The automorphic-forms tag has no wiki summary.

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### Refining the moderate growth condition on automorphic forms

Let $G$ be a simple algebraic group defined over $\mathbb Q$. In their Corvallis article (automorphic forms and automorphic representations), Borel and Jacquet define an automorphic form to be a ...

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**1**answer

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### A stupid question on automorphic l-function

This may be a silly question for experts in this area. But I am really suffering for not being able to compute local-L function of some automorphic representation.
So, I post it hoping some ...

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**1**answer

981 views

### Understanding the “idea” behind Langlands

Apologies in advance if this is a bit too simple to ask here, but I think I'm probably more likely to get an answer here than at stackexchange.
I've been trying to learn the basics of the Langlands ...

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192 views

### On the theta lift and its L-function

I am wondering how the relation is between of the automorphic L-function and its lift's.
More precisely,
Let $E/F$ be a quadratic extension of number fields. Let $W$ be a hermitian space over E of ...

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**1**answer

226 views

### On the restriction of cusp. irr. representation and period.

Let's consider only global case.
Let $G_n$ be classical algebraic group over global # field (eg, $GL(n),SO(n), U(n)$...) and let $\pi_n$ be its irr. cusp. reps of $G_n$.
Then we can define the ...

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239 views

### on the fundamental lemma

I consider the fundamental lemma for the spherical Hecke algebra.
Let $G$ a connected reductive quasisplit group on $F$, a local field of equal characteristic $p$.
and $H$ an endoscopic group.
Can ...

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**1**answer

155 views

### Global square integrability ensures local sq. integrability?

This might be a stupid question for expert in this area.
I am considering automorphic representation of algebraic group.
In studying it, local tempered, local square integrable representations ...

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**2**answers

323 views

### a question about the proof of analytic continuation of Eisenstein series for GL(2)

I'm reading Gelbart and Jacquet's paper 'forms of GL(2) from the analytic point of view', and was confused at a point in the proof of analytic continuation of Eisenstein series. On the top of page ...

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220 views

### Extending cuspidal representation to more bigger group.

I am thinking of extending an irreducible cuspidal representation to more bigger group. My question is almost same with the earlier one posted by Neal Harris except the only one.
Let me first ...

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**1**answer

457 views

### Is a reductive adelic group a Type I group?

I foresee that to experts of automorphic forms this question will sound unimportant or useless or even not worthy of an answer; but none of these are going to stop me from asking it!
The question is ...

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**1**answer

335 views

### On the Weil representation of unitary groups.

I suppose I am the first one who asked about Weil representation here.
In studying Weil representation, I fell into a slough and so determined to ask you for shedding a light. I think your responses ...

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204 views

### On the L-function of unique subrepresentation of induced representation.

In studying the L-functions of induced representation, it is not easily come up with me the papers or books dealing the L-function of irreducible subrepresentation of induced representation, while the ...

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287 views

### What is the support of the Whittaker function of a new vector on GL(2)?

Let $W$ be the normalized Whittaker function associated to a new vector in an irreducible generic representation $\pi$ of $G=GL_2(k)$, where $k$ is a $p$-adic field. Let $c$ be the conductor of $\pi$, ...

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139 views

### On Langlands Pairing and transfer factors

In the paper "On the definition of transfer factors" Langlands and Shelstad define a certain number of factors $\Delta_{I}$, $\Delta_{II}$,$\Delta_{III,1}$,$\Delta_{III,2}$, which are roots of unity.
...

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158 views

### On the Weil representation of U(1) and U(3).

Let $E/F$ quadratic extention number fields.
Let $V$ be the $m$-dimension hermition vector space over $E$.
Let $W$ be the $2n$-dimension skew hermitian vector space over $E$ and $Y_n + Y_n^*=W$ be ...

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218 views

### Reference on Casselman-Shalika formula for GL(n) and PGL(n)?

I am looking for reference on Casselman-Shalika formula for GL(n) and PGL(n) at finite place p.

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576 views

### Why only half-integral weight automorphic forms?

Why is that the theory of automorphic forms concentrates on the case of half-integral weight? I read in Borel's book "Automorphic forms on $SL_2$" (Section 18.5) that by considering the finite or ...

