Questions tagged [automorphic-forms]
An automorphic form is a well-behaved function from a topological group $G$ to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup $\Gamma \subset G$ of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups.
51 questions from the last 365 days
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Explicit Jacquet-Langlands correspondence for real reductive groups
Let $G$ be a connected reductive group over $\mathbb R$. Let $G'$ over $\mathbb R$ be an inner form of $G$ with ${}^LG={}^LG'$. By local Langlands correspondence over $\mathbb R$, if a $L$-packet of $...
- 6,622
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Galois representations attached to discrete automorphic representations
Let $F$ be a totally real field. Let $G$ be a (split) connected reductive group over $F$. Let $\pi$ be an irreducible automorphic representation of $G$.
Recall in the work of Buzzard and Gee "The ...
- 6,622
3
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Voronoi formula on $\mathrm{GL}_4$ in the level aspect with ramification
$\DeclareMathOperator\GL{GL}$Let $f$ be an automorphic form on $\GL_3$ for $\Gamma_0(p)$ with $p$ being a prime (see Bump or Goldfeld's books for definitions). Recall that, in this paper-"The ...
- 41
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124
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Relation of automorphic representation and its constant term
Let $\pi$ be an irreducible non-cuspidal automorphic representation of a classical group $G$ defined over a number field $F$. Let $P$ be a maximal parabolic subgroup of $G$. Let $\pi'$ be the ...
- 1,019
3
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0
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136
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Field of definition of automorphic Galois representation
Let $\pi$ be a regular, cuspidal, algebraic automorphic representation of $GL_n(\mathbb{A}_K)$ for a totally real field $K$. Then for every embedding $\lambda$ of $E=\mathbb{Q}(\pi)$ in $\overline{\...
- 190
2
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164
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Bounds of modular functions on the Ford circles
Assume a holomorphic function from a product of two upper half planes $Z: \mathbb{H}_+\times \mathbb{H}_+\rightarrow \mathbb{C}$ with an expansion of the form
$$
Z(\tau,\tau') = \sum_{(h,h')\in S} a_{...
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283
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Why are there so few irreducible admissible representations of $\text{GL}(n,\mathbb{R})$ (up to infinitesimal equivalence)?
Studying Langlands's classification of irreducible admissible representations, I have been rather stunned by the following:
Theorem
Up to infinitesimal equivalence, all irreducible admissible ...
- 511
1
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0
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125
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When is a vector bundle on a Shimura variety an automorphic vector bundle?
Let $(G, X)$ be a Shimura datum, let $K \subset G(\mathbb{A}_f)$ be an open compact subgroup, and denote by $\text{Sh}_K(G,X)$ the Shimura variety whose complex points are given by $G(\mathbb{Q})\...
- 159
1
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0
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77
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Distinguishedness of discrete series induction
Let $D_k$ denote a discrete series representation of $\text{GL}_2(\mathbb{R})$ of weight $k\geq 2$. Consider the parabolically induced representation $D_k \times D_k$, which is a representation of $\...
- 145
10
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2
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404
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Impact of the squarefreeness of the level for modular forms
I often notice papers and results that assume that the level is squarefree in the setting of modular forms, but have a hard time figuring how where this impacts or simplifies the argument. Is there in ...
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5
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Reference Request: Completeness of the space of all Whittaker models(a lemma in JPSS1981)
$\DeclareMathOperator\GL{GL}$There is a lemma in the proofs of local converse theorem stated as
Suppose $F$ is a non-archimedean local field, $\psi$ is a non-trivial addtive character on $F$. $N_n$ ...
- 53
8
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1
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356
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Average bounds on Rankin-Selberg coefficients for modular forms
Let $f$ be a cuspidal Hecke newform of weight $k$ and level $N$, and denote by $a_f(n)$ its $n$-th Fourier coefficient. The newform $f$ is normalized so that $a_f(1) = 1$. As a consequence of Rankin-...
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3
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1
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On the local factor of Rankin-Selberg L-functions
I have a puzzle on the local factors of Rankin-Selberg $L$-functions. Consider two newforms on $\text{GL}_2$. Let $f$ be a newform of square-free level $N$, and $g$ a newform of trivial level. As ...
- 353
2
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0
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99
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Factorization of global Waldspurger's integrals and connection to central L-values
Let $\pi$ be the irreducible cuspidal automorphic representation of $\mathrm{GL}_2$. Let $E/F$ be a quadratic extension with given embedding $E^{\times} \to \mathrm{GL}_2(F)$.
For $f_1 \in \pi$, $f_2 \...
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4
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Automorphic forms on $\mathrm{GL}_{2}$, $\mathrm{SL}_{2}$, and $\mathrm{Mp}_{2}$ — classical counterparts
I asked exact same question on MSE but haven't got answer yet, so asking here, too. I may erase the original one once I got an answer here.
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I'm confusing about automorphic representations of $\...
- 2,215
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Connection between a special integral transform and averages of L-functions
Let $\Gamma = \operatorname{SL}_2(\mathbb{Z})$ and $\mathcal{H}$ be the upper half-plane. For $A>1$, define the truncated Eisenstein series $E_A(z,s)$ as $$E_A(z,s) = \begin{cases} E(z,s), & \...
