Higher reciprocity laws

**5**

votes

**1**answer

301 views

### Local Langlands correspondence and Galois equivariance

The local Langlands correspondence $\text{rec}$ for $\text{GL}_{n}$ itself is not Galois equivariant (i.e. invariant under automorphisms of its field of definition) but rather its twist by ...

**35**

votes

**0**answers

877 views

### What is the status of Arthur's book?

Arthur's long-awaited book project is now published (The endoscopic classification of representations: orthogonal and symplectic groups). However, in the book he takes some things for granted:
The ...

**12**

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. ...

**10**

votes

**0**answers

168 views

### Propagation of modularity and the Artin conjecture

The (still incomplete) solution of the Artin conjecture on dimension $\leq2$ has been a massive research effort that has spanned (knowingly or not) around a century.
A very natural question is, what ...

**10**

votes

**0**answers

306 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 ...

**9**

votes

**0**answers

251 views

### Automorphic factorization of Dedekind zeta functions

It is well known that for abelian number fields, the factorization of its Dedekind zeta function goes like this:
$$\zeta_K(s)=\zeta(s)\prod_{\chi \neq 1} L(s,\chi)$$
with the Dirichlet characters ...

**9**

votes

**0**answers

439 views

### Langlands program beyond CM fields?

I apologize since this is a quite vague question. And I am personally at an expert in these fields at all.
It seems to me that there are two main directions of the Langlands program, namely, ...

**8**

votes

**0**answers

250 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, ...

**8**

votes

**0**answers

499 views

### questions on Deligne's letter to Piatetski-Shapiro

Now, that the letter is to be found here: http://www.math.ias.edu/~jaredw/DeligneLetterToPiatetskiShapiro.pdf
,I'd have a couple of questions concerning it. Actually, I have a problem only with the ...

**8**

votes

**0**answers

497 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 ...

**7**

votes

**0**answers

413 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 $G^\vee$ ...

**6**

votes

**0**answers

175 views

### Rankin-Selberg for Maass form GL(3)xGL(2)

Let $F$ be a Maass cusp form for $\mathrm{SL}(3,\mathbb{Z})$ (level 1 trivial character).
Let $g$ be a Maass cusp form for $\Gamma_0(N)$ with character $\chi$ mod $N$. For convenience, you may assume ...

**6**

votes

**0**answers

209 views

### mod $p$ Jacquet-Langlands correspondence

Let $F$ be a local field of characteristic $0$. Let $D$ be division algebra over $F$ of dimension $n^2$. The construction of irreducible complex representations of $D^*$ is known by Howe, Zink, and ...

**6**

votes

**0**answers

304 views

### What can we say about the Local Langlands Correspondence for GL_n without using Bernstein-Zelevinski?

I have two specific questions regarding the LLC for $GL_n$, and in particular, what we can say about the conjecture if we don't have the ideas of Bernstein and Zelevinski, which reduce the problem to ...

**5**

votes

**0**answers

375 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\; ...

**5**

votes

**0**answers

518 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 ...

**4**

votes

**0**answers

247 views

### Why Whittaker functions are useful?

Whittaker functions appears in Langlands program. Recently, it is shown that some Whittaker functions can be obtained by integrating a function related to decoration over a geometric crystal in ...

**4**

votes

**0**answers

126 views

### parametrization of irreducible finite dimensional representation of Weil group

Let $F$ be a p-adic field, with p a prime denoting the residue field characteristic. Let $\mathcal{W}_F$ be the Weil group. In the local Langlands correspondence for $GL(n,F)$, it is important to know ...

**4**

votes

**0**answers

141 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

**0**answers

128 views

### Functoriality for triple product GL(2) x GL(2) x GL(2)

Let $f$, $g$ and $h$ be three general automorphic forms on GL(2).
Do we know that $L(s, f\times g\times h)$ comes from an automorphic form on GL(8)?

**3**

votes

**0**answers

132 views

### Do local L-functions/epsilon factors vary continuously with the Fell topology?

Edit due to the comment.
Consider $G=GL(2)$ over a local field $F$. The Fell topology on the unitary dual of $G(F)$ is seperable.
Given a sequence of irreducible unitary representations $(\pi_n)$ of ...

**3**

votes

**0**answers

253 views

### The operator \boxtimes and \boxplus in automorphic representations

Given two automorphic representations $\pi_1, \pi_2$ of $GL_2(\mathbb A_Q)$ and $GL_3(\mathbb A_Q)$ respectively. Let $\pi_i =\otimes_v \pi_{i, v}$.
Now, for each $v$, let $\pi_{1, v}\boxtimes ...

**3**

votes

**0**answers

369 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 $\mathbb{F}_p$. ...

**3**

votes

**0**answers

333 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 ...

**2**

votes

**0**answers

207 views

### algebraic representation over $\mathbb{C}$

In reading the Harris-Taylor book, I encounter expressions like "Let $\xi$ be an algebraic representation of $G$ over $\mathbb{C}$". What does this mean? Here $G$ is a reductive group over ...

**2**

votes

**0**answers

158 views

### Functoriality for non-split orthogonal groups

I am trying to understand the functoriality conjectures of Langlands. We know that the functoriality conjectures imply that automorphic $L$-functions of a connected reductive group are equal to ...

**2**

votes

**0**answers

209 views

### Automorphicity of L-Factors of Zeta Functions

Associated to a variety over a number field $K$, one has a family of ``Hasse--Weil'' L-functions, which can be combined (as an alternating product) to give the Hasse--Weil zeta function of the ...

**2**

votes

**0**answers

176 views

### Conceptual reason behind Shimura lifts

Shimura lifts are correspondence between integer weight and half-integral weight automorphic forms. Half integral weight things are associated to representation of a double cover of $G ...

**2**

votes

**0**answers

267 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

**0**answers

386 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).

**2**

votes

**0**answers

186 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 ...

**2**

votes

**0**answers

225 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 ...

**1**

vote

**0**answers

339 views

### Langlands reciprocity for C*-algebras

I just came across this paper which, judging by what I understood, establishes the Langlands reciprocity conjecture for a certain Shimura variety. My question, regardless of the validity of the proof, ...

**1**

vote

**0**answers

163 views

### Langlands correspondence for reducible representations

The Langlands correspondence over a function field matches irreducible $n$-dimensional Galois representations with cuspidal irreducible automorphic representations.
My question is: Is there any idea ...

**1**

vote

**0**answers

341 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 ...

**0**

votes

**0**answers

377 views

### Local Langlands conjecture for GL(2)

Let $F_v$ be a local field. Let $\sigma_v$ be a two-dimensional representation of $Gal(\overline{F}_v:F_v) \rtimes W_{F_v}$. Now, there exists an infinite-dimensional representation $\pi_v$ of ...