# Tagged Questions

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### On the computation of Asai L-function

I want so compute some simple twisted Asai L-function.
Let $E/F$ be a quadratic extemsion of number fields and $v$ a finite place of $F$.
Let $\chi$ be a unitary automorphic character of ...

**1**

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**1**answer

96 views

### On the pole of local L-function

Let $F$ be a number field and $v$ a finite place of $F$.
Let $\chi_v$ be a unramified unitary character of $F_v$.
Then we define local L-function $L_v(s,\chi_v):=\frac{1}{1-\chi_v(\omega)q^{-s}}$ ...

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56 views

### nonvanishing of global theta lifting from U(1) to U(1?)

I understanding nonvanishing of theta lifting, either global or local, is a difficult and open problem. But I wanna know if there is an answer for the following simplest case.
Let $E/F$ be a ...

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**0**answers

102 views

### Multiplicity of automorphic representation

Let $\pi$ be an automorphic subrepresentation of a reductive group $G$. Here by this, I mean an irreducible representation realized in a subspace of the space of automorphic forms on $G$.
Let $m_\pi$ ...

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**0**answers

96 views

### Modular form, number of divisors [duplicate]

The Fourier expansion of Eisenstein series $E_k$ $(k \ge 4)$, which are modular forms, as well as the quasimodular $E_2$, involves powers-of-divisors $\sigma_{k-1}(n) = \sum_{d|n} d^{k-1}$.
Is there ...

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**1**answer

262 views

### Hecke-module structure implicit in definition of automorphic forms in Borel-Jacquet's Corvallis article

Let $G$ be a connected reductive group over a number field $F$, $G_\infty=\prod_{v\mid\infty} G(F_v)$, $\mathbf{A}$ the adèles of $F$, $\mathbf{A}_f$ the finite adèles of $F$. Fix a maximal compact ...

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72 views

### Base change of discrete series

Let $\pi_f$ be an automorphic representation of $GL_2(A_Q)$ (attached to a modular form $f$), and suppose we want to look at its base change lift to say a quadratic imaginary field. Which are the ...

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241 views

### Which L-functions are not “Langlands-Shahidi L-functions”?

The Langlands-Shahidi method, among other things, obtains certain L-functions from the constant term of Eisenstein series attached to so-called $(G,M)$ pairs, where $G$ is a reductive group, $M$ a ...

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154 views

### Symmetric Fifth Power Lift of GL(2) Automorphic Form

Let $\pi$ be an automorphic representation of $GL(2)/\mathbb{Q}$. For simplicity, you can take it to be a Maass form for $SL(2,\mathbb Z)$. Kim, Shahidi, Gelbart-Jacquet prove that
$$L(s, \pi, ...

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**1**answer

185 views

### Self-dual automorphic forms on GL(4)

as is known among experts, all self-dual automorphic form on GL(3) comes from symmetric square lift from GL(2). You can find this from Ramakrishnan ...

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**1**answer

170 views

### Newform and Galois representation (Shimura-Deligne Reciprocity Law)

Shimura-Deligne Reciprocity Law implies that to a newform $f \colon =f(z) \in S^{\mathrm{new}}_k(\Gamma_0(N))$ of weight $k$, one can associate Galois representation $\rho_{f,\lambda} \colon ...

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114 views

### Order of individual Fourier coefficient of a Maass form

Let $D$ be a definite quaternion division algebra over $\mathbb{Q}$ and $\mathcal{O}$ be an Eichler order of $D$. Let $F$ be a Maass form in $L^2(PGL_2(\mathcal{O})\backslash ...

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**1**answer

179 views

### looking for reference on dihedral, tetrahedral, or octahedral forms

I am looking for a reference on dihedral, tetrahedral, or octahedral forms. As far as I read, they are some cuspidal automorphic forms on $GL(2)$ induced from $GL(1)$. Dihedral is from $GL(1)/K$ to ...

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**1**answer

241 views

### Is the adjoint L-function on GL(m) holomorphic?

Let $\pi$ be an automorphic representation on GL($m$)/$\mathbb{Q}$.
Define $$L(s,\pi,Ad):=\frac{L(s,\pi\times\overline{\pi})}{\zeta(s)}.$$ This is an L-function with euler products of degree $m^2-1$.
...

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**1**answer

207 views

### absolute convergence of Rankin-Selberg series

Let $\pi$ and $\pi'$ be two general automorphic representation on $GL(n)$ and $GL(n')$ over $\mathbb{Q}$.
I heard that the rankin-selberg convolution L-function $L(s,\pi\times\pi')$ is absolutely ...

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**0**answers

120 views

### Eisenstein series over a definite division algebra

Let $D$ be the definite quaternion division algebra over $\mathbb{Q}$. $\mathcal{O}$ is a maximal order inside $D$, let's fix $\mathcal{O}$ to be the Hurwitz quaternion. Let ...

