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.

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Intertwining operator is not an isomorphism?

Let $F$ be a number field and $G$ a symplectic group over $F$. Let $P=MN$ is a maximal parabolic subgroup of $G$ and $W_M=N_G(M)/M$. Since $P$ is maximal, $W_M \simeq S_2$. Let $w$ be a non-trivial ...
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A question related to newform and irreducible cuspidal representation of $\operatorname{GL}_n$

I was reading adelization of classical automorphic forms and learnt that each cusp form corresponds to an automorphic representation of $\operatorname{GL}_n(\mathbb{A}_\mathbb{Q})$. I understood the ...
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Residual and continuous spectra of $L^2( G(k) \backslash G(\mathbb A) ; \omega)$, and cuspidal automorphic data

Let $G$ be a connected, reductive group over a number field $k$. Let $\mathbb A$ be the ring of adeles of $k$, $\omega$ be unitary character of $Z_G(\mathbb A)/Z_G(k)$, and $V = L^2(G(k) \backslash G(...
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Equivalence between Ramanujan and Selberg conjectures

At first the Ramanujan conjecture for automorphic forms and the Selberg conjecture appear to be understood as totally independent. However, they are now known to be tighyly connected once viewed in ...
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A weakly holomorphic modular form is a harmonic maass form

It is known that for $\Gamma_0(N)$, a weakly holomorhpic modular form is a harmonic maass form. Here is the definitions. A weakly modular form $f$ for $\Gamma_0(N)$ is a meromorphic function on the ...
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Meaning of extended principal part of weakly holomorhpic modular forms

In p.312 of 'Rhoades, Robert C., Linear relations among Poincaré series via harmonic weak Maass forms. Ramanujan J. 29 (2012), no. 1-3, 311–320', the author defines the extended principal part at ...
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Left translation of automorphic form satisfies $K$-finiteness?

Does a left translation of an automorphic form satisfy left $K$-finiteness? Let $F$ be a number field and $G$ is an algebraic group. Let $\phi$ is an automorphic form on $G$. Let $K$ be a maximal ...
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Restriction of product of automorphic forms

Let $W \subset V$ be quadratic spaces over a number field $F$. Let $G_n=SO(V)$ and $G_m=SO(W)$ and we consider $G_m$ as a subgoup of $G_n$ via a diagonal embedding. Let $f$ be an automorphic form of ...
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Global Arthur packet consist of only globally generic representations?

I would like to ask very stupid two questions to experts. I am wondering whether every globally generic automorphic representation of unitary groups are contained some global Arthur packet associated ...
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Semidirect product of metaplectic group and Heisenberg group

I know that Symplectic group has an action on Heisenberg group. I am wondering how to extend this to non-trivial two fold metaplectic covering? Thanks in advance!
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How can I see the relation between shtukas and the Langlands conjecture?

The following bullet points represent the very peak of my understanding of the resolution of the Langlands program for function fields. Disclaimer: I don't know what I'm writing about. Drinfeld ...
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Bounds on Fourier coefficients for $GL(3)$

I am referring for instance to this question about coefficients of automorphic forms on $GL(3)$. I know that the Ramanujan on average bound is known and gives $$\sum_{n^2 m < x} |\lambda(n,m)|^2 \...
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Under Ramanujan conjecture, is primitivity equivalent to cuspidality and irreducibility?

Lemma 4.2 in M. Ram Murty, Selberg conjectures and Artin L-functions(1994), states that under Ramanujan conjecture, an irreducible cuspidal automorphic representation of $\operatorname{GL}_{n}(\mathbb{...
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Is an automorphic form of $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ determined by its L-function?

To an automorphic representation $\pi$ of $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ one can associate its L-function $s\mapsto L_{\pi}(s)$. Is the map $\pi\mapsto L_{\pi}$ bijective? Edit March ...
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Degree of automorphic forms, SL(3,Z), and the elliptic Gamma function

In this article, the authors interpret a certain special function, the elliptic Gamma function, defined as $$ \Gamma(z,\tau,\sigma)=\prod_{j,k=0}^\infty\frac{1-e^{2\pi i((j+1)\tau+(k+1)\sigma-z)}}{1-...
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Iwasawa decomposition on unitary group of anisotropic kernel

Let $E/F$ be a quadratic extension of number fields. If $V$ is a hermitian space over $E$, let $V=X+V_0+Y$ be its Witt decomposition, where $X,Y$ are maximal totally isotropic subspaces and $V_0$ is ...
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Metaplectic group $Mp(2n)(\mathbb{A}_F)$ splits over $Sp(2n)(F)$?