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### Explicit examples of algebraic Hecke characters with infinite image?

Jerry Shurman has a lovely set of notes explaining the classical definition of Hecke characters, the idelic definition of Hecke characters, their relationship, and the classification of algebraic ...

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1k views

### Why doesn't functoriality immediately imply the modularity theorem?

Let $E/\mathbb{Q}$ be an elliptic curve. By the modularity theorem, the prime indexed coefficients of its $L$-function agree with those of a weight $2$ cusp eigenform $f$ with integer coefficients. ...

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**1**answer

239 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.

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### Restriction of representations from $SO_{2n}$ to $SO_{2n-1}$ and $K$-fixed vectors

Let $F$ be a $p$-adic field, $G=SO_{2n}(F)$ the split special orthogonal group and $H=SO_{2n-1}(F)$, taken as a subgroup of $G$. Assume that we have an irreducible, admissible representation $\pi$ of ...

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### Half integral weight Hecke operators

I would like to find a source giving the exact formula for the product of two Hecke operators $T_{\kappa}(n^2)$ and $T_\kappa(m^2)$ of half integral weight. That is, $\kappa \in \frac 12 \mathbb{Z} - ...

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218 views

### Writing a basis of a representation for $GL_2(\mathbb Q_p)$ in terms of the new vector

For an irreducible smooth (generic) representation $\pi$ of $G=GL_2(k)$ with central character $\omega$, where $k$ is a $p$-adic field, we define the conductor of a vector $v\in\pi$ as follows. Let ...

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### Modularity of higher dimensional abelian varieties

In another question I asked about strategies for giving an effective version of the Shafarevich conjecture for abelian varieties over $\mathbb{Q}$.
For elliptic curves, one can give a proof using ...

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### Difference between automorphic forms for SL(2) and GL(2)?

Hi,
Let $A$ denote the adeles of $Q$.
I know how to decompose $L^2(SL(2,A)/SL(2,Q))$ into irreducible $SL(2,A)$-representations. What is the difference between this decomposition and the ...

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**1**answer

541 views

### Special value of $L$-function

Let $p$ be a prime number. Let $f$ be a newform of weight 2 on $Î“_0(p)$, and $E_f$ denote the associated newform quotient of $J_0 (N)$ over $\mathbb{Q}$. Is there a way to express the
algebraic part ...

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437 views

### Base-Change and Automorphic-Induction for $GL_1$

Dear all, I try to understand the base-change and automorphic-induction in the theory of automorphic forms, for the simplest case: $GL_1$. Both are implied by Langlands conjectures
Base-Change
Let ...

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**1**answer

268 views

### Weyl law for SL(2,C)

Are there any estimates for the eigenvalues of the Laplace operator for $\Gamma \backslash SL(2, \mathbb{C})/SU(2)$ known beyond the main term? Here, $\Gamma$ should be congruence subgroup in ...

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893 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 ...

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276 views

### What is an automorphic representation of CM type ?

In a recent paper of BL-Gee-Geraghty: "Sato-Tate for Hilbert modular forms" (JAMS 2011), a theorem is proved for regular algebrai cuspidal automorphic representation of $GL_2(\mathbb A_F)$ with $F$ a ...

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### Are there Maass forms where the expected Galois representation is $\ell$-adic?

Recall that by theorems of Deligne and Deligne--Serre, there is the following dichotomy:
Modular forms on the upper half plane of level $N$ and weight $k\geq 2$ correspond to representations ...

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234 views

### Pseudo coefficients and orbital integrals

I am looking for a reference/idea, how this passage from Labesse's Snowbird Lecture "Introduction to endoscopy" pg.5 can be explained:
"We shall denote by $f_\pi$ a pseudo-coefficient for $\pi$, ...

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### Product of two cuspforms is not a cuspform

Let $f$ and $g$ be two cuspforms on $\Gamma \backslash \mathbb{H}$. They could be Maass cuspforms, or holomorphic modular forms. Let us say that they are holomorphic and also that $\Gamma = ...