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A question on vanishing of Godement–Jacquet-like zeta integral
$\DeclareMathOperator\GL{GL}$The classical Godement–Jacquet zeta integral is of this form:
$f$ is a matrix coefficient of a cuspidal automorphic representation of $\GL_n(\mathbb{A}_\mathbb{Q})$, and $\...
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Which L-functions are not known to be automorphic for $\mathrm{GL}_n$?
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\sym{sym}$I would like to compile a list of primitive L-functions which satisfy
the usual axioms (Dirichlet series with an Euler product,
and a ...
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4
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Reference Request: on an explicit formula for class-1 Whittaker functions on split reductive groups over p-adic fields
The 1978 preprint by S.Kato 'On an explicit formula for class-1 Whittaker functions on split reductive groups over p-adic fields' is cited by papers involving unramified computations of local ...
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Casimir eigenvalues of p-adic automorphic representations
In the context of p-adic local Langlands correspondence:
Is it possible to define Casimir eigenvalues for p-adic automorphic representations? If a local representation arises from a global Galois ...
- 2,907
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Why the residues of Siegel Eisenstein series becomes constant at certain point?
$\DeclareMathOperator\GL{GL} \DeclareMathOperator\SO{SO} \DeclareMathOperator\Ind{Ind}\DeclareMathOperator\B{B}$Let $F$ be a number field and $V$ be a $(2n+1)$-dimensional quadratic space over $F$. ...
- 1,019
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1
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642
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Generalizations of Hamburger's Theorem
(Despite the name, the theorem in question is not a joke nor is it a statement about a delicious food).
An old theorem of Hans Hamburger from 1921, as stated in Marvin Knopp's paper "On Dirichlet ...
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2
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107
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A nonzero cuspidal automorphic representation has a nonzero Fourier-coefficients?
$\DeclareMathOperator\GL{GL} \DeclareMathOperator\Sp{Sp} \DeclareMathOperator\Ind{Ind}\DeclareMathOperator\B{B}$Let $F$ be a number field and $G_n$ the symplectic group over a $2n$-dimensional ...
- 1,019
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98
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Morphisms on L^2(G) induced by morphisms of LCA groups
I am looking for a good reference to understand the space $L^2(G)$ for a locally compact abelian (LCA) group $G$.
In particular, I would like to understand when $L^2(-)$ is functorial, so that if $\...
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Is "self-dual" equivalent to "dihedral" for Maass forms on $\mathrm{GL}(2)$?
Suppose $f$ is a Maass cusp form on $\Gamma_0(D)$. The associated symmetric square $L$-function $L(\mathrm{sym}^2 f, s)$ has a pole at $s = 1$ if and only if $f = \overline{f}$ (if $f$ is self-dual).
...
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9
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680
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Roadmap to Carayol-Deligne-Langlands
Having begun self-study of Fermat's Last Theorem a few years ago, I have only recently begun to understand and appreciate the theorem of Carayol-Deligne-Langlands on local-global compatibility for ...
- 283
1
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136
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Notion of "Hodge bundle" for abelian type Shimura varieties
For a Siegel type Shimura datum $(\text{GSp}_{2g}, \mathcal{H}^{\pm})$ and level $K$, we construct the Shimura variety $S_{g,K} := \text{Sh}_K(\text{GSp}_{2g},\mathcal{H}^{\pm})$. We have a universal ...
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Counting local representations for $\mathrm{GL}_2$
$\DeclareMathOperator\GL{GL}$Some context.
In number theory, it is natural to study distribution questions for the family of elliptic curves over $\mathbb{Q}$ (or any fixed number field for that ...
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3
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The lower bound for the automorphic $L$-function $L(s,\pi)$ at the edge of the critical strip $\Re s=1$
Let $\pi$ be any automorphic Maass form on $\text{GL}_m$ of level $N$, say. Assume that the associated $L$-function $L(s,\pi)$ satisfies some good conditions; for example, it satisfies the functional ...
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5
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A question on hybrid subconvexity for individual L-functions
Sorry to disturb. I have a question need some explanations from the experts on the MO-website.
As usual, we let $L(f,s)$ be the corresponding $L$-function associated to the newform $f$ on $SL_2(\...
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shifted convolution in arithmetic progressions
Let $r(n)$ be the number of ways of writing $n$ as the sum of two integer squares. Asymptotics for the shifted convolution problem $$ \sum_{n\in \mathbb N\cap[1,x]}r(n) r(n+1)$$ are quite classical; a ...
- 3,062
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73
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Number of rational points of a connected reductive group in a compact subset
Let $G$ be a connected reductive $\mathbb{Q}$-group. Let $\mathbb{A}$ denote the ring of adèles of $\mathbb{Q}$. Let $B \subset G(\mathbb{A})$ be a compact, let $x \in G(\mathbb{A})$ and consider the ...
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Rankin-Selberg convolutions with mixed integral and half-integral weights
Let $f(z)$ denote a weight $0$ Hecke-Maass form of level $N$ and let $\theta(z)$ denote the Jacobi theta function. Then $y^{1/4} f(z) \overline{\theta(z)}$ transforms as an automorphic form of weight $...