My question is the title. In some literature, authors seem to use this without assumption. Is it ture in general?
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Do we know absolute bounds for the norm of Satake parameters?

If we consider the set of all unramified Satake parameters $S$ of all automorphic representations of $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ as $n$ varies, do we know absolute (that is, ...
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Paramodular forms with level and Iwahori subgroups?

Given an integer $N>0$, not necessarily prime, we have the paramodular group $K(N) \subset \text{Sp}_{4}(\mathbb{Q})$, which consists of matrices of the form $$\begin{bmatrix} * & *N & * &...
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Intuition about how Voronoi formulas change lengths of sums

In reading the literature one encounters countless examples of Voronoi formulas, i.e., formulas that take a sum over Fourier coefficients, twisted by some character, and controlled by some suitable ...
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Subquotient of principal series

Let $F$ be a local field of characteristic 0. I am wondering whether an unramified principal series representation of $\operatorname{GL}_n(F)$ can have 1-dimensional quotient when $n>1$. In some ...
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Distribution of signs of automorphic forms

Let's say we have an automorphic form $f$ on $GL(2)$ that is self-dual. In particular, the associated L-function $L(s,f)$ satisfies a functional equation with sign $\varepsilon_F = \pm 1$. Is it ...
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Relations between spectral parameters of automorphic representations

Let $\pi$ be an automorphic representation (say, trivial central character) of $GL(2)$. Let $\alpha(p)$ and $\beta(p)$ denote its spectral parameters at the place $p$, that is to say the associated ...
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For what automorphic representations is Ramanujan-Petersson known?

I had in mind that Ramanujan-Petersson conjecture was essentially unknown in the case of number fields. I however recently heard that If an automorphic representation on $GL(2)$ is ramified at a ...
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Jacquet module of irreducible principal series

Let $F$ be a local field of characteristic zero and $G=\operatorname{GL}_n(F)$. Let $B=UT$ be a Borel subgroup of $G$ and $\chi=(\chi_1,\cdots,\chi_n)$ is an unramified character of $B$. Consider ...
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Jacquet module of unramified principal series representaion with respect to parabolic subgroup of $GL_n(F)$

Let $F$ be a local field of characteristic zero and $G=\operatorname{GL}_n(F)$. Let $B=UT$ be a Borel subgroup of $G$ and $\chi=(\chi_1,\cdots,\chi_n)$ is an unramified character of $B$. Consider ...
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Irreducibility of parabolic induction on unitary group

Let $F$ be a local field of characteristic 0. I know that $\pi=\text{Ind}_{B_k(F)}^{GL_k(F)}(\chi_1 \boxtimes \cdots \chi_k)$ for some unramified characters $\chi_i$'s is irrducible if there is no $\...
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Central character of some conjugate self dual representation

I want to ask some question on conjugate self dual representation. Let $E/F$ be a quadratic extension of number fields and $c:GL_n(\mathbb{A}_E) \to GL_n(\mathbb{A}_E)$ be a automorphism induced by ...
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Analogous theorem for Hilbert modular forms

I have studied modular forms and saw a correspondence like a newform correspond to a automorphic representation of $\mathrm{GL}_n(\mathbb{A_Q})$. Does any similar result holds for Hilbert modular ...
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Are the L-functions of a normalized newform and the corresponding cuspidal representation equal?

Let $f \in S_k(\Gamma_0(N))$ be a normalized newform with Fourier expansion $$f(z) = \sum\limits_{n=1}^{\infty} a_n e^{2\pi i z n}$$ and $a_1 = 1$. Then $f$ is an eigenfunction of all Hecke ...
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Rankin-Selberg convolution and product of degrees as of Christmas 2019

Almost 5 years ago (time flies), I asked in Rankin-Selberg convolution and product of degrees whether the Rankin-Selberg convolution of two automorphic representations of respectively $\operatorname{...
Sylvain JULIEN's user avatar
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Conductor of Principal series representation

Let $\mathbb{F}$ be a local field and let $\pi$ be a principal series representation of $GL_2(\mathbb{F})$ that is $\pi=Ind_B^{GL_2}(\chi_1\otimes\chi_2)$ for two characters $\chi_1$ and $\chi_2$ of ...
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conductor formula

Let $\pi_p$ be an irreducible representation of $GL_2(\mathbb{Q}_p)$. Assume $\pi_p$ is ramified,hence it will have a positive conductor. Consider $sym^3(\pi_p)$ which is a representation of $GL_4(\...
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Is weakly holomorphic modular form finitely generated as module of modular function?

Let's use $M_k^{!}(\Gamma_{0}(N))$ to denote weakly holomorphic modular form with weight k, level $\Gamma_0(N)$(in particular, k might be negative). Then obviously $M_0^{!}(\Gamma_{0}(N))$ acts on it ...
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Does the symmetric square L-function vanish at one?

