Questions tagged [automorphic-forms]
Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups.
250
questions
0
votes
2
answers
64
views
Definition unramified local representation
We can define unramified representation of $GL(2,\mathbb Q_p$) as a representation $(\pi , V) $ such that $V^K\neq \{0\}$ for $K$ the maximal compact subgroup i.e. $GL(2,\mathbb Z_p)$.
However, I do ...
- 2,865
2
votes
0
answers
11
views
Computing the local $\gamma$-factor for the local zeta integral (Goldfeld-Hundley)
Let $\nu \leq \infty$ and $\mathbb{Q}_\nu$ denote a local field. Suppose $s \in \mathbb{C}$ with $\Re(s) > 0$. Let $\Phi: \mathbb{Q}_\nu \to \mathbb{C}$ be a Schwartz-Bruhat function. Let $\omega : ...
- 643
6
votes
0
answers
65
views
Relationship between scalar and vector valued automorphic forms
Let $F$ be a number field with signature $(r_1,r_2)$. There are two definitions of automorphic forms on $GL_{2,F}$ that I am aware of. Let $K=O_2(\mathbb R)^{r_1}\times U_2(\mathbb C)^{r_2}$ be a ...
- 303
1
vote
1
answer
31
views
Does the order of vanishing at infinity for an automorphic form depend on the chosen period?
In Diamond and Shurman's A First Course in Modular Forms, the authors define meromorphy at infinity as follows:
Let $f$ be a meromorphic function on the upper half plane that is weakly modular with ...
- 1,521
3
votes
0
answers
38
views
Selberg's Trace Formula for Hecke Eigenvalues
I am looking for a reference (if one exists) for an application of Selberg's trace formula after the Hecke operators have been applied. Perhaps to give some context and notation to this, so my ...
- 1,921
0
votes
0
answers
16
views
Surjectivity of Weil representation
Let $F$ be a local field (e.g. $\mathbb{R}$, $\mathbb{C}$, or finite extensions of $\mathbb{Q}_p$).
Let $X$ (resp. $Y$) be a non-degenerate quadratic (resp. symplectic) space over $F$.
Then $\mathrm{...
- 14.7k
1
vote
0
answers
19
views
Definition of Bessel/Fourier-Jacobi/Whittaker models
I've seen the notion of the models in the title a lot in the context of automorphic forms and representations, but I wonder if there's any nice reference for the definition of them for general ...
- 14.7k
1
vote
0
answers
32
views
Reference for classical Ramanujan conjecture for number fields
The classical Ramanujan conjecture is given in terms of bounds for the Hecke eigenvalues of normalized eigenforms. The modern formulation of Ramanujan conjecture is given in terms of Satake parameters ...
- 797
1
vote
0
answers
44
views
Fricke involution’s effect on character
I’m using the definition $W_Nf(\tau)=i^kN^{-k/2}\tau^{-k}f(-1/N\tau)$. Now suppose $f\in M_k(\Gamma_1(N),\chi)$, show $W_Nf\in M_k(\Gamma_1(N),\chi^{-1})$.
I know this is pure calculation but I’m ...
- 149
2
votes
0
answers
102
views
Classification of automorphy factors for $\mathrm{SL}_2(\mathbb{R})$ on the upper half-plane
By an automorphy factor (or a factor of automorphy) for $\mathrm{SL}_2(\mathbb{R})$ on the upper half-plane $\mathbb{H}$, I mean a continuous map
$$j \colon \mathrm{SL}_2(\mathbb{R}) \times \mathbb{H} ...
- 861
0
votes
0
answers
95
views
Proof of Eichler-Shimura isomorphism
For a congruence subgroup $\Gamma \subseteq \mathrm{SL}_2(\mathbb{Z})$ we have the Eichler-Shimura isomorphism
$$
M_k(\Gamma) \oplus \overline{S_k(\Gamma)} \cong H^1(\Gamma,V_k)
$$
with $V_k$ a ...
anon
1
vote
0
answers
66
views
Automorphic representations which are trivial at the archimedean place
Consider the Hilbert space $V = L^2(Z(\mathbb{A})\mathrm{GL}_2(\mathbb{Q})\backslash \mathrm{GL}_2(\mathbb{A}))$. This is a unitary representation of $\mathrm{GL}_2(\mathbb{A})$, acting by right ...
