Higher reciprocity laws

**13**

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**3**answers

812 views

### Number theory and physics

I was following some lectures by Edward Frenkel about Langlands correspondence. He was describing some analogies between number theory and theoretical physics (Mirror symmetry). At some point ( my ...

**7**

votes

**1**answer

408 views

### Lindelof Hypothesis implying Selberg Eigenvalue Conjecture?

The Generalized Lindelof Hypothesis says that for the $L$-function of an automorphic form we have
$$L(1/2+it)\ll Q(t)^{\epsilon}$$
for any $\epsilon>0$ where $Q(t)$ is the conductor of $L(s)$ at ...

**4**

votes

**1**answer

223 views

### Is the twisted symmetric fifth power $L$-function holomorphic?

Let $\pi$ be a Maass cusp form for SL($2,\mathbb Z$). Let $\omega$ be a primitive Dirichlet character.
Let us consider the $L-$ function
$$L(s,Sym^5 \pi \times \omega)$$ or $L(s,Sym^6 \pi \times ...

**10**

votes

**1**answer

573 views

### Relation between Weil Conjecture and Langlands Program

Recently I read Gelbart's An Elementary Introduction To The Langlands Program, which explained the origin of the program, and this question came to me. For an elliptic curve over finite field, the ...

**13**

votes

**2**answers

576 views

### The status of automorphic induction

Background: Let $K/F$ be a degree $r$ extension of number fields. It is conjectured that an automorphic representation of GL$_n$ associated to $K$ induces an automorphic representation of GL$_{rn}$ ...

**56**

votes

**7**answers

5k 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 ...

**9**

votes

**0**answers

226 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 ...

**10**

votes

**0**answers

156 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 ...

**6**

votes

**1**answer

195 views

### Simplest case of Langlands-Shahidi method

I would like to read the simplest examples of Langlands-Shahidi method carried out to prove the functional equation of $L$-function.
Could the constant term of $\mathrm{GL}(2)$-Eisenstein series be ...

**2**

votes

**1**answer

292 views

### Two questions about Whittaker functions

I am watching the video: Modeling p-adic Whittaker functions, Part I. I have two questions about Whittaker functions in the video.
From 33:00 to 37:00, it is said that after changing of variables, ...

**23**

votes

**2**answers

2k views

### Mochizuki's “phenomena in number theory” outside the scope of Langlands

(Crossposted from math.stackexchange by suggestion)
On page 12 of Shinichi Mochizuki's "On the Verification of Inter-universal Teichmuller Theory: A Progress Repor", he writes
"The ...

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vote

**2**answers

193 views

### Generic irreducibility of parabolic induction

In J.Bernstein's notes: REPRESENTATION OF P-ADIC GROUPS, he remarked the following result(see P.88):
Let $G$ be a reductive group defined over nonarchimedean local field $F$, $P$ parabolic subgroup of ...

**12**

votes

**1**answer

244 views

### To what extent are modular parametrizations expected to generalize?

By the Modularity Theorem (a.k.a. the Shimura--Taniyama--Weil Conjecture), if $E$ is an elliptic curve over $\textbf{Q}$ with conductor $N$, then there exists a “modular parametrization” $\psi: X_0(N) ...

**12**

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**2**answers

536 views

### Explicit Chebotarev and Langlands - irreducibility of X^5-X-1 mod primes

Is there an explicit infinite set of primes, modulo which $X^5 - X - 1$ is irreducible?
Since our polynomial's Galois group over $\mathbb{Q}$ is $S_5$, Chebotarev's density theorem implies that ...

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votes

**1**answer

154 views

### The infinity-type of automorphic representations in the Langlands correspondence

Let $K$ be a number field, $\rho\colon \mathrm{Gal}_K\to \mathrm{GL}_n(\overline{\mathbf{Q}_p})$ a geometric (i.e.: unramified a.e., de Rham above $p$) irreducible Galois representation. One piece of ...

**0**

votes

**1**answer

129 views

### Clifford-Mackey theory, references

I am working on a problem related to the local Langlands correspondence and I am interested in certain smooth representations of locally profinite groups (in particular of the Weil group of a local ...

**2**

votes

**1**answer

219 views

### A computation about Whittaker functions and Eisenstein series

I have some questions about the computation of Eisenstein series and Whittaker functions in the book. The question is on page 29, Theorem 4.3.
My questions are in the following.
(1) I think that ...

**6**

votes

**0**answers

173 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 ...

**2**

votes

**0**answers

327 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, ...

**0**

votes

**1**answer

95 views

### Weil group of a local field, small notational problem

In Bushnell and Henniart, The Local Langlands conjecture for GL(2), there is a proposition on p. 184 in which they prove the following:
Let $F$ be a non-archimedean local field, $\mathcal W_F$ its ...

**13**

votes

**1**answer

742 views

### Where did (or will) appear: “Funktorialität in der Theorie der automorphen Formen” by Langlands?

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 ...

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votes

**1**answer

219 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 ...

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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 ...

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votes

**2**answers

722 views

### What's the status of Arthur's announced classification for GSp(4)?

In "Automorphic representations of GSp(4)" (2004) (see http://www.math.toronto.edu/arthur/), James Arthur announces a classification of discrete automorphic representations of GSp(4). There are no ...

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votes

**1**answer

340 views

### Self-dual automorphic forms on $GL(4)$

As is known among experts, all self-dual automorphic forms on $GL(3)$ come from symmetric square lifts from $GL(2)$. You can find this in Ramakrishnan ...

**2**

votes

**0**answers

153 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 ...

**22**

votes

**2**answers

1k views

### Have we ever proved any non-solvable case of reciprocity without the Langlands program ?

The reciprocity of the title is the following not completely well-posed problem:
Fix $P(X)$ a monic irreducible polynomial of degree $n$, with coefficients in $\mathbb Z$. "Describe"
(in some sense) ...