Take a cuspidal automorphic representation $\pi$ of $GL(3)$ over a number field. My question is quite straightforward and can be related to this one : Can $L(1, \pi, \mathrm{sym}^2)$ be zero? If ...
TheStudent's user avatar
5 votes
1 answer
309 views

Proving automorphy of the Galois representations of number fields without considering the residual representation

All the papers proving automorphy of the representations of Galois groups of number fields that I have come across seem to first reduce the representation modulo a prime, prove the automorphy of the ...
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10 votes
2 answers
949 views

Why is Langlands functoriality usually related with period integral in a third group?

In the introduction of "PERIODS OF AUTOMORPHIC FORMS "by HERVE JACQUET, EREZ LAPID, and JONATHAN ROGAWSKI, they said "In many cases, it should be possible to characterize the $H$-distinguished ...
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Fundamental lemma and transfer of characteristic functions of congruent subgroups

Let $E/F$ be an unramified quadratic extension of $p$-adic fields. The Jacquet-Rallis fundamental lemma states that $1_{S_n(O_F)}$ and $(1_{U_n(O_F)}, 0)$ are transfer of each other, see ”On the ...
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Archimedean L-factors for symplectic group

Let $\pi$ be an automorphic representation of $GSp(4)$. Provided a representation $r$ of the Langlands dual group of $GSp(4)$ (namely, the standard or the spinor one), it is possible to define a ...
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Langlands Reciprocity and Fermat's Last Theorem

Question: Can Langlands Reciprocity be used to prove Fermat's Last Theorem? Background A few years ago I was reading a book on the Langlands Program and the introduction provided a list of ...
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On Fourier expansions of Siegel automorphic forms

Please tell me references on Fourier expansions of (non-holomorphic) automorphic forms on the symplectic group of matrix size > 2. I can find formulas of "Fourier coefficients'' of non-holomorphic ...
mis's user avatar
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When does a locally symmetric space have no odd degree Betti numbers?

Let $G$ be a semisimple real lie group, $K$ be a maximal compact subgroup of $G$, $\Gamma$ be a torsion-free cocompact discrete subgroup. The Betti number the locally symmetric space $X_{\Gamma}:=\...
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System of eigenvalues of partial Laplacians and the Jacquet Langlands correspondence

I'm looking for a precise reference. I could not dig it up in the original paper of Jacquet-Langlands. Let $F$ be a totally real number field with $[F:\mathbb{Q}]=r$, let $B_1=M_2(F)$ be the split ...
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Possible Context for this "Siegel-like" Modular Form Construction?

The following construction of something very nearly a Siegel modular form of degree 2 arose in my research. I'm outside the worlds of automorphic forms and number theory, so I'm wondering if it ...
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Index and congruence subgroup from scaling variables of Jacobi form

Let $J_{k,m}(N)$ be the space of Jacobi forms of weight $k$, index $m$, and congruence subgroup $\Gamma_{0}(N) \rtimes \mathbb{Z}^{2}$. I do not believe it is relevant here to specify what type of ...
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4 votes
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L and epsilon factors of Gelbart-Jacquet lifts

I would like to understand better the L-function and epsilon factors attached to a Gelbart-Jacquet lift. Does the fact of being a Gelbart-Jacquet lift translates into strong properties concerning ...
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Does weight $2$ cuspidal Bianchi modular form have infinitely many zero Fourier coefficient at prime ideals?

Let $K$ be an imaginary quadratic field. Let $f \in S_2(\mathfrak{n})$ be a weight $2$ cuspidal cof level $\Gamma_0(\mathfrak{n})$ over $K$ (for definitions one can see http://www.lmfdb.org/knowledge/...
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6 votes
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Decay of matrix coefficients of non-tempered representation

A theorem of Cowling--Haagerup--Howe gives an effective decay rate of the matrix coefficients of a tempered representation $\pi$ of a semi-simple algebraic $G$ in terms of Harish-Chandra $\Xi$ ...
Subhajit Jana's user avatar
6 votes
2 answers
404 views

Reduction to Lie algebra version of fundamental lemma?

Ngo famously proved the Langlands-Shelstad fundamental lemma for Lie algebras using the geometry of the Hitchin fibration. For the purposes of the trace formula, one actually needs the fundamental ...
Spencer Leslie's user avatar
3 votes
1 answer
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Shimura correspondence for automorphic forms on other groups

I've been told recently that the Shimura correspondence does not fit into Langlands functoriality, i.e. does not have a natural generalization to other groups. However, it should have some ...
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