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**1**answer

173 views

### standard zero free region of automorphic L-function on GL(N)

Let $L(s,\pi)$ be the standard(Godement-Jacquet) $L$-function of $\pi$, where $\pi$ is a cuspidal automorphic represetation of $GL(m,A_Q)$.
What's the standard zero-free region for $L(s,\pi)$? any ...

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**1**answer

156 views

### Euler product of Asai L-function?

Let $\pi$ be an automorphic form of GL(n)/$\mathbb{Q}$ with standard $L$-function
$$L(s,\pi)=\prod_p \prod_{i=1}^n(1-\frac{\alpha_{p,i}}{p^s})^{-1},$$
where $\{\alpha_{p,i}:i=1,\dots,n\}$ are the ...

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124 views

### Is there an integral pairing between quaternionic Hecke algebras and cusp forms?

Let $F$ be a totally real number field with integers $\mathcal{O}_F$ and $B$ a quaternion algebra over $F$ split at exactly one infinity place.Fix $n\geq 1$ and like in the special case $F=\mathbb{Q}, ...

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**1**answer

201 views

### Regularity assumption in the simple trace formula

In the simple trace formula of Deligne Kazhdan one assumes that the test function is supported at the elliptic regular elements at one place and is a supercusp form at another place. Why can't one ...

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195 views

### Irreducibility of Galois representations attached to unitary groups

If $G$ is a unitary group in $n$ variables over $\mathbb Q$, attached to an hermitian form for an imaginary quadratic extension $E/\mathbb Q$ and if we suppose that the hermitian form is definite over ...

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129 views

### Notation in Shimura “Arithmetic of Automorphic …”

I can't find the following (abuse of?) notation explained in Shimura's "Introduction to the Arithmetic of Automorphic Forms", I'm hoping someone can clarify it:
Chapter 7, section 4, page 177:
...

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**2**answers

228 views

### Real character modular forms: Fourier coefficient real?

Let $f$ be a modular form of level $N$ and real character $\chi$ of mod $N$ and weight $k$.
Does the Fourier coefficient or hecke-eigenvalue of $f$ have to be real?
What I knew is that if $N=1$ and ...

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318 views

### What's the status of Arthur's announced classification for GSp(4)?

In "Automorphic representations of GSp(4)" (2004) (see http://www.math.toronto.edu/arthur/), James Arthur announces a classification of discrete automorphic representations of GSp(4). There are no ...

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**1**answer

289 views

### Generalization of Watson's triple product

In Watson's thesis (page 51) we can find his beautiful triple product formula. My question is that does there exist a generalization of this formula? By generalization, I mean:
If $\phi_n$'s are ...

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**0**answers

184 views

### Why Whittaker functions are useful?

Whittaker functions appears in Langlands program. Recently, it is shown that some Whittaker functions can be obtained by integrating a function related to decoration over a geometric crystal in ...

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225 views

### Gelfand pair and double coset decomposition

Let $F$ be a non-Archimedean local field with ring of integers $O$, $\pi$ be a uniformizer. Let $\tilde{G}$ be a connected algebraic group over $F$ and splits over $F$, fix a split maximal torus ...

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72 views

### On the local theta correspondence between U(1) and U(2)

Let $E/F$ be a quadratic extension of number fields and $V,W$ hermitian and skew-hermitian vector spaces of dimension 1,2 respectively. Let $v$ be a place of $F$ and $\chi$ be a fixed character of ...

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**0**answers

102 views

### Does the global theta lift performed twice yield the identity when it doesn't vanish?

Let $E/F$ be a quadratic extension of number fields and $V,W$ are hermitian and skew hermitian vector space over $E$ whose dimension is $n,m$ respectively.
Let $\pi$ be a irreducible tempered ...

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**0**answers

79 views

### Bounding global matrix coefficient for PGL_2

I'm trying to find a reference that gives a bound for the adelic matrix coefficient for $\text{PGL}_2$ using the bound towards Ramanujan conjecture. More specifically:
Let $G=\text{PGL}_2$. Let $F$ ...

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**1**answer

286 views

### Order of some $L$-function at $s=0$

Sorry, I asked this two days ago, but this time I modified it to be easily read and added more specific explanation. I hope to get your illuminating comment on whether my approach is right.
I am ...

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**1**answer

184 views

### The effect of base change on the L-function of GL(2)?

Let $F$ be a local field (whose residue field is $q$) and $E$ its quadratic extension. Let $\pi$ be a irreducible principal series representation $\pi(\chi_1, \chi_2)$ of $GL_F(2)$ especially where ...

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**1**answer

248 views

### An application of Strong Approximation

I am trying to read a paper in which the authors claim that a certain map between vector spaces is injective, and this follows from the strong approximation theorem. I do not understand how the ...