- 11
2
votes
0
answers
69
views
Is the natural homomorphism $SL_2(\mathbb{Z})\to SL_2(\mathbb{Z}/N\mathbb{Z})$ onto? [duplicate]
EDIT: Let $N\geq 2$ be a natural number. Consider the natural group homomorphism
$$SL_2(\mathbb{Z})\to SL_2(\mathbb{Z}/N\mathbb{Z})$$
given by reduction modulo $N$.
Is it onto? If not is anything ...
- 243
1
vote
1
answer
32
views
Subspace of newforms one-dimensional with CM $\implies$ unique newform a Poincare series.
Let $k \geq 2$. Say $S_{k}^{\text{new}}(\Gamma_{0}(N), \chi)$ is one-dimensional and spanned by a newform with CM, and $S_{k}^{\text{old}}(\Gamma_{0}(N), \chi)$ has positive dimension. Must it be true ...
- 1,719
1
vote
0
answers
22
views
Reduced row echelon basis of weakly holomorphic modular forms have algebraic coefficients?
Let $S^{\infty}_{k}(\Gamma_{0}(N))$ be the space of weakly holomorphic modular functions for $\Gamma_{0}(N)$ whose only possible poles lie at the cusp $\infty$ and vanish at all other cusps. There is ...
- 1,719
0
votes
0
answers
36
views
What it means when automorphic function has " nebentypus" $\psi$ [duplicate]
I am reading a research paper and I am not able to understand what could be the definition of " nebentypus" here : " We study automorphic functions $f$ on $\Gamma= \Gamma_0(N)$ of ...
- 57
4
votes
1
answer
90
views
$N$-fixed vectors in smooth representations are also fixed by ${\rm SL}_2(\mathbf{Q}_p)$.
Let $N = \left(\begin{matrix} 1 & * \\ 0 & 1 \end{matrix}\right)$ be the upper triangular unipotent subgroup in ${\rm GL}_2(\mathbf{Q}_p)$ and $K_n = 1 + p^n M_2(\mathbf{Z}_p)$ the usual ...
- 43
0
votes
1
answer
63
views
Normalisation of $L$-function for classical modular forms and automorphic representations
I found that the normalisation of $L$-functions of classical modular forms and corresponding automorphic representations is somewhat confusing for me. Recall that if $f\in S_k(\Gamma_0(N))$ is a ...
- 1,323
1
vote
0
answers
79
views
How to define a measure on a quotient space
I have been trying to understand the following fact, consider $\mathbb{H}$ the upper half plane of the complex numbers. And let $\Gamma_0 = SL_2(\mathbb{Z})$ act on $\mathbb{H}$ we know that there is ...
2
votes
0
answers
66
views
Contragredient of an automorphic representation
I found that I didn't quite understand how to think about the contragredient of an automorphic representation. I have read this post on Mathoverflow, which is helpful:
https://mathoverflow.net/...
- 1,323
1
vote
1
answer
44
views
Basis for modular form over full modular group
It is well known that a basis of the modular forms of weight $k$ over $\text{SL}_2(\mathbb{Z})$ is $\{E_{4}^iE_{6}^j: 4i + 6j = k\}$. Moreover, let
$$d=\dim\mathcal M_k(\text{SL}_2(\mathbb{Z})),$$
...
- 746
3
votes
0
answers
26
views
$(\mathfrak{g}, K)$-module for covering group of $\mathrm{GL}_2(\mathbb{R})$
Harich-Chandra introduced the notion of $(\mathfrak{g}, K)$-module to study representations of a real Lie group $G$ via its Lie algebra $\mathfrak{g}$ and a maximal compact subgroup $K$.
One also has ...
- 14.7k
2
votes
0
answers
59
views
Adelic theta function over function fields
I saw the following claim on Godement-Jacquet's classical book "Zeta functions of simple algebras": (on page 153) Let $F$ be a global function field, $\Phi$ a Schwartz function on $\mathbb{A}...
- 1,323
7
votes
1
answer
166
views
Why does a cusp form correspond to holomorphic differential form?
Consider a cusp form $f \in S_2(\Gamma)$ for a certain congruence subgroup $\Gamma$. I would like to understand why $f(z)dz$ is a holomorphic 1-form on the curve $X_\Gamma$. Its invariance is exactly ...