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5
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Reference Request: Test vectors for local Rankin-Selberg L-factors in ramified cases
Let $F$ be a global number field, i.e. a finite extension of the field of rational numbers. Let $\sigma$, $\pi$ be automorphic representations of $\mathrm{GL}_n(F)$ and $\mathrm{GL}_{n+1}(F)$ ...
- 639
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79
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Logarithm map for groups defined over adelic ring
I've been reading the book Eisenstein series and automorphic representations and I am struggling to understand the definition of a logarithm map $H:G(\mathbb{A})\rightarrow \mathfrak{h}(\mathbb{R})$ (...
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Genericity of local representation with a non-generic local A-parameter
Let $\pi$ be an irreducible smooth representation of a classical $p$-adic group. Suppose that $\pi$ has a local L-parameter associated to some non-generic local A-parameter $\psi$. Then I am wondering ...
- 1,019
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91
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Question on the genericity 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 $T_n$ ...
- 1,019
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What is the "weight" of an automorphic form for $\mathrm{PGL}_2$?
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PGL{PGL}$I'm trying to understand what the notion of "weight" is for automorphic forms over $\GL_2(F)$ where $F$ is some number field, in ...
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3
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1
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98
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Reference Request: Possible generalizations of the stability of $\gamma$-factors
$\DeclareMathOperator\GL{GL}$
Let $F$ be a nonarchimedean local field. Suppose $\pi, \sigma$ are irreducible admissible representations of $\GL_{n}(F)$ and $\GL_{m}(F)$ respectively, with $n \geq m$. ...
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Question on generic A-packet
Let $G$ be a classical group and $\phi$ be a generic $A$-parameter of $G$.
I am wondering whether each automorphic representations in the $A$-packet associated to $\phi$ are locally generic at almost ...
- 1,019
2
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76
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Simple question on the genericity of induced representation
$\DeclareMathOperator\GL{GL} \DeclareMathOperator\Sp{Sp} \DeclareMathOperator\Ind{Ind}$
Let $F$ be a $p$-adic field and $\Sp(2n)$ symplectic group over 2n dimensional symplectic space over $F$.
Let $B=...
- 1,019
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99
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Question on the unramified representation
$\DeclareMathOperator\GL{GL}$Let $F$ be a $p$-adic field and $\chi$ be an unramified character of $\GL_1(F)$.
Consider an induced representation $\pi$ of $\GL_2(F)$ induced from the character $\chi|\...
- 1,019
8
votes
1
answer
567
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Symmetric power lift of modular forms
Let $f_1$ and $f_2$ be two cuspforms of weights $k_1$ and $k_2$ and nebentypus $\epsilon_1$ and $\epsilon_2$ respectively such that $f_1 \neq f_2 \otimes \chi$ for some Dirichlet character $\chi$ of ...
- 424
1
vote
1
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Iwahori action on the $p$-ordinary line of a principal series representation
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\diag{diag}\DeclareMathOperator\Ind{Ind}\newcommand\Iw{\mathrm{Iw}}\DeclareMathOperator\ord{ord}$Let $F$ be a $p$-adic local field, i.e. a finite ...
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6
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1
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536
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How to see that Eisenstein series are eigenfunctions of the laplacian?
Let $\Gamma$ be a discrete subgroup of $PSL_2(\mathbb{R})$ of finite type. Let $c_1,\ldots,c_h\in\mathbb{R}\cup\{\infty\}$ be a set of representatives of the $\Gamma$-equivalence classes of cusps. For ...
- 7,527
3
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0
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122
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Is there any notion of Poincaré series for Hermitian modular forms?
I have been studying modular forms and their generalisations for a year or so. It is a very interesting fact that the space of cusp forms $S_k$ is generated by the Poincaré series of exponential type (...
3
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0
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117
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Reference Request: Local decomposition of GGP period integrals of cuspidal forms on unitary groups
Setup: Let $E/F$ be a CM-extension of global number fields. Let $(V,\phi)$ be an Hermitian space of dimension $n$ over $E$. Let $(V^{\flat}, \phi^{\flat})$ be a subspace of $V$ of dimension $n-1$ on ...
- 639
2
votes
0
answers
141
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Analyticity of unramifed part of Rankin-Selberg $L$-functions on $\Re(s)=1$
I have only a little knowledge about automorphic representations and $L$-functions. Now I am reading the textbook of Goldfeld and Hundley on automorphic representations, and also planning to read the ...
- 663
3
votes
1
answer
214
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$p$th Fourier coefficients of newforms for ramified primes $p$
This question is about some basic(classical) results on Atkin-Lehner-Li theory of newforms. Let $f$ be a (normalized) newform of level $N$ and character $\epsilon$. Denote the $n$th Fourier ...
- 663
2
votes
1
answer
264
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'$\times$' or '$\otimes$' when writing $L$-functions?
Recently, I came across the Langlands correspondence theorem, there is the following line:
$$L(s,\pi(\sigma) \times \pi(\tau)) = L(s,\sigma \otimes \tau), $$
where $\sigma$ and $\tau$ are ...
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