Higher reciprocity laws

**4**

votes

**2**answers

283 views

### vanishing of spectral term in Arthur-Selberg trace formula for GL(2)?

Hi,
In the Arthur-Selberg trace formula for $G = GL(2)/\mathbf Q$ (as seen for example in
Gelbart's "Lectures on the Trace Formula"), the spectral side includes terms
like:
$$
\int_{-\infty}^\infty ...

**17**

votes

**2**answers

1k 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 ...

**19**

votes

**1**answer

512 views

### Décomposition des nombres premiers dans des extensions non abéliennes

Gauß famously determined the cubic character of $2$ in his Disquisitiones : $2$ is a cube modulo a prime number $p\equiv1\mod3$ if and only if $p=x^2+27y^2$ for some $x,y\in\mathbf{Z}$. This implies ...

**2**

votes

**0**answers

241 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 ...

**2**

votes

**2**answers

460 views

### Classification of quasi-split unitary groups

Let $U$ be a unitary group defined with respect to an extension $E/F$ of non-archimedean local fields, and assume it is realised with respect to a pair $(V,q)$, where $V$ is an $n$-dimensional vector ...

**0**

votes

**1**answer

229 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.

**2**

votes

**0**answers

263 views

### Complex Finite Dimensional Representation of GL(N,C)

What are all the complex finite dimensional linear representation of $GL(N,\mathbb{C})$?
We already know all the complex finite dimensional linear representation of SU(N).

**10**

votes

**0**answers

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. ...

**4**

votes

**0**answers

121 views

### Is there an arithmetic analogue of Drinfeld's count of a number of 2d irreps of fundamental group of a curve ?

There is a paper by V. Drinfeld 1981, which title is "Number of two-dimensional irreducible representations of the fundamental group of a curve over a finite field".
It gives a formula for this ...

**3**

votes

**1**answer

437 views

### What do orbital integrals have to do with reciprocity?

Hi, this is my first question (of many). I am blogging for the Fields Medal Symposium and would like to get into the mathematics involved with our program.
In an attempt to sort through the articles ...

**9**

votes

**0**answers

453 views

### Funktorialität in der Theorie der automorphen Formen

In 2010 Langlands wrote an article with the title Funktorialität in der Theorie der automorphen Formen: Ihre Entdeckung und ihre Ziele. On the IAS website, he says that
This note ... was ...

**3**

votes

**0**answers

316 views

### local deformation rings and Hecke algebras

Let $\bar{\rho}_p$ be a two-dimensional irreducible local Galois representation of $Gal(\bar{\mathbb{Q}}_p / \mathbb{Q}_p)$ on a $k$-vector space, where $k$ is a finite extension of $\mathbb{F}_p$. ...

**10**

votes

**0**answers

266 views

### Is there a theory of Maaß forms over finite fields ?

Here is a somewhat naïve question which must have occurred to many people, so it would be nice to record here the attempts at an answer :
Is there a theory of Maaß forms over ...

**13**

votes

**1**answer

581 views

### The simplest even Artin representations of degree 2 and the corresponding Maaß forms

What are the simplest numerical examples of even dihedral (resp. tetrahedral, resp. octahedral, resp. icosahedral) representations
...

**10**

votes

**1**answer

507 views

### The first odd degree-2 Artin representation for which the Artin conjecture was proved

At the DeKalb conference on Hilbert's problems, John Tate gave a masterly survey of Problem 9, the General Reciprocity Law. He ends with a discussion of the Langlands Programme, especially the case ...

**4**

votes

**1**answer

282 views

### semisimplicity of automorphic Galois representations

Is it known that the Galois representation constructed by Harris and Taylor in their book is semisimple? I can't see this proven in the book, but on the other hand, everywhere else the representation ...

**2**

votes

**1**answer

424 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

**1**answer

220 views

### Representation theory of G1 versus G/Z

Let $G$ be an locally compact group $G$, then every irreucible representations $\pi$ is isomorphic to $\omega_{\pi} \otimes \pi'$, where $\omega_{\pi}$ is the central character of $\pi$ and $\pi'$ an ...

**2**

votes

**0**answers

166 views

### CM abelian variety from an algebraic Hecke character?

Hi,
Given an algebraic Hecke character $\chi$ of a number field $k$ there should be a
"rank 1 CM-motive" $M$ with
$\overline Q$-coefficients such that $L(s,M) = L(s,\chi)$. This follows from the ...

**12**

votes

**1**answer

972 views

### Galois representations attached to Maass form

So, how does one construct a galois representation from a Maass form?
For a modular cusp eigenform, I am familiar with the work of Eichler-Shimura, Deligne, Deligne-Serre, and realize these are ...

**28**

votes

**1**answer

1k views

### How would Hilbert and Weber think about the Langlands programme?

Explanations to a general mathematical audience about the Langlands programme often advertise it as "non-abelian class field theory". They usually begin as follows: a modern style formulation of ...

**3**

votes

**2**answers

479 views

### Representations of GL(2, Q_p) and GL(2, Z_p)

The cuspidal representations of $GL_n(F)$ a non archimedean field $F$ with ring of integers $o$ can be classified by inducing irreducible representation from $Z GL_n(o)$.
The general question:
...

**2**

votes

**0**answers

200 views

### Local counterpart of the NON-Hitchin Hecke eigen-sheaves ?

Insight of Beilinson and Drinfeld at early 90-ies - that Hitchin's D-modules are Hecke eigen-D-modules. However they are NOT all Hecke-eigensmodules and actually they are only the half-dimensional ...