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**1**answer

156 views

### Can the isometry group of the set of zeros of an L-function $F$ be used to make $F$ automorphic?

I'm still trying to understand the notion of automorphic (L-)function. Due to my lack of knowledge of the subject, this question may appear pretty vague and therefore may not be suitable for MO. I ...

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**2**answers

435 views

### Using Eichler-Selberg trace formula to compute dimension of modular forms?

Is it possible to use Eichler-Selberg trace formula to compute the dimension of modular forms of weight $k$ for $SL(2,\mathbb Z)$? This was computed by classical methods such as Riemann-Roch.

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**1**answer

111 views

### degree of an isobaric sum

I'm trying to understand a few things about automorphic L-functions. In page 5 of http://arxiv.org/pdf/1401.0390.pdf, the author mentions the isobaric sum decomposition ...

**1**

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**1**answer

195 views

### Local component of global irreducible representation of GL_2(A_F)

In studying automourphic representation, I want to know whether my understanding is on the right way.
Let $\pi$ be a irreducible cuspidal representation of $GL_2(A_F)$.
Then $\pi_v$, the local ...

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**0**answers

133 views

### Are quantities involved in Generalized Ramanujan Conjecture eigenvalues of some unitary operator?

If I'm not mistaken, every automorphic L-function $L(s,\pi)$ verifies $\displaystyle{L(s,\pi)=\prod_{p}L_{p}(s,\pi_{p})}$ where ...

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**3**answers

416 views

### Definition of Hecke operators

I am confused about the definition of Hecke operators. It will be great if someone provides some references.
Shimura's 'Arithmetic Theory of Automorphic forms' says: Let $\Gamma$ be acting in the ...

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**2**answers

197 views

### On size of Hecke algebras.

Let $G$ be a subgroup in $SL_2(\mathbb{Z})$ and $S_k(G)$ be the space of cusp (automorphic?) forms invariant by any element of $G$ of weight $k$.
Question 1: Generally for two arithmetic subgroups ...

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**1**answer

81 views

### Name or references for minimal $N$ such that $\left(\frac{a}{b}\right)_n = \left(\frac{a}{b'}\right)_n$ whenever $b \equiv b' \bmod (N)$

Let $\left( \dfrac{a}{b} \right)_n$ denote the nth power residue symbol, a generalization of the Legendre symbol. I have recently seen it quoted that there is a minimal ideal $N$ (minimal by ideal ...

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**1**answer

328 views

### Is the universal elliptic curve $\overline M_{1,2}$ a toric stack?

It is well-known that the compactification $\overline M_{1,1}$ of the moduli space of elliptic curves over $\mathbb C$ is a weighted projective line with
weights $4$ and $6$. As far as I can tell, ...

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**2**answers

309 views

### Adelic methods for classical modular forms

Many conjectures about properties of automorphic forms on $\mathrm{GL}(2)$ can be formulated in the basic language of classical modular forms (i.e. Hecke forms that are holomorphic on $\mathbb{H}$ or ...

**3**

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**1**answer

203 views

### On a unitary automorphic representation

I sometimes come across this notion called "unitary automorphic representation". But I have never seen the precise definition. When they say $(\pi, V)$ is a unitary automorphic representation of a ...

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**5**answers

477 views

### Is a unitary representation always semisimple?

I have been reading the online lecture notes by Fiona Murnaghan
http://www.math.toronto.edu/murnaghan/courses/mat1197/notes.pdf
The first lemma in p.35 says that every unitary representation of ...

**1**

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**1**answer

368 views

### stationary phase method in analytic number theory

I hope someone can tell me something about the error term in the formula calculating the oscillatory integral like $\int_a^b g(x)e(f(x))d x$. Specially, the exact formula on page 114 of M. Huxley's ...

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**0**answers

189 views

### What is a “cohomological type” automorphic representation?

Sorry for asking such a question. This is supposedly a well known definition for experts. However as an non-expert, I tried (very hard) google, Mathscinet etc, but still couldn't find a satisfactory ...

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**1**answer

210 views

### Is $(G,K)$ a strong Gelfand pair?

Let $F$ be a $p$-adic field with ring of integers $\mathcal{O}$. When $G={\rm GL}_n$, it is a classical result that $(G(F),G(\mathcal{O}))$ is a Gelfand pair. Is it actually a strong Gelfand pair? I ...

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**1**answer

533 views

### best record toward Selberg's eigenvalue conjecture?

What's the best record toward Selberg's eigenvalue conjecture:
Maass forms on $\Gamma_o(N)$ has eigenvalue greater than 1/4?

**4**

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**1**answer

274 views

### To which automorphic forms/rep's over a function field can we associate a Galois representation?

As far as I understand it, by the work of Lafforgue (cf. Laumon, "Cohom. of Drinfeld ... II", Thm 12.4.1) there is a Galois representation associated to an irreducible cuspidal automorphic ...