The automorphic-forms tag has no wiki summary.

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### strong approximation and one-dimensional automorphic representations

Hi,
Let $D$ be a quaternion algebra over $\mathbf Q$ such that $D\otimes\mathbf R = M_2(\mathbf R)$.
Let $\pi = \pi_\infty \otimes_p \pi_p$ be an irreducible automorphic representation of ...

**6**

votes

**2**answers

726 views

### Decomposition of $L_0^2(GL_2({\mathbb{Q}}) \backslash GL_2(A), \psi)$

Two questions concerning the decomposition of $L_0^2(GL_2({\mathbb{Q}}) \backslash GL_2(A), \psi)$, where $\psi$ is a Hecke character on the adelic ring $A$:
It is known that when $\psi$ is trivial ...

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**2**answers

2k views

### A stupid question about Automorphic forms

Okay, so an automorphic form $f$ on a reductive group $G/ \mathbb{Q}$ and arithmetic subgroup $\Gamma$ is a smooth function satisfying the following conditions:
(a) invariance with respect to left ...

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**2**answers

792 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 ...

**4**

votes

**1**answer

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### Why is the Arthur trace formula so powerful?

Considering the Arthur trace formula, why are the sort of convolution operators, whose "normalized traces" are given in geometric terms and spectral terms, actually able to distinguish all automorphic ...

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**2**answers

635 views

### Why is the simple trace formula a weaker tool than the Arthur trace formula?

What are some concrete examples of theorems which can be deduced from the Arthur trace formula, which do not follow from the simple trace of Kazdhan and Flicker?
(So I do not mean weaker in the sense ...

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votes

**2**answers

628 views

### Unitary groups over number fields

When defining unitary groups over number fields, one usually takes $F$ to be a totally real number field, $E$ a CM quadratic extension of $F$, and $V$ a hermitian space attached to $E/F$. Then $U(V)$ ...

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342 views

### Non-vanishing of twists of L functions for GL(4)

Hello,
This is a question in the spirit of
Nonvanishing of central L-values of quadratic twists?
and the application I have in mind is to p-adic L-functions a la Ash-Ginzburg.
The question is ...

**4**

votes

**1**answer

654 views

### Why is the cuspidal spectrum discrete?

Hi, I have a short question concerning the spectral theory of automorphic forms. What is the main property of the unipotent group $N$, which consist of matrices in the form ...

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votes

**3**answers

680 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}) ...

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1k views

### Non-vanishing of p-adic L-functions

In Non-vanishing of L-series of modular forms (easy case?) it was answered that for a cuspidal newform $f$ of weight strictly greater than 2, then $L(f,1)$ is non-zero. (Here the $L$-series is ...

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**2**answers

1k views

### Relation between Theta series and Eisensteinseries

In "Mackey - Unitary Group Representation in Physics, Probability and Number Theory" on page 326, George Mackey mentions a result of Ludwig Siegel, which was later generalized to semi-simple Lie ...

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votes

**2**answers

584 views

### Are the $\Gamma(N)$ the only normal congruence subgroups of $\mathrm{SL}_2(\mathbb{Z})$?

The answer to the original question is no, see JSE!
Are the $\Gamma(N)$ the only normal congruence subgroups of $\mathrm{PSL}_2(\mathbb{Z})$ with no finite subgroups (elliptic elements)? What about ...

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**1**answer

593 views

### Double coset spaces of reductive groups and integral representations of L-functions

Let $G$ be a reductive group over a number field $k$, with center $Z$. Let $P$ be a parabolic subgroup. Let $H$ be a reductive subgroup of $G$. To what extent can we understand the double coset space ...

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**4**answers

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### Meromorphic continuation of Eisenstein series

I am interested what kind of (different) proofs of meromorphic continuation Eisenstein series (for general parabolic subgroups) exist in the literature? The only one I understand well,
is Bernstein's ...

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**3**answers

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### Nonvanishing of central L-values of quadratic twists?

Let $\pi$ be a cuspidal automorphic representation of GL(2) over a number field (if you want, assume it's $\mathbb Q$ and $\pi$ comes from a holomorphic modular form).
In the case $\pi$ has trivial ...

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votes

**2**answers

434 views

### Orbits of SL_n acting on matrices of determinant p

Fix a positive integer $n$ and let $S$ be the set of $n$ by $n$ matrices
with entries in $\mathbf{Z}_p$ (the $p$-adic integers) whose determinant is $p$.
The group $G:=\mathrm{SL}_n(\mathbf{Z}_p)$ ...

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votes

**2**answers

868 views

### Automorphic forms and Galois representations over imaginary quadratic fields: generalizing Taylor's theorem?

Let $K/\mathbf{Q}$ be an imaginary quadratic field, with $\sigma \in \mathrm{Gal}(K/\mathbf{Q})$ a generator. Suppose $\pi$ is a cuspidal automorphic representation of $GL_2 / K$ with central ...

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votes

**2**answers

579 views

### Automorphic form encoding the orders of $N$ modulo $p$.

Let $N$ be a nonzero rational number. For every prime number $p$ with $v_p(N)=0$, let $a_p$ denote the index in $\mathbb Z/p\mathbb Z$ of the subgroup generated by $N$ modulo $p$. So we have $a_p=1$ ...

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**0**answers

253 views

### spectral decomposition for elliptic surfaces?

I'm looking for explicit formulae for the spectral decomposition of $L^2(S)$, where $S$ is an elliptic surface (of complex dimension 2).
To be precise, the elliptic surface I'm looking at is the ...