- 255
5
votes
0
answers
58
views
Irreducibility of smooth representations and Hecke modules
Let $(\pi, V)$ a (smooth) admissible representation of a locally profinite topological group $G$ and fix $K \le G$ an open compact subgroup. These notes claim that if $V^K$ is a nonzero irreducible ...
- 1,454
3
votes
0
answers
32
views
Do $(\mathfrak{g},K)$-modules only deal with real Lie groups?
In Bump's Automorphic Forms and Representations, p. 200, he gives the definition of a $(\mathfrak{g},K)$-module for $\mathfrak{g}=\mathfrak{gl}_n\mathbb{R}$ and $K=O(n)$ being the maximal compact ...
- 1,257
2
votes
2
answers
67
views
Rankin-Selberg bound on coefficients
For an automorphic L-functions, often there is a bound refereed to by Ramanujan on average, where the coefficients satisfy
$$\sum_{n<x} \lambda(n) \ll x^{1+\varepsilon}$$
Why is this bound true and ...
- 1,067
1
vote
0
answers
28
views
How to compute that the volume of $Z(\mathbb{A})\backslash \frak{S}$ for a Siegel set is finite?
I found it quite hard to find a complete reference for the (adelic) reduction theory of algebraic groups used in automorphic representations. I want to show that for $G=GL(n)$ over a number field $F$, ...
- 1,323
1
vote
1
answer
53
views
Reference request: multiplicative group of a central simple algebra, their reductivity and parabolic subgroups
During my study of the theory of automorphic forms and $L$-functions, I never found any literature dealing with the following:
Suppose $D$ is a central division algebra over a local or global field $F$...
- 1,323
3
votes
1
answer
53
views
A measure theoretic problem related to induced representations
I met such a rather concrete measure theoretic problem while dealing with induced representations of $p$-adic groups.
So let $G$ be a unimodular locally compact group, with $P$ its closed subgroup (...
- 1,323
6
votes
1
answer
98
views
A convergence lemma for adelic zeta function in automorphic forms
I'm reading Godement-Jacquet's classic Zeta functions of simple algebras (1972, Springer). On page 153, the first line: " We also $\textbf{take for granted}$ the following lemma (numbered 11.3)......
- 1,323
0
votes
1
answer
83
views
Intuitive understanding of Tamagawa measure and its relationship between local measures?
Weil's book Basic Number Theory, (1973, second edition) pp. 113 mentioned the Tamagawa measure on $k_\mathbb{A}$, where $k$ is a global field and $k_\mathbb{A}$ its ring of adeles (an old fashioned ...
- 1,323
3
votes
0
answers
82
views
Writing the automorphic form $\phi_f$ of a newform $f$ as a pure tensor
Let $\mathbb{A}$ be the adele ring of $\mathbb{Q}$.
Let $f\in S_k(\Gamma_0(N),\chi)$ (possibly a newform) and $\phi_f:\text{GL}_2(\mathbb{Q})\backslash\text{GL}_2(\mathbb{A})\to \mathbb{C}$ be its ...
- 1,257
1
vote
0
answers
78
views
Smooth functionals in the contragredient representation of a locally profinite group
I'm reading Bump's book Automorphic Forms and Representations and getting stuck in Section 4.2 about contragredient representation of smooth representations.
Setup: Let $G$ be a locally profinite ...
- 2,035
1
vote
1
answer
65
views
Whittaker at Archimedean Test vector
Let $\pi$ be a generic irreducible Casselman Wallach representation of $GL_n(\mathbb{R})$. Let $\phi$ be a test vector for the archimedean local L function $L(s,\pi)$. Is there an explicit expression ...
- 1,155
4
votes
1
answer
139
views
Integrating an additive character over a local field
Let $F$ be a non-Archimedean local field, $\psi$ a non-trivial additive character of $F$. Let $\mathfrak{o}$ be the ring of integers of $F$, and $\mathfrak{p}$ be the maximal ideal of $F$. Endow $F$ ...
- 1,257
3
votes
0
answers
152
views
Application of Selberg Pre-Trace formula
I am trying to solve the following problem:
Prove the following estimate:
$$\sum_{\vert t_i\vert<T}\vert u_j(z)\vert^2\ll T^2$$
by choosing appropriate $h$ in the Selberg's (pre)-trace formula.
I ...
- 1,921
4
votes
1
answer
86
views
Why are Rankin-Selberg convolutions different when $n=m$?
