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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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Discreteness of Langland's parameter and its centralizer

Let $G$ be a connected reductive group over a $p$-adic field $F$. Let ${^LG}=\widehat{G}\rtimes \Gamma$ ($\Gamma=\Gamma_F$ the absolute Galois group of $F$) be the L-group of $G$, where $\widehat{G}$ ...
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If $f$ is $K$-finite, then $Xf$ is also $K$-finite

This problem comes form bump, Automorphic forms and representations, p,300. It's about the ''K-finiteness'' conditions in the definition of the automorphic forms of $GL(2,A)$, where A is the adele ...
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If $χ_p$ is ramified then $χ(p) = 0$

Consider suppose that $χ_{idelic}$ is the idelic lift of the Dirichlet character $χ$. In my text, it says that if the local character $χ_p$ is ramified that is there exists some u $\in Z_p^{x}$ such ...
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On the right regular representation $\rho:GL(n,A)\rightarrow\mathrm{End}_{\mathbf{C}}(L^2(GL(2,F)\backslash GL(2,A),\omega))$

Let $F$ be an algebraic number field, $A$ its adele ring. Let $\omega$ be a unitary character, that is, a continuous group homomorphism $\omega:A^\times/F^\times\rightarrow C^\times$. Let $L^2(GL(2,F)\...
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The space of cuspidal elements $L_{0}^2(GL(2,F)\backslash GL(2,A),\omega)$ is a closed subspace of $L^2(GL(2,F)\backslash GL(2,A),\omega)$.

Let $F$ be an algebraic number field, $A$ its adele ring. Let $\omega$ be a unitary character, that is, a continuous group homomorphism $\omega:A^\times/F^\times\rightarrow C^\times$. Let $L^2(GL(2,F)\...
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On the definition of an admissible representation of a $p$-adic group

Let $G$ be a reductive over a $p$-adic field $F$ and $(\pi,V)$ a (complex) smooth representation of $G(F)$. I know that the notion of admissible is defined as follows: for any compact open subgroup $...
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Functional equation of the Hecke L function in ideal term and "ideal number" term (Neukirch Chapter VII)

$\def\A{\mathbb{A}} \def\B{\mathbb{B}} \def\C{\mathbb{C}} \newcommand{\Cx}{\mathbb{C}^{\times}} \def\F{\mathbb{F}} \def\G{\mathbb{G}} \def\H{\mathbb{H}} \def\K{\mathbb{K}} \def\M{\mathbb{M}} \def\N{\...
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The idempotented algebra

Let $k$ be a field, An idempotented algebra over a field k is a k-algebra H together with a collection E of idempotents of H satisfying the conditions: For all $e_1, e_2 ∈ E$ there exists $e_0 ∈ E$ ...
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Complete proof for the shape of quasicharacters of $\mathbb{R}^{\times}, \mathbb{C}^{\times}$

Quasicharacters (:=continuous group homomorphism to $\mathbb{C}^{\times}$) of $\mathbb{R}^{\times}, \mathbb{C}^{\times}$ seems to be known to be following forms (the following is quoted from [Raghuram,...
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Holomorphic non vanishing modular forms without regularity at infinity

Let $\mathcal{O}(\mathcal{H})^\times$ be the multiplicative group of holomorphic functions on the Poincaré half-plane $\mathcal{H}$ that do not vanish there. Let $j(g,z)=(cz+d)$ and $gz=(az+b)/(cz+d)$ ...
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Jacquet module of a parabolic induction

Let $G=Sp(2)$ be a symplectic group over a $p$-adic field and $P$ be the Siegel parabolic subgroup of $G$. Let $\text{Ind}_P^G$ be the normalized induction functor from smoothe representations of $P$ ...
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sum of rapidly decreasing functions over $S$-integers minus origin.

Let $F$ be a number field and $S$ a finite set of places including all infinity places. Let $f\in \mathcal{A}(F_\infty)$ be a rapidly decreasing function (in the usual sense), and $K\subseteq F_{S\...
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On the computation of local adjoint $L$-function of unramified representation

Let $F$ be a p-adic local field and $W$ be a 2n-dimensional symplectic space over $F$. Let $G_n$ be the isometry group of $W$ and $B_n$ be the Borel subgroup of $G_n$. Then the maximal torus $T_n$ of $...
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(Homogeneous spaces) Moduli Space of Lattices

I'm looking at the "moduli space of $n$-dimensional lattices", which should be the double quotient $$ \mathcal{M} = \text{GL}_n(\mathbb{Z}) \backslash \text{GL}_n(\mathbb{R}) / (\mathbb{R}^\...
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Construct an explicit outer automorphism for unitary groups (MVW involution)

Let $E/F$ be a quadratic field extension with Galois group $\{1,c\}$, where $c$ is the nontrivial automorphism of $E$. Consider $V$ to be a $E$-vector space of dimension $n$. We equip it with a $\...
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Proof that principal congruence subgroups $\Gamma(N)$ are torsion free.