**2**

votes

**1**answer

125 views

### A certain idele class character

Let $E/K$ be a cubic extension of number fields, $\nu$ be a Grossencharacter of the idele class group $\mathbb{I}_{E}/E^{\ast}$ such that $\nu^2$ is trivial and $\nu$ restricted to the idele class ...

**3**

votes

**1**answer

291 views

### Arthur's refinement of parameters for unitary automorphic representations

In his work on the classification of automorphic representations of a group $G$, Arthur has conjectured that the parameterization of such representations involves a homomorphism $\rho : SL_2 \times ...

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votes

**2**answers

717 views

### Are there any simple, interesting consequences to motivate the local Langlands correspondence?

Let's pretend that we know local Langlands at a fairly high level of generality... i.e. we know something along the lines of:
Let $G=\mathbf{G}(F)$ be the group of $F$-points of a connected ...

**18**

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 ...

**6**

votes

**0**answers

200 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 ...

**5**

votes

**1**answer

523 views

### Serious introduction to the Langlands program for nonspecialist

I recently became interested in the Langlands program and hope to learn more.
For context, I am an analytic number theorist but have some light background in algebraic number theory and modular ...

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votes

**1**answer

479 views

### Request for errata for Automorphic Forms on GL(2)

Edit (7/21/2014): We have finished proofreading Jacquet-Langlands and posted it to Robert Langlands's publications site. If you would like a copy, please download it from here:
...

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**0**answers

242 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, ...

**2**

votes

**1**answer

474 views

### GL(2) Local Langlands and Artin's L-function

The context I am thinking of mainly is GL(2), and accordingly, the degree 2 Artin L-function. But comments about GL(n) in general are also welcome.
In light the local Langlands correspondence, what ...

**4**

votes

**1**answer

206 views

### looking for reference on dihedral, tetrahedral, or octahedral forms

I am looking for a reference on dihedral, tetrahedral, or octahedral forms. As far as I read, they are some cuspidal automorphic forms on $GL(2)$ induced from $GL(1)$. Dihedral is from $GL(1)/K$ to ...

**9**

votes

**1**answer

295 views

### Is the adjoint L-function on GL(m) holomorphic?

Let $\pi$ be an automorphic representation on GL($m$)/$\mathbb{Q}$.
Define $$L(s,\pi,Ad):=\frac{L(s,\pi\times\overline{\pi})}{\zeta(s)}.$$ This is an L-function with euler products of degree $m^2-1$.
...

**5**

votes

**1**answer

244 views

### absolute convergence of Rankin-Selberg series

Let $\pi$ and $\pi'$ be two general automorphic representation on $GL(n)$ and $GL(n')$ over $\mathbb{Q}$.
I heard that the rankin-selberg convolution L-function $L(s,\pi\times\pi')$ is absolutely ...

**11**

votes

**2**answers

454 views

### Langlands' original observation about Ramanujan conjecture

Obviously functoriality of arbitrary high symmetric power lifts of automorphic forms on GL(2) will lead to the Ramanujan conjecture. But I guess that is too strong for Ramanujan. I came across some ...

**7**

votes

**1**answer

254 views

### standard zero free region of automorphic L-function on GL(N)

Let $L(s,\pi)$ be the standard(Godement-Jacquet) $L$-function of $\pi$, where $\pi$ is a cuspidal automorphic represetation of $GL(m,A_Q)$.
What's the standard zero-free region for $L(s,\pi)$? any ...

**0**

votes

**1**answer

214 views

### Euler product of Asai L-function?

Let $\pi$ be an automorphic form of GL(n)/$\mathbb{Q}$ with standard $L$-function
$$L(s,\pi)=\prod_p \prod_{i=1}^n(1-\frac{\alpha_{p,i}}{p^s})^{-1},$$
where $\{\alpha_{p,i}:i=1,\dots,n\}$ are the ...

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votes

**2**answers

639 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 ...

**14**

votes

**1**answer

797 views

### What is the Twisted Trace Formula?

I am studying the trace formula using "An Introduction to the Trace Formula" by James Arthur. I would like to understand the twisted trace formula, but unfortunately I never came across a good ...

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votes

**0**answers

118 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)?

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votes

**2**answers

245 views

### Which Weil group over a $p$-adic field?

For simplicity, call the Weil group of a local nonarchimedean field $F_v$ to be the following extension:
$$1\longrightarrow F^\times_v\longrightarrow W_{F_v}\longrightarrow\text{Gal}(F_v/\mathbb ...

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votes

**2**answers

486 views

### References for particular topics related to Langlands

I have never really concentrated on Langlands, which explains my poor level of understanding of it. But I have read quite a few introductory papers related to Langlands, and to the circle of ideas ...

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votes

**3**answers

657 views

### What is the intuition behind the definition of cuspidal representations?

Let $\mathbb{G}$ be a reductive group defined over a number field $K$, let $Z$ be its center, and let $\mathbb{A}:=\mathbb{A}_K$ be the ring of adeles of $K$. Reasonably, we care about the ...

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votes

**2**answers

2k views

### Current Status on Langlands Program

The Langlands Program was launched almost fifty years ago, and progress has been made gradually, much of it hard earned. Langlands himself wrote a survey on the functoriality conjecture in 1997, Where ...

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votes

**1**answer

498 views

### Local Langlands for $GL(2,\mathbf{C})$ and reducible principal series

My naive picture of the local Langlands correspondence for $GL(2,\mathbf{C})$ is this. The Weil group of $\mathbf{C}$ is canonically $\mathbf{C}^\times$. On the Galois side then we're looking at ...