# Tagged Questions

**5**

votes

**0**answers

240 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 ...

**8**

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**0**answers

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, ...

**1**

vote

**0**answers

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 ...

**4**

votes

**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 ...

**7**

votes

**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 $m^2-1$.
...

**5**

votes

**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 ...

**6**

votes

**0**answers

119 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 ...

**7**

votes

**1**answer

172 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 ...

**5**

votes

**1**answer

106 views

### Functional equation and conductor for a Rankin-Selberg convolution

Let $f$ be a Modular form/Maass form on $GL(2)$ with level $N$ and character $\eta$ and Fourier coefficients $a(n)$.
The Rankin-Selberg convolution
$$L(s,f\times\bar f)=\sum ...

**3**

votes

**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 ...

**6**

votes

**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 ...

**1**

vote

**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 ...

**8**

votes

**1**answer

532 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?

**3**

votes

**1**answer

231 views

### Classical Lower Bound of L(1) assuming GRH

Let $L(s)$ be an automorophic $L$-function with conductor $C$ defined by Iwaniec and Sarnak.
What's the classical lower bound of $L(1)$ assuming Riemann Hypothesis?
And what's the reference?
Is ...

**4**

votes

**1**answer

353 views

### subconvexity problem for $GL(3) × GL(2)$ $L$-function without involving in symmetric lift

A question in study of subconvexity topic puzzles me for a long time, which mabe a stupid question for many experts. I really wish someone to help me out, and any advice will be highly appreciated.
...

**5**

votes

**0**answers

361 views

### a generalization of a formula of Shimura

Let $\phi$ be a $GL(2)$ automorphic form with Fourier coefficients $a(n)$ and $a(1)=1$.
Obviously we have $L(s,\phi)=\sum \frac{a(n)}{n^s}$.
Shimura have the following formula
$L(s, Ad\; ...

**1**

vote

**0**answers

112 views

### Off critical line zeros for half integer weight $L$-functions

Let $f(z) = \sum_{n=1}^\infty A(n)n^{\frac{k-1}{2}}e(nz)$ be a modular form of weight $k$ for a half integer $k$. Put
$$L(s,f) = \sum_{n=1}^\infty \frac{A(n)}{n^s} $$
to be the $L$-function.
Further ...

**1**

vote

**1**answer

240 views

### Does FE of Selberg Zeta function imply Trace formula?

Does the functional equation of the Selberg Zeta function imply the Selberg trace formula?
BTW, the trace formula implies the functional equation.

**7**

votes

**3**answers

901 views

### Sato-Tate measure for GL(3) Automorphic forms

As we have known, the Sato-Tate measure for GL(2) turned out to be the half circle measure
$\frac{1}{2\pi} \sqrt{4-x^2}dx$ on [-2,2],
which appears in various versions of equi-distribution problems ...

**2**

votes

**1**answer

299 views

### Dual Maass form for level=N in GL(2)

Let $\Gamma=\Gamma_o(N)$ be the congruence subgroup.
Let $f\in C^\infty(\Gamma\backslash GL(2,R)/SO(2,R)R^*)$ be a Maass form. How shall we define its dual(contragredient) Maass form $f'$?
If ...

**2**

votes

**1**answer

478 views

### Different cuspidal automorphic representations with same representations at infinity

Let us fix a representation $\pi_\infty$ of GL(n,$\mathbb R$).
Let us fix a character $\chi$ of K, where K is a compact subgroup of $GL(n,\mathbb A_{finite})$.
$$K=\Pi_{v<\infty}K_v$$
$K_v$ is ...

**4**

votes

**2**answers

515 views

### What is the relation of the Kuznetsov-Bruggeman trace formula and the Selberg trace formula?

I have read that there is an elementary way to show that the above mentioned trace fromulas are equivalent in the sense, that each of them can be derived directly from the other. There should exist a ...

**13**

votes

**2**answers

770 views

### Why isn't meromorphic continuation enough for converse theorems?

This is a very naive question which really does little more than highlight my ignorance of how converse theorems really work.
Take an algebraic gadget which should be conjecturally associated to an ...

**12**

votes

**3**answers

660 views

### There is no lattice in PSL(2,R) which contains PSL(2,Z) properly?

How can I see that there is no lattice in $G=\mathrm{PSL}_2( \mathbb{R})$ which contains $\Gamma_1=\mathrm{PSL}_2( \mathbb{Z})$ properly, or equivalently, that $X_1 =\mathrm{PSL}_2(\mathbb{Z}) ...