# 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 ...

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### Symmetric powers of Ramanujan tau-function

Let $\Delta(z)$ be the modular form associated with Ramanujan $\tau$-function.
For any $k=2,3,...$, $Sym^k\Delta$ is conjectured to be an automorphic form on $\mathrm{GL}(k+1)$ and $L(s, Sym^k\Delta)$...

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### Haar measure on $\mathrm{SL}_3(\mathbb{Z}) \backslash \mathrm{SL}_3(\mathbb{R}) / \mathrm{SO}_3(\mathbb{R})$

The bi-invariant Haar measure on the quotient $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathrm{SL}_2(\mathbb{R}) / \mathrm{SO}_2(\mathbb{R})$ represents the moduli space of rank two real lattices modulo ...

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### Dimension of space of K-fixed vectors

If $G$ is an unramified group over an $p$-adic field $F$, the Satake isomorphism identifies the spherical Hecke algebra with respect to a special maximal compact subgroup $K$. In particular,
(1) $H(G(...

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### Equivalence of formulations of Ihara's lemma

I'm wondering about the relationship between two formulations of Ihara's lemma for $\text{GL}_2$ I've seen:
(1) the "concrete" version given in, for example, Darmon, Diamond, and Taylor, which says ...

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### What is a Hilbertian stack?

In describing spectral decompositions of some $L^2$ spaces, Moeglin-Waldspurger (Spectral Decomposition and Eisenstein Series) keep using the term "Hilbertian stack" (e.g. the Corollary in V.3.14) but ...

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### Orbits of arithmetic subgroups intersection a compact set

Let us suppose we have $G$ a connected reductive group over a number field $F$. Consider $G(\mathbb{A})$ the group over the adeles and $G(\mathbb{Q})$ embedded discretely. For $\gamma \in G(\mathbb{Q})...

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### Example of infinite automorphic multiplicity

Let $G$ be a locally compact group and $\Gamma$ a lattice in $G$. For an irreducible unitary representation $\pi$ of $G$ let
$$
m_\Gamma(\pi)=\dim\mathrm{Hom}_G(\pi,L^2(\Gamma\backslash G))
$$
be its ...

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### What motivations for automorphic forms?

Automorphic forms are ubiquitous in modern number theory and stands as a mysterious Graal lying at the intersection of many fields, if not building valuable bridges between them. However, since this ...

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### Automorphic quotient for quaternion algebras

Are automorphic quotient for quaternion algebras always compact (safe the totally split case)?
Is there any good reference for proof of this fact, or easy arguments to say do?

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### Projection onto locally constant function

I am asking a question which looks very elementary to experts.
Let $F$ be a number field and $\mathbb{A}_F$ its adele ring. Let $\omega$ be a unitary central character of $GL_2(\mathbb{A}_F)$,
$X_{...

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### Fourier Transforms of Convolutions

Straightforward computations lead to the following standard property of Fourier transformation: it transforms convolutions into products, i.e. for functions $f$ and $g$ Schwartz class we have
$$\...

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### Volumes of Hecke operators

Let $G=GL(2, F)$ and $K$ a maximal compact subgroup. Unramified Hecke operators are defined by the action of the double cosets
$$T(n) = \bigcup_{\substack{ad=n, a>0 \\ a|d}} K \left( \begin{array}{...

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### Expression of the root number for Maass forms

Take a holomorphic cusp newform, say $f \in S_k(N)^\mathrm{new}$, for a squarefree level $N$. It is an eigenvalue of the Atkin-Lehner operator, and this feature allows to express its root number as
$$\...

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### Examples of automorphic forms over $\mathbb{H}^3/\text{PSL}_2(\mathbb{Z}[i])$

If I understood my automorphic forms correctly, at least cusp forms can be thought of as elements of $L^2(G/\Gamma)$ for a $G = \text{SL}_2(\mathbb{R})$ and $\Gamma = \text{SL}_2(\mathbb{Z})$ or a ...

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121 views

### About transfer from Hilbert modular forms to Siegel modular forms

Suppose $F$ is a totally real field of degree $d$. Is there an explicit way (like theta series or so) to construct automorphic forms on $\mathrm{Gsp}(2d)$ from Hilbert modular forms of ${\rm GL}_2(F)$?...

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### Functional equation and contragredient

I am often annoyed by the presence of the contragredient $\tilde{\pi}$ of the representation we consider, when we write the functional equation of its L-function:
$$L(s, \pi) = \varepsilon(s, \pi) L(1-...

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### Archimedean Langlands classification

I am trying to clarify how the archimedean admissible dual is classified in the $G=GL(n, \mathbf{R})$ case.
Fix a semistandard Levi subgroup $M$ in $G$, $\delta$ a square-integrable representation of ...