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### Questions about the Bernstein center of a $p$-adic reductive group

Dear all,
The "Bernstein center" of a $p$-adic reductive group appears frequently in the literature of automorphic forms, often without a precise definition. For example, in page 233 of ...

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310 views

### Any reference on Eisenstein Series for \Gamma_o(N) in GL(2)

What's the best reference on Eisenstein Series for $\Gamma_o(N)$ in GL(2,R)?
For fixed $\Gamma_o(N)$, should there be several Eisenstein series(corresponding to each cusp)?

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298 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 ...

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420 views

### When is compact induction in GL(2) from an open compact group admissible?

Let $G$ be a locally profinite group and $K$ an open compact subgroup (mod the center), then Bushnell has shown that the following three statements are equivalent for a finite dimensional ...

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242 views

### What is different about the Residual Spectrum

In the context of spectral decomposition of functions in $L^2(\Gamma \backslash \mathfrak{h})$, or Selberg trace formula, we come across three different types of spectrum.
First off there is the ...

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284 views

### Atkinâ€“Lehner operator for GL(3)?

Let $f$ be an automorphic form for $\Gamma_0(N)\subset SL(3,\mathbb{Z})$.
$\Gamma_0(N)=(a,b,c;d,e,f;g,h,i)\in SL(3,\mathbb{Z})|g=h=0(mod N)$
Is there any Atkin-Lehner operator for $\Gamma_0(N)$ ...

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230 views

### Double coset decomposition of symplectic group over a quadratic extension

I'm trying to understand the double coset decomposition of $G(F)\setminus G(E)/K_E$ , where $G = \mathrm{GSp}_{2n}$ is the rank $n$ group of symplectic similitudes, $E/F$ is a quadratic extension of ...

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### What is the dual of principal series of GL(3,R)?

It is common to construct principal series by induction from Borel subgroup. Say $H_1$ and $H_2$ are dual representations. Both are induced representation from Borel subgroups.
Is the integration ...

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### An interesting double coset in the theory of automorphic forms

Dear all,
Does anyone have some idea to describe the double coset $P(F)\\backslash G(F)/H(F)$ , say using Weyl group elements ? Here $G=GL_n\times GL_{n-1}$ is defined over a number field $F$ , ...

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### Tamagawa number for functional fields

Let $G$ be a split semi-simple simply connected group over a global field $F$ and let
$\omega$ be a top-degree differential form on $G$ without zeroes (defined over $F$). It is well
known that ...

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349 views

### Non congruence subgroups containing congruence subgroups.

Does there exist Fuchsian groups, which is not conjugated in $SL(2, \mathbb{R})$ to a subgroup of $SL(2, \mathbb{Z})$, but still contains a congruence subgroup?

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862 views

### Automorphic Forms on product of groups $G\times H$

Dear all, I have some difficulty in understanding the notion of automorphic forms on product of groups.
Let $G$, $H$ be two reductive groups defined over a number field
$F$. Let $\mathcal{A}(G)$ be ...

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220 views

### Generating function related to 2-residues of partitions

Question
Express the following power series in two commuting variables $x,y$ as an infinite product, or a short sum of infinite products:
$$
...

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### Square integrable functions on $\Gamma \backslash G$

I am trying to understand proposition 2.1.6 in Bump's book Automorphic forms and Representations.
Let $G=GL(2,\mathbb{R})^+$ and define $G_1=G/Z^+$, where $Z^+$ denotes the center, and define ...

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394 views

### Growth of smallest closed geodesic in congruence subgroups?

Let $\Gamma$ be one of the classical congruence subgroups $\Gamma_0(N)$, $\Gamma_1(N)$ and $\Gamma(N)$ of $SL(2, \mathbb{Z})$.
How does the lower bound for the length of primitive geodesics on ...

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750 views

### The historical development of automorphic geometry

Background:
Today the notion of automorphic geometry is often framed in the context of the Langlands program, in particular what is sometimes called the Langlands reciprocity conjecture. This is ...

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569 views

### Elliptic orbital integral

Let $F$ be a local field and $G = GL(n,F)$. Let $f$ be an element $C_c^\infty(G)$.
Let $\gamma$ be an elliptic element of $G$ with irreducible characteristic polynomial.
What are strategies to ...