Let $\pi$ an automorphic cuspidal representation on $GL_n(\mathbb{A})$ (and similarly $\pi'$ on $GL_m$). For $m=n-1$, it is standard to introduce the Rankin-Selberg L-function as (the gcd of the ...
- 1,177
2
votes
1
answer
184
views
The notions of $K$-finite vectors in admissible representations
Let $G$ be a real Lie group and $K$ be a maximal compact subgroup. Let $(\pi, V)$ be a Hilbert space continuous representation of $G$. It can be restricted to $K$ as a representation of $K$. For each ...
- 2,035
1
vote
1
answer
92
views
About holomorphic functions on the upper half plane with respect to SO(2).
Let $\mathbb{H}$ be the upper half plane. For $g=\left(\begin{array}{cc}a&b\\c&d\end{array}\right) \in SL_2(\mathbb{R})$ and $z \in \mathbb{H}$, set $J(g,z)=cz+d$. Let $SO(2)$ be the special ...
- 11
2
votes
0
answers
64
views
*Why* the converse theorem (conjecture) should be true?
Currently, the following $\mathrm{GL}(n)$ converse theorem due to Cogdell and Piatetski-Shapiro is known: For an admissible irreducible representation $\pi$ of $\mathrm{GL}(n, \mathbb{A})$, $\pi$ is ...
- 14.7k
4
votes
1
answer
127
views
Relations between different zeta functions for a simple algebra
I'm trying to understand the classical works of Eichler, Shimura and many others (especially Shimizu and Tamagawa's annals papers) on the "classical" (I'm a newcomer and I'm not sure whether ...
- 1,323
3
votes
1
answer
177
views
Explicit definition of the cusps of a congruence subgroup of the symplectic group
I'll begin by defining the notion of a cusp of a congruence subgroup of $\textrm{SL}_2(\mathbb{Z})$:
$\textrm{SL}_2(\mathbb{Z})$ has a natural action on the compactified upper half complex plane $\...
- 2,877
1
vote
0
answers
97
views
Background behind Eichler's discovery of the relationship of a modular form with an elliptic curve
In Fraenkel's Love and Math (and Richard Taylor's Modular Arithmetic IAS Post https://www.ias.edu/ideas/2012/taylor-modular-arithmetic), specifically in Chapter 8 Magic Numbers, page 88., Fraenkel ...
- 316
3
votes
1
answer
104
views
Why does the "automorphic factor" exist?
In the theory of automorphic forms, functions we consider are not totally invariant (why? is it because otherwise there would be too few of them, like none except constants?), but "twisted" ...
- 109
0
votes
0
answers
72
views
Reference request: Katz modular forms modulo $p$ at cusps vs classical modular forms modulo $p$ at cusps
I, like the author of this post, am severely lacking the background to make the connection between
reducing modular forms' $q$-expansions modulo $p$ at various cusps, and
$q$-expansions of modular ...
- 1,719
2
votes
1
answer
134
views
Compactness of automorphic quotient - quaternion group and $\mathrm{GL}_{2}$
Let $\mathbb{A} = \mathbb{A}_{\mathbb{Q}}$ be an adele ring over $\mathbb{Q}$ and $G_{1} = \mathrm{GL}_{2}$, $G_{2} = \mathrm{Res}_{D/\mathbb{Q}}\mathrm{GL}_{1}$ be two groups over $\mathbb{Q}$, where ...
- 14.7k
0
votes
0
answers
54
views
Classification of Representation of $GL(2,\mathbb{R}^+)$
$GL^+(2,\mathbb{R})$ is the notation for positive non-zero determinant square matrices of order two.
I am reading Bump's book on Automorphic Forms and Representations. One of the major part of second ...
- 59
1
vote
0
answers
37
views
F-split $SO(2)$ is isomorphic to $GL(1)$?
Let $F$ be a number field and $\chi$ is a character of $\mathbb{A}^{\times}/F^{\times}$.
In some paper, it is written that we can consider $\chi$ as a representation of $F$-split $SO(2)$.
I am ...
- 357
5
votes
1
answer
226
views
Explicit Jacquet-Langlands correspondence
Jacquet-Langlands correspondence gives a 1-to-1 correspondence between automorphic forms on $\mathrm{GL}_{2}(\mathbb{Q})$ and automorphic forms on $\mathrm{GL}_{1}(D)$, where $D$ is a division algebra ...
- 14.7k