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### Hecke eigensystem in cohomology vs. compactly supported cohomology

What follows is a question that's probably well-known to experts, but I haven't been able to find a reference.
Let $\mathrm G$ be a connected, semisimple $\mathbb Q$-group. Let $K \subset \mathrm G(\...

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305 views

### Potential automorphy of abelian varieties

Let $A$ be an abelian variety over $\mathbb Q$. One could ask
(1) is there a finite extension $K$ of $\mathbb Q$ such that the L-function $L(A/K,s)$ is the L-function of an automorphic form?
or
...

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### Special values of adjoint $L$-functions of automorphic representations of $\mathrm{GSp}(4)$ as Petersson norms

Here I consider cuspidal automorphic representations $\pi$ over the similitude group $\mathrm{GSp}(4,\mathbb{A}_\mathbb{Q})$. Let $f$ be a non-zero vector in the representation $\pi$. I want to know ...

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### Relation between Hecke operators and coefficient of L-functions

This question has its seed in this one by Gory, which found an enlightening answer but one of the comments kept me wondering. I am beginning to discover Hecke operators, and there appears to be an ...

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### Modular forms for different groups than $SL(2,\mathbb Z)$

I know some theory of "classical" modular forms, that is functions in the complex upper-half plane satisfying
$f(\frac {az+b} {cz+d})=(cz+d)^kf(z)$
I know one can study modular forms on finite-...

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### Poles of $L$-functions associated to Maass forms

Let $\pi$ be an automorphic representation of $GL_2$ over a number field. What can I say concerning the order of the pole at $1$ of the $L$-function $L(s, \pi)$? Can we say more about $L(s, \mathrm{...

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### Conjectures that can be tested with large numbers of Hecke eigenvalues of GSp(4) automorphic forms

As part of my thesis work I have proved Ibukiyama's conjecture implies something about $\mathrm{SO}(5)$ forms associated to certain lattices lifting to $\mathrm{GSp}(4)$ (This was originally a ...

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### Ramanujan conjecture in terms of representations

Given an automorphic representation, I would like to bound $\alpha_1^\nu(p) + \alpha_2^\nu(p)$ where the $\alpha_i$ are the Satake parameters of an automorphic form $f$ of, say, $GL_2$. So that $\...

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### Eisenstein series for non congruence subgoups

What is the present status of the Eisenstein series for noncongruence subgroups?
I am aware of work of A. Scholl and Rohrlich work on the subject.
Is there any specific examples that has been ...

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### How does Riemann hypothesis implies estimates?

In Iwaniec, Luo and Sarnak article (precisely (4.23)), it is said that GRH for $L(s, \mathrm{sym}^2(f))$, for a holomorphic cusp newform $f$ of level $N$ and weight $k$, implies
$$\sum_{p \nmid N} \...

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### Local L-factors for automorphic representations

For Hecke L-functions associated to a holomorphic cusp form $f$ of level $N$, the local factors can be decomposed into
$$L_p(s, f) = (1-\lambda_f(p)p^{-s} + \chi_N(p)p^{-2s})^{-1}$$
where $\chi_N$ is ...

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### What is the matter with Hecke operators?

This question is inspired by some others on MathOverflow. Hecke operators are standardly defined by double cosets acting on automorphic forms, in an explicit way.
However, what bother me is that ...

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### Automorphic quotients for inner forms or $GSp(4)$

For a quaternion algebra $D$, introduce the quaternionic similitude unitary groups:
\begin{equation}
\mathrm{GU}_D = \left\{ g \in \mathrm{GL}(D) \ : \ g^\star
\left(
\begin{array}{cc}
& 1 \\
1 &...

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### A question on representation theory of p-adic groups

Let $V$ be a complex vector space of infinite dimension and let $(\pi,V)$ be a representation of the $p$-adic group $G:=GL_2(\mathbb{Q}_p)$. From representation theory, we know that if the ...

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### Solutions of $a_1Y_1 + a_2Y_2 + a_3Y_3 = 0$

For three variables $Y_1,Y_2,Y_3$, we consider the linear equation
$(\sharp) \quad X_1Y_1 + X_2Y_2 + X_3Y_3 = 0$.
We shall consider the solutions of $(\sharp)$ within the ring $K[[X_1,X_2,X_3]]$. ...

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### Are theta functions cuspidal representations?

After reading many textbooks I still can't get the jargon correct. Given a spherical harmonic $u \in L^2(S^n)$ one could construct a theta function:
$$ \theta (z;u) = \sum_{ m \in \mathbb{Z}^3} u (m)...

