0answers
109 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 ...
2
votes
1answer
66 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 ...
4
votes
1answer
92 views

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 ...
1
vote
0answers
39 views

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 ...
1
vote
0answers
44 views

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 ...
1
vote
0answers
33 views

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 ...
1
vote
1answer
31 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. ...
0
votes
1answer
36 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.
5
votes
1answer
121 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 ...
0
votes
1answer
37 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 ...
0
votes
1answer
50 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 ...
2
votes
1answer
96 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 ...
1
vote
2answers
168 views

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 ...
2
votes
1answer
100 views

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 ...
2
votes
0answers
22 views

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 + ...
3
votes
2answers
415 views

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 ...
0
votes
1answer
45 views

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 ...
2
votes
0answers
48 views

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 ...
1
vote
0answers
50 views

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 ...
1
vote
1answer
76 views

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 ...
4
votes
0answers
121 views

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 ...
2
votes
1answer
102 views

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 ...
2
votes
1answer
66 views

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 ...
-1
votes
1answer
59 views

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 ...
5
votes
2answers
164 views

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 ...
3
votes
2answers
94 views

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 ...
1
vote
2answers
83 views

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 ...
3
votes
1answer
134 views

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 $$ M = \begin{pmatrix}A & B\\ C & D \end{pmatrix} \in \operatorname{Sp}_n(\mathbb{Z})$$ ...
4
votes
2answers
203 views

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 $F$, ...
12
votes
1answer
1k views

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
votes
1answer
108 views

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 ...
4
votes
2answers
130 views

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 ...
2
votes
0answers
121 views

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 ...
7
votes
2answers
425 views

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} ...
6
votes
1answer
260 views

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 $$ ...
2
votes
1answer
80 views

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 ...
5
votes
1answer
216 views

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 ...
4
votes
1answer
84 views

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. ...