Questions tagged [langlands-conjectures]
Higher reciprocity laws
368
questions
22
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1
answer
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What is the $p$-adic Langlands conjecture for $\mathbf{GL}_1$?
In the Boston conference on Fermat's Last Theorem (Summer 1995), Barry Mazur said (around 15m into the video) about class field theory that
If you are a number-theorist and you want to cheer ...
23
votes
1
answer
2k
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Any open Langlands Conjectures for GL_1?
Are there any general conjectures/properties (in the Langlands Program) for automorphic representations of $GL_n$ which are still open for $n=1$?
3
votes
2
answers
319
views
Poles of Rankin-Selberg $L(s,\pi\times\tilde \pi)$?
Let $F$ be a number field and let $\pi$ be a cuspidal automorphic representation of $GL_n(\mathbb{A_F})$.
Do we know that $L(s,\pi\times \tilde\pi)$ has a simple pole at $s=1$?
Do we know that $\...
5
votes
1
answer
503
views
Local Langlands Conjecture for p-adic SO(4), reference request
In section 10 of Gan-Gross-Prasad's paper "Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups" http://arxiv.org/pdf/...
9
votes
2
answers
2k
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Langlands program vs Shimura-Taniyama-Weil conjecture
Edward Frenkel said that "we can see Langlands program as a generalization of Shimura-Taniyama-Weil conjecture in the case of elliptic curves"
I hope I'm not distorting his phrase, can someone ...
22
votes
1
answer
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What do Hecke eigensheaves actually look like?
Let $\mathbb F_q$ be a finite field, $C$ a curve over $\mathbb F_q$ of genus $g\geq 2$, $\rho: \pi_1(C) \to GL_2(\overline{\mathbb Q}_\ell)$ an irreducible local system. The geometric Langlands ...
-1
votes
1
answer
733
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Consequences of Langlands functoriality conjecture
I would like to know whether Langlands' functoriality conjecture implies that the Selberg class coincides with the class of automorphic L-functions and, if so, whether this class is closed under ...
7
votes
1
answer
538
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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 ...
43
votes
1
answer
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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 ...
2
votes
0
answers
247
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 $\mathbb{Q}$...
43
votes
7
answers
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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 ...
16
votes
1
answer
2k
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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
0
answers
282
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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 ...
8
votes
1
answer
456
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 ...
28
votes
2
answers
4k
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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 representation-...
3
votes
2
answers
555
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 ...
18
votes
1
answer
538
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) ...
15
votes
2
answers
956
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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 there ...
16
votes
1
answer
2k
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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 ...
4
votes
1
answer
308
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 \...
5
votes
1
answer
424
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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
247
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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
411
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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 $B(...
2
votes
1
answer
387
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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, ...
9
votes
0
answers
423
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 ...
1
vote
0
answers
436
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, ...
2
votes
1
answer
151
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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 ...
2
votes
0
answers
268
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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 ...
9
votes
1
answer
813
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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 $...
2
votes
1
answer
231
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 ...
6
votes
1
answer
1k
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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 ...
7
votes
0
answers
481
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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
2k
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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 forms....
9
votes
0
answers
395
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
$$L(s, \pi, Sym^m)$...
9
votes
1
answer
695
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Self-dual automorphic forms on $GL(4)$
As is known among experts, all self-dual automorphic forms on $GL(3)$ come from symmetric square lifts from $GL(2)$. You can find this in Ramakrishnan (http://www.math.caltech.edu/~dinakar/papers/...
5
votes
1
answer
323
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looking for reference on dihedral, tetrahedral, or octahedral forms
I am looking for a reference on dihedral, tetrahedral, or octahedral forms. As far as I read, they are some cuspidal automorphic forms on $GL(2)$ induced from $GL(1)$. Dihedral is from $GL(1)/K$ to $...
19
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2
answers
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Are there any simple, interesting consequences to motivate the local Langlands correspondence?
Let's pretend that we know local Langlands at a fairly high level of generality... i.e. we know something along the lines of:
Let $G=\mathbf{G}(F)$ be the group of $F$-points of a connected reductive ...
14
votes
1
answer
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Is the adjoint L-function on GL(m) holomorphic?
Let $\pi$ be an automorphic representation on $\mathrm{GL}(m)/\mathbb{Q}$.
Define $$L(s,\pi,\mathrm{Ad}):=\frac{L(s,\pi\times\overline{\pi})}{\zeta(s)}.$$ This is an $L$-function with Euler product of ...
8
votes
1
answer
625
views
Absolute convergence of Rankin–Selberg series
Let $\pi$ and $\pi'$ be two general automorphic representations on $\operatorname{GL}(n)$ and $\operatorname{GL}(n')$ over $\mathbb{Q}$.
I heard that the Rankin-Selberg $L$-function $L(s,\pi\times\pi')...
12
votes
2
answers
756
views
Langlands' original observation about Ramanujan conjecture
Obviously functoriality of arbitrary high symmetric power lifts of automorphic forms on GL(2) will lead to the Ramanujan conjecture. But I guess that is too strong for Ramanujan. I came across some ...
9
votes
1
answer
513
views
standard zero free region of automorphic L-function on GL(N)
Let $L(s,\pi)$ be the standard(Godement-Jacquet) $L$-function of $\pi$, where $\pi$ is a cuspidal automorphic represetation of $GL(m,A_Q)$.
What's the standard zero-free region for $L(s,\pi)$? any ...
0
votes
1
answer
807
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Euler product of Asai L-function?
Let $\pi$ be an automorphic form of GL(n)/$\mathbb{Q}$ with standard $L$-function
$$L(s,\pi)=\prod_p \prod_{i=1}^n(1-\frac{\alpha_{p,i}}{p^s})^{-1},$$
where $\{\alpha_{p,i}:i=1,\dots,n\}$ are the ...
3
votes
1
answer
890
views
GL(2) Local Langlands and Artin's L-function
The context I am thinking of mainly is GL(2), and accordingly, the degree 2 Artin L-function. But comments about GL(n) in general are also welcome.
In light the local Langlands correspondence, what ...
3
votes
0
answers
211
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Functoriality for triple product GL(2) x GL(2) x GL(2)
Let $f$, $g$ and $h$ be three general automorphic forms on $\operatorname{GL}(2)$.
Do we know that $L(s, f\times g\times h)$ comes from an automorphic form on $\operatorname{GL}(8)$?
3
votes
2
answers
575
views
Which Weil group over a $p$-adic field?
For simplicity, call the Weil group of a local nonarchimedean field $F_v$ to be the following extension:
$$1\longrightarrow F^\times_v\longrightarrow W_{F_v}\longrightarrow\text{Gal}(F_v/\mathbb Q_v)\...
14
votes
2
answers
826
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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 ...
14
votes
3
answers
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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 $\mathbb{G}(...
19
votes
3
answers
2k
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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 ...
40
votes
2
answers
7k
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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 ...
15
votes
1
answer
986
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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 2-...