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8
votes
1answer
392 views

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 ...
7
votes
1answer
421 views

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 ...
5
votes
1answer
545 views

Special value of $L$-function

Let $p$ be a prime number. Let $f$ be a newform of weight 2 on $Γ_0(p)$, and $E_f$ denote the associated newform quotient of $J_0 (N)$ over $\mathbb{Q}$. Is there a way to express the algebraic part ...
3
votes
0answers
565 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 ...
3
votes
1answer
272 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 ...
7
votes
3answers
927 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 ...
1
vote
1answer
287 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 ...
14
votes
3answers
1k views

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 ...
10
votes
1answer
384 views

Reference for: CM Hilbert Modular forms arise from Hecke characters

For classical modular forms, the correspondence between the form having CM by an imaginary quadratic field $K$ and it being induced from a Hecke character on $K$ is well-known. (Ribet's paper is a ...
2
votes
1answer
252 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 $\pi$, ...
4
votes
2answers
439 views

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 = ...
14
votes
3answers
1k views

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 ...
2
votes
2answers
340 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)?
2
votes
1answer
309 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
1answer
442 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 ...
2
votes
0answers
257 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 ...
9
votes
1answer
302 views

Atkin–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 $\Gamma_0(N)$ ...
5
votes
1answer
275 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 ...
3
votes
2answers
308 views

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 ...
7
votes
1answer
940 views

An interesting double coset in the theory of automorphic forms

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 $F$ , $H=GL_{n-1}$ ...
2
votes
0answers
537 views

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 ...
5
votes
1answer
363 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?
2
votes
3answers
872 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 ...
4
votes
0answers
233 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: $$ ...
1
vote
1answer
429 views

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 ...
11
votes
4answers
403 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 ...
4
votes
3answers
762 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 ...
2
votes
2answers
580 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 ...
3
votes
1answer
384 views

Sums of Kloosterman sums over primes

For $m,n,c\in\mathbb{N}$ let $S(m,n;c)$ be the Kloosterman sum $$S(m,n;c)=\sum_{a=1, \gcd (a,c)=1}^ce\left(\frac{ma+n\overline{a}}{c}\right).$$ The Kuznetsov Trace Formula allows us to obtain bounds ...
9
votes
1answer
825 views

Is this a subcase of the fundamental lemma?

Let $F$ be a local field and $G= GL(n,F)$. Assume that $\gamma$ is an element of $G$ and $G_\gamma$ is its centralizer. The orbital integral is defined as $$ O_\gamma^G( \phi) = ...
12
votes
4answers
1k views

Where do the real analytic Eisenstein series live?

In obtaining the spectral decomposition of $L^2(\Gamma \backslash G)$ where $G=SL_2(\mathbb{R})$, and $\Gamma$ is an arithmetic subgroup (I am satisfied with $\Gamma = SL (2,\mathbb{Z})$) we have a ...
10
votes
2answers
736 views

modular form Fourier coefficients and associated automorphic representation

Hi, Let $f$ be a cuspidal modular form of some weight and level $N$. Then it determines an irreducible automorphic representation $\pi = \bigotimes'\pi_p$ of $GL_2(\mathbf Q)$. Let $f = \sum_i a_i ...
2
votes
1answer
490 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 ...
3
votes
1answer
492 views

classification of irreducible admissible (g,K)-module for GL(3,R)

classification of irreducible admissible (g,K)-module for GL(3,R) Is there a classification of irreducible admissible (g,K)-module for GL(3,R)? For GL(2,R) we have principal series, discrete series ...
2
votes
2answers
492 views

What is the relationship between (g,K)-module and Maass forms?

What is the relationship between (g,K)-module and Maass forms for GL(2)? (g,K)-modules are defined in chapter 2 of Bump, Automorphic forms and representations. There is a classification of ...
7
votes
2answers
1k views

What is the non-motivic motivation behind automorphic representations?

In one of my last questions: What is the "reason" for modularity results? it was pointed out to me that "the notion of automorphic representation developed independently of any concern with ...
11
votes
1answer
1k views

Which Shimura varieties are known to be automorphic?

This seems like something that should be well-known, but as an outsider to the field, I'm having trouble locating precise statements. Hasse-Weil zeta functions of Shimura varieties should be ...
18
votes
4answers
1k views

How badly can strong multiplicity one fail in the theory of automorphic representations?

Let $G$ be a connected reductive group over a global field $k$, and let $\pi=\otimes_w\pi_w$ and $\pi'=\otimes_w\pi'_w$ be two automorphic representations for $G$, where here of course $w$ is ranging ...
3
votes
1answer
406 views

What is the nature of the locus in the eigencurve associated to some conditions on the associated automorphic representation (at $p$)?

I've chatted informally with some folks about this question before and gotten some very nice insights, but I thought I'd toss it out to a wider audience because it is a continuing curiosity of mine. ...
5
votes
2answers
439 views

embedding of local tempered representation into cuspidal automorphic representation

Let v be a finite place of a number field F. Let $\pi_{v}$ be an irreducible tempered representation of $ GL_{n}(F_v)$. Is it true that we can find some irreducible cuspidal automorphic representation ...
2
votes
0answers
1k views

Corvallis 1979 proceedings

These proceedings have long been freely available on the AMS website, but now it seems we can't even find them anymore (e.g. http://www.ams.org/publications/online-books/pspum331-index and ...
6
votes
1answer
378 views

embedding of local representation into automorphic representation

Assume $v$ is a place of a number field $k$, finite or not. Let $\pi_v$ be an irreducible admissible generic representation of $GL_n(k_v)$. Is it always true that we can find some irreducible generic ...
6
votes
0answers
253 views

Is the space of global Whittaker functions complete?

Let $f$ be a complex valued function of $GL_n(\mathbb{A})$, where $\mathbb{A}$ is the adeles of some number field. Assume $f(ug)=\psi(u)f(g)$ for any $u$ in the standard maximal unipotent subgroup ...
1
vote
1answer
174 views

non-holomorphic index-raising operator for Jacobi-Maass forms?

Is there a well-defined non-holomorphic index-raising operator $V_l$, taking index-$m$ Jacobi-Maass forms to index-$ml$ Jacobi-Maass forms (same weight), analogous to the holomorphic operator defined ...
7
votes
1answer
431 views

How does the right regular of GL(n, R) and GL(n,Qp) decompose?

The question is contained in the title. I would guess that this question is already answered in the literature. Given the reductive group $GL(n)$ over a complete local field, how does the right ...
4
votes
2answers
556 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 ...
3
votes
1answer
899 views

Jacquet Langlands correspondance

I have one issue with the Jacquet Langlands correspondance. The Weyl law for $H$ modulo a congruence subgroup and the Weyl law for cocompact groups are different. So why does this not contradict this ...
5
votes
1answer
569 views

Base Change for Eigenvarieties

Let $E/F$ be a Galois extension of number fields, and $G$ a reductive group over $F$. If Langlands Base Change is known for $G/F$ and $G/E$, and moreover the eigenvarieties for $G/F$ and $G/E$ have ...
4
votes
0answers
519 views

base change and Langlands' combinatorial exercise

Hi, Is it correct that Langlands' combinatorial exercise (as he terms it in his paper "Shimura varieties and the Selberg trace formula") is to establish base change identities between orbital ...
1
vote
2answers
800 views

Decomposition of Artin L functions

The Dedekind zeta function of an abelian extension $E$ of $\mathbb{Q}$ factors as a product of Artin L function $L(s, \chi)$, where the product runs over all irreducible representations $\chi$ of ...