**11**

votes

**1**answer

1k views

### What is the relation between L. Lafforgue and Frenkel-Gaitsgory-Vilonen results on Langlands correspondence ?

What is the relation between Lafforgue's result on Langlands
and Frenkel-Gaitsgory-Vilonen ? ( http://arxiv.org/abs/math/0012255 , http://arxiv.org/abs/math/0204081 )
Does one imply other ? If not ...

**2**

votes

**3**answers

863 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

**3**answers

752 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

**2**answers

572 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 ...

**5**

votes

**0**answers

482 views

### The Shafarevich Conjecture and motivic Langlands stacks.

Hi, I recently learned about an amazing conjecture of Shafarevich
(proved by Faltings) about the finiteness of the number
of curves of a fixed genus with good reduction outside a
finite number of ...

**6**

votes

**1**answer

530 views

### Followup questions about the relationship between modular forms and motives

It occurs more and more that I ask a question on math stackexchange and then realize that it is more appropriate to mathoverflow. Hopefully this reflects well on myself... In any case, I copy here ...

**8**

votes

**1**answer

811 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) = ...

**1**

vote

**0**answers

335 views

### General cohomology groups and motives

Let $X$ be a variety over $\mathbb{Q}$. Let $\mathcal{F}$ be a sheaf on $X$. Then we have an action of $Gal(\mathbb{Q})$ on $H_{et}^i(X,\mathcal{F})$. In certain cases we can say a lot about this ...

**2**

votes

**1**answer

741 views

### In what way do the Weil Conjectures pertain to Langlands?

For a relative variety $X$ over a ring of integers $O_K$, we can define a zeta function. This zeta function is defined as the product of the zeta functions of the variety specialized to ...

**11**

votes

**2**answers

609 views

### Insolvable number fields ramified only at one (small) prime

In his first Eilenberg Lecture at Columbia, Benedict Gross says that only recently have we been able to give examples of finite galoisian extensions $K$ of ${\bf Q}$ which are ramified only at $2$ ...

**7**

votes

**2**answers

2k views

### Consequences of the Langlands program

In the one-dimensional case the Langlands program is equivalent to the class field theory and the two-dimensional case implies the Taniyama-Shimura conjecture.
I would like to know are there any ...

**2**

votes

**1**answer

480 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

**1**answer

457 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 ...

**13**

votes

**5**answers

2k views

### What is the “reason” for modularity results?

The question is a little wishy-washy, but I take my cues from other popular questions that relate to the philosophy behind the mathematics as Why do Groups and Abelian Groups feel so different? .
I ...

**5**

votes

**2**answers

541 views

### Is there for every variety X an abelian variety A such that their 1st l-adic cohomologies are isomorphic?

This question is somewhat inspired by Kevin Buzzard's answer to What is the interpretation of complex multiplication in terms of Langlands? and somewhat from my own curiosity about such topics.
Let ...

**3**

votes

**1**answer

694 views

### What is the interpretation of complex multiplication in terms of Langlands?

I'm trying to interpret things in the following terminology:
Assume the standard conjectures, the existence of the conjectural Langlands group, and anything else you wish.
I assume the following ...

**3**

votes

**0**answers

160 views

### Finite field analogue of representations in same packet have equal central character

In Kevin Buzzard's recent question, a warm up question was: if two automorphic representations are nearly equivalent, then are the central characters of their local components equal?
Working my way ...

**7**

votes

**1**answer

632 views

### How does the conjectural Langlands group fit into the Tannakian point of view?

I've read that one way to formulate the Langlands program is the following:
Let $\mathcal{L}_ {\mathbb{Q}}$ be the conjectural Langlands group. Then the category of semi-simple (continuous) ...

**16**

votes

**4**answers

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 ...

**11**

votes

**3**answers

2k views

### What makes Langlands for n=2 easier than Langlands for n>2?

I must confess a priori that I haven't read the proof of Taniyama-Shimura, and that my familiarity with Langlands is at best tangential.
As I understand it Langlands for $n=1$ is class field theory. ...

**2**

votes

**2**answers

561 views

### Are the $L$-functions of $X_0(N)$ automorphic?

This question, like all of my previous questions regarding Langlands, is very naive.
All $g\geq 1$ curves come from quotients of the upper half plane. The curves $X_0(N)$ come from quotients of ...

**3**

votes

**0**answers

312 views

### Motivic interpretation of genus 2 Siegel forms induced by lifts of Maass and Skoruppa

Background: There are several known lifts from integral weight modular forms to Siegel forms of genus 2, among them the Saito-Kurokawa lift. Another lift construction that is important for ...

**12**

votes

**2**answers

1k views

### What is the strongest, most natural, conjectural form of Langlands?

This is inspired by my previous question:
What is the precise relationship between Langlands and Tannakian formalism?
As well as the excellent link that Tom Leinster put in a comment to that thread: ...

**13**

votes

**1**answer

1k views

### What is the precise relationship between Langlands and Tannakian formalism?

As anyone who's been reading the forums closely can see, I've been averaging a question a day about Tannakian formalism for the past few days. It's quite an interesting concept!
In any case, I wish ...

**4**

votes

**2**answers

517 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 ...

**7**

votes

**0**answers

390 views

### Is endoscopy interesting in simply-laced cases?

Let $G$ be a complex algebraic group, and write $Z(g)$ for the centralizer of a semisimple element $g$ in $G$. I will assume $G$ is simply connected, in which case $Z(g)$ is connected.
Let $G^\vee$ ...

**3**

votes

**1**answer

887 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 ...