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votes

**3**answers

712 views

### Local to Global principle for reductive groups

Let $G$ be a reductive group over an algebraic number field $k$. Denote with $k_v$ a local field and with $A$ the ring of its adeles, let $G_k$, $G_{k_v}$ resp. $G_A$ be the group of its $k$- resp. ...

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### Automorphic forms on GL(3)

We know that the classical Maass forms on GL(3) are depicted, for instance, in D.Goldfeld's book. I wonder that if there exists "holomorphic" automorphic forms on GL(3) as an analogue of GL(2) case. ...

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### Langlands in dimension 2: the Yoshida conjecture

Background:
One prominent part of the Langlands program is the conjecture that
all motives are automorphic.
It is of interest to consider special cases that are more precise, if less
sweeping. ...

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votes

**1**answer

614 views

### Finite dimensional automorphic representations of a definite quaternion with prime discriminant and Hecke action

Before stating the questions that I have, which are very specific and probably not so interesting to someone who has never thought about these things, I need to introduce some notation.
Let $p$ be ...

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**2**answers

892 views

### Arithmetic geometry from a bird's-eye view

Is ist true that Arithmetic Geometry can roughly be separated into two areas:
1) Showing that motivic $L$-functions are automorphic.
2) Calculating special values of these $L$-functions.

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### Are there motives which do not, or should not, show up in the cohomology of any Shimura variety?

Let $F$ be a real quadratic field and let $E/F$ be an elliptic curve with conductor 1 (i.e. with good reduction everywhere; these things can and do exist) (perhaps also I should assume E has no CM, ...

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votes

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### How is representation theory used in modular/automorphic forms?

There is certainly an abundance of advanced books on Galois representations and automorphic forms. What I'm wondering is more simple: What is the basic connection between modular forms and ...

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**1**answer

614 views

### $L$-functions for $\Theta$-lifts

Let $E/F$ be a quadratic extension of number fields. Let $W$ be a hermitian space over $E$ of dimension $2,$ and let $V$ be a skew-hermitian space of dimension $3$ over $E.$ Consider the associated ...

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478 views

### Automorphic representations attached to abelian varieties

Let $A$ be an abelian variety defined over $\mathbb{Q}$, of dimension $d$. It is widely expected that there is an automorphic representation $\pi_A$ of $GL(2d)/\mathbb{Q}$ whose L-function agrees ...

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### Simple explicit example of local Jacquet-Langlands theorem for inner forms of GL(n), and consequences

This one will be very easy for the experts.
Let $F$ be a nonarch local field, let $n\geq1$ be an integer, choose $0\leq d<n$ and let $D$ be the central simple algebra over $F$ with invariant $d/n$ ...

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### Constructing coherent sheaves on Shimura varieties.

Let me first run through the setting of my question in an example I understand well; that of modular curves. If $Y_1(N)$ denotes the usual modular curve over the complexes, the quotient of the upper ...

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**1**answer

616 views

### Rallis inner product formula for U(2,2) and U(3)

Victor Tan has a couple of papers on a regularized Siegel-Weil formula for U(2,2) and U(3). The papers I'm talking about are:
"A Regularized Siegel-Weil Formula on U(2,2) and U(3)", Duke, 1998.
"An ...

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votes

**2**answers

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### modularity of algebraic varieties

Hello,
Are there any examples of varieties which are not Shimura varieties or abelian varieties
and whose L-functions have been shown to be a product of automorphic L-functions?
Thanks.
N

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2k views

### p-adic L-functions

For modular forms, it is known that you can construct p-adic L-functions by interpolating (p-power conductor) twists of their associated L-functions at special values. Similarly, Kubota-Leopoldt's ...

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### Terminology occuring in automorphic representation and relationship between them

When one tries to read about automorphic representation few terms come up more than others namely,
1.Cuspidal
2.Square Integrable
3.Absolutely Cuspidal
4.Super Cuspidal
My understanding about ...

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528 views

### Is there a canonical notion of “mod-l automorphic representation”?

As the title says.
In particular, I am interested in the story for a general reductive group $G$, say defined over $\mathbb{Q}$. I can imagine that mod-$\ell$ (algebraic) automorphic representation ...

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votes

**2**answers

440 views

### Conductor of monomial forms with trivial nebentypus

Is it true that the conductor of a holomoprhic or a Maass cusp form with trivial nebentypus corresponding to a two-dimensional dihedral representation (over $\mathbb{Q}$ )is non-square-free?

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**1**answer

408 views

### extending cusp forms

Let $E/F$ be a quadratic extension of number fields, and let $V$ be an $n$-dimensional Hermitian space over $E$.
Let $\tilde{G} := GU(V)$ and $G := U(V)$. Suppose that $(\pi, V_{\pi})$ is an ...

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votes

**7**answers

4k views

### What are the local Langlands conjectures nowadays, for connected reductive groups over a $p$-adic field?

Let me stress that I am only interested in $p$-adic fields in this question, for reasons that will become clear later. Let me also stress that in some sense I am basically assuming that the reader ...

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**2**answers

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### Overview of automorphic representations for $SL(2)/{\mathbf{Q}}$?

In short: what does Labesse-Langlands say?
Slightly more precise: what are the cuspidal automorphic representations of $SL_2(\mathbf{A}_{\mathbf{Q}})$, together with multiplicities? Let's say that I ...

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4k views

### Where stands functoriality in 2009?

Robert Langlands is famous in number theory for making famous and deep conjectures about very abstract things called automorphic forms, somewhere in the 60s.
There's a very interesting article by ...