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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107 views

Question on the proper sub-representation of induced representation

$\DeclareMathOperator\Ind{Ind}$Let $G$ be a reductive group over a $p$-adic local field $F$, and $P=MN$ a parabolic subgroup. Let $\sigma$ be an irreducible representation of $M(F)$ and consider its ...
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The Fontaine Mazur conjecture for $\text{GL}_1$ over $\mathbf{Q}$

The Fontaine-Mazur cojectures says that if $\chi : \text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \to \mathbf{Q}_p^\times$ is a character which is unramified almost everywhere and de Rham at $p$ then it ...
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Level vs. conductor of a supercuspidal representation

What is the relation between level and conductor of a supercuspidal representation of $\operatorname{GL}_2(\mathbb{Q}_p)$ for some prime $p$? Proposition 3.4 in Loeffler and Weinstein - On the ...
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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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Global tempered representation vs representation whose global A-parameter is tempered

My question is the title! I thought that the global automorphic representation whose $A$-parameter is tempered is the global tempered representation and every global tempered representation is ...
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Globally generic automorphic representation is contained in some generic A-packet?

I am wondering whether every globally generic representation of special orthogonal group or unitary group is contained in some global generic A-packet. I think it would be not because the Ramanujan ...
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225 views

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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203 views

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?
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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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108 views

Sets of L-functions being “almost bimonoids”

Let $\mathcal{M}$ be a set of L-functions (where by L-function I mean any L-function associated to an automorphic representation of $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ which is an element ...
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103 views

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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1answer
136 views

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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61 views

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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174 views

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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Some question on parabolic induction

$\DeclareMathOperator\GL{GL}$Let $F$ be a local field of characteristic zero and $G=\GL_n(F)$ and $P_{k,n-k}$ is a parabolic subgroup of $G$ whose Levi part is $\GL_{k} \times \GL_{n-k}$. Let $a$, $b$...
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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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1answer
158 views

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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308 views

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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1answer
204 views

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{...
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198 views

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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Conductor formula for symmetric square transfer

Let $\pi$ be an automorphic cuspidal representation of $\operatorname{GL}_2(\mathbb A_{\mathbb Q})$, and let $\Pi = \operatorname{sym}^2(\mathbb\pi)$, which is a representation of $\operatorname{GL}_3(...
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138 views

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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366 views

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 ...
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Modular form attached to a hecke character of cubic/quartic extension of rational numbers

We know that for each Hecke character of a quadratic extension of $\mathbb{Q}$, we can define a modular form (in fact a cusp form). We can find this construction in the book Topics in Classical ...
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1answer
283 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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1answer
382 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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66 views

Double quotients and the isomorphism theorems

I am working on some first aspects of modular forms and automorphic representations, and would like to understand better the formal dictionary between both. It seems to be essentially based on the ...
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1answer
107 views

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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1answer
469 views

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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