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### Exceptional primes

Let $f$ be a cuspidal Hecke Eigenform of weight $k \geq 2$ and let $\rho_{f, \lambda}:G_{\mathbb{Q}} \rightarrow GL_2(E_{\lambda})$ be the corresponding Galois representation with $2 \mid \lambda$ ...

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### Meaning of Ramanujan-Petersson conjecture? [closed]

I found it very hard to explain the Ramanujan-Petersson conjecture in a straightforward way.
I can only say now: think about automorphic forms as sound waves, and then the conjecture predicts that ...

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### Global Langlands function fields

Has V. Lafforgue proved the automorphic-to-Galois direction in the Global Langlands conjectures for general reductive groups over function fields?
What is the current status, more generally?
Related ...

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### Non-vanishing of left- vs. right-averages over lattices in $SL(2,\mathbb{R})$

I asked the same question on MSE one week ago, but it has not received any answers.
Background. Let $G=SL(2,\mathbb{R})$, let $K=SO(2)$, and let $\Gamma$ be a lattice in $G$, e.g. $SL(2,\mathbb{Z})$...

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### Where stands functoriality in 2017?

In 2002, R. Langlands put forward a new strategy to prove the general functoriality conjecture in the Beyond endoscopy paper. The main purpose of this strategy is to detecting the automorphic ...

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### $GSp(4)$ vs $PSp(4)$

After some months wandering through examples of algebraic groups in the theory of automorphic forms and number theory, I wonder why so many efforts are spent in understanding $GSp(4)$ (local newforms, ...

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### What condition makes unitary reductive group unramified?

I am a little bit confused with the definition of an unramified unitary group.
Let $F$ be a local field of characteristic zero whose residue field is finite field of characteristic $p$.
Then for a ...

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### Cohomology of adelic locally symmetric spaces

I am most probably wrong in asserting as follows.
Let $G$ be a connected reductive group over $\mathbb{Q}$, and $S_{K_f} = G(\mathbb{Q}) \backslash G(\mathbb{A}/K_\infty Z(\mathbb{A}) \cdot K_f $ be ...

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### Alternative way to prove the functional equation for Eisenstein series?

Let $E(z,s):=\pi^{-s}\Gamma (s) \sum_{(m,n)=1}\frac{y^s}{|mz+n|^{2s}}$ be the real-analytic Eisenstein series.
It satisfies the functional equation $E(z,s)=E(z,1-s)$ with two poles at $s=0,1$.
The ...

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**1**answer

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### cohomological representations of GL(N)

I am trying to understand cohomology of $G := GL(N)$. For this I need to understand representations of $G(\mathbb{R})$ with nontrivial $(\mathfrak{g},K_\infty)$-cohomology, where $\mathfrak{g}$ is the ...

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### Reference for shtuka and trace formula

I really want to learn the work of Laurent Lafforgue and the joint work of Zhiwei Yun and Wei Zhang. They both involve shtuka and trace formula, which I only know the basic idea. So I would like to ...

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### When will the value of automorphic function $f(x)$ satisify an algebraic equation?

When will the value of automorphic function $f(x)$ satisfy an algebraic equation? Or what is the value of $x$ such that the value of automorphic function $f(x)$ is algebraic?
If the question is too ...

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### Residue of Eisenstein Series on GL(n)

Reference: Mœglin, C.; Waldspurger, J.-L.: Le spectre residuel de GL(n)
On GL($n$), the result of Waldspurger shows that if an automorphic representation $\pi$ is non-cuspidal and in the discrete ...

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### What kind of non-cuspidal automorphic representation are not isobaric sums?

Let's say $\pi$ is an automorphic representation on $GL_3(A_{\mathbb Q})$ (or $GL_n(A_{\mathbb Q})$).
If $\pi$ is not cuspidal, what $\pi$ can be other than isobaric sums?
If there is such a thing, ...

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### Rankin-Selberg integral for GL(3) form with Odd Maass form on GL(2)

Let $F$ be a Hecke-Maass cusp form for $SL_3(\mathbb Z)$.
Let $u$ be a Hecke-Maass cusp form for $SL_2(\mathbb Z)$.
The following integral
$$\mathcal L(s,F\times u)=\int_{{SL}(2,\mathbb{Z})\...

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### infinitesimal character of Langlands quotient for GL(n,R)

Let $G = GL(n,\mathbb{R})$. Consider a Langlands data $(Q_F, \sigma, \lambda)$ with $F \subset \Delta$ (the set of simple roots), $Q_F$ the associated standard parabolic subgroup, $\sigma$ an ...

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### Is there a definition of supercupidal parameter in the Local Langland correspondence?

By the recent works of Mok, and Kaletha, Shin, White, James, I know that there is a notion of tempered $L$-parameter, square integrable $L$-parameter and generic $L$-parameter of unitary groups.
...