I have read the fact that the principal congruence subgroups $\Gamma(N)$ of $\mathrm{GL}_n({\mathbb{Z}})$ are torsion free for $N \geq 3$ several times, but only saw proofs for very specific ...
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Why $(\mathfrak g_{\mathbb C},K)$-modules?

Let $G$ be a reductive group over $\mathbb Q$, $\mathfrak g=Lie(G)$ be its Lie algebra, $\mathfrak g\otimes_{\mathbb R} \mathbb C$ be its complexification. A definition of automorphic representation ...
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Question on the unique irreducible quotient of a standard module

Let $F$ be a local field of characteristic zero and $\chi$ be a character of $GL_1(F)$. Let $\pi$ be the normalized parabolic induction from the character $\chi |\cdot|^{\frac{n-1}{2}} \times \chi |\...
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Reference request: functional equation of real analytic Eisenstein series of weight $k$

I want to do some computations which require the functional equations of the following real analytic Eisenstein series of weight $k$: $$E_k(z, s)=\sum_{(c, d) \in \mathbf{Z}^2 \backslash(0,0)} \frac{y^...
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Daniel Bump's Automorphic Forms and Representation Exercise 4.7.2 (Page 520)

Exercise 4.7.2 of Daniel Bump's Automorphic Forms and Representations (Page 520) (a) Let $F$ be a local field, either Archimedean or non-Archimedean. By a finite function, we mean a function $\phi$ ...
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Twists of a modular form

Let $f$ be a cusp form and $\pi_f$ be the corresponding cuspidal representation of ${\mathrm{GL}}_2(\mathbb{A}_\mathbb{Q})$. If $\chi$ is a Dirichlet character modulo $M$, then we know that $f \otimes ...
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Why is $GSp(4)$ an interesting group?

In various settings (geometry, automorphic forms) the symplectic group does appear. In my mind, it may be realized formally as the group of matrices $$GSp(4) = \left\{ g \in GL(4) \ : \ g^T J g = \...
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Certain Automorphic Forms

Assuming that $f(z)$ is a non-zero weight-$k$ automorphic form (weakly holomorphic modular form), is it possible for $f$ to satisfy $$f|_kS=-f,\quad f|_kT=f,$$ where $f|_kS=z^{-k}f(-1/z)$ and $f|_kT=f(...
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Twists and conjugates of automorphic representations

I am trying to understand the definition of RAECSDC (regular, algebraic, essentially conjugate self-dual, cuspidal), which appears in many papers on automorphy lifting. Let $\pi$ be an automorphic ...
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An automorphic function with no poles is constant.

Daniel Bump calls $f$ an automorphic function if it satisfies the formula $$f\left(\frac{az+b}{cz+d}\right)=f(z)$$ where $\begin{pmatrix} a&b\\ c&d \end{pmatrix}\in\mathrm{SL}(2,\mathbb{Z})$ ...
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Compactness of $\mathbb{A}^1/\mathbb{Q}^\times$

I am working through the proof of Theorem 5.3.3 in the following text Deitmar's Automorphic Forms. The subgroup $\mathbb{Q}^\times$ is discrete in $\mathbb{A}^\times$ and the quotient $\mathbb{A}^1/\...
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Iwasawa Integral formula and modular function

Question: I am working through the proof of the following theorem 3.1.19 (Page 86) from Deitmar's Automorphic Forms, (Iwasawa Integral formula) Let $G = SL_2(\mathbb{R})$ and $A,N,K$ be subgroups of $...
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Show that $f(\rho)=0$ for any modular form of weight $k$ if $\rho=e^{\frac{2\pi i}{3}}$ and $3\nmid k$.

In Exercise 1.3.3 from Automorphic Forms and Representations, Daniel Bump gives the hint as follows: Observe that $\gamma(\rho)=\rho$ where $\gamma= \begin{pmatrix} 1&1\\ -1& \\ \end{pmatrix}...
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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 ...
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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 : ...
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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 ...
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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 ...
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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 ...
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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{...
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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 ...
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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 ...
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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 ...
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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} ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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$N$-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 ...
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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 ...
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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 ...
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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/...
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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 $$d=\dim\mathcal M_k(\text{SL}_2(\mathbb{Z})),$$ ...
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$(\mathfrak{g}, K)$-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 ...
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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}...
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