Questions tagged [langlands-conjectures]
Higher reciprocity laws
367
questions
3
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1
answer
310
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Example of a non-odd motive appearing in cohomology of intermediate degree
I would like to know an example of a projective variety over a totally real field where a complex conjugation is not odd on some of its étale cohomology.
Edit: I am looking for the most interesting ...
3
votes
0
answers
83
views
Recovering a $G$-valued representation/parameter
Number theoretic phrasing
Let $G$ be a connected reductive group over a characteristic $0$ field $F$. Associated to $G$ is its Langlands dual group $^{L}G$. For every dominant cocharacter $\mu$ of $...
2
votes
0
answers
112
views
relative rank two group: structure of parabolic subgroup-- high-level Jacobson--Morozov sl_2 triple
Given a parabolic subgroup $P=MN$ of a connected reductive group $G$ defined over a local field $F$, let $W_M$ be the relative Weyl group of $M$ in $G$, assume that the reduced roots relative to $M$ ...
5
votes
0
answers
187
views
Globalizable Galois representations
Let $\rho$ be a $p$-adic representation of $G=\text{Gal}(\bar{\mathbb{Q}_p}/\mathbb{Q}_p)$.
When does $\rho$ extend to a representation of the global galois group? What can be said about the locus ...
10
votes
2
answers
940
views
Why is Langlands functoriality usually related with period integral in a third group?
In the introduction of "PERIODS OF AUTOMORPHIC FORMS "by
HERVE JACQUET, EREZ LAPID, and JONATHAN ROGAWSKI, they said
"In many cases, it should be possible to characterize the $H$-distinguished ...
3
votes
1
answer
318
views
branching laws for $p$-adic representations of reductive groups
There are many papers studying branching laws of irreducible admissible complex representations of classical groups over local fields, are there some analogues for $p$-adic representations?
For ...
3
votes
1
answer
312
views
A detail in Bushnell and Henniart's book, “The local Langlands conjecture for GL(2)”
I am recently troubled with a computational detail in Bushnell and Henniart's book, "The local Langlands conjecture for Gl(2)". Let $(\mathfrak{A},n,\alpha)$ be a simple stratum, and define $K_\...
2
votes
1
answer
278
views
A problem in Bushnell and Henniart's book, "The local Langlands conjecture for GL(2)"
On page 123 of Chapter 5 in Bushnell and Henniart's book The Local Langlands Conjecture for GL(2), they state
an elementary property of tamely ramified extension of local fields, which is as follows,
...
6
votes
1
answer
843
views
Langlands Reciprocity and Fermat's Last Theorem
Question:
Can Langlands Reciprocity be used to prove Fermat's Last Theorem?
Background
A few years ago I was reading a book on the Langlands Program and the introduction provided a list of ...
5
votes
0
answers
165
views
Dependence of X in definition of Shimura variety
(Disclaimer: this question is related to this question, but is different enough that it warrants (in my opinion) a separate question)
Let $G$ be a connected reductive group over $\mathbb{Q}$. To $G$ ...
4
votes
0
answers
86
views
"Generic member" in a nontempered L-packet
It is a standard conjecture that there is a unique generic member in a tempered L-packet. Is there an analogue of this for non-tempered ones, namely does one expect that there is a unique "most ...
4
votes
0
answers
130
views
System of eigenvalues of partial Laplacians and the Jacquet Langlands correspondence
I'm looking for a precise reference. I could not dig it up in the original paper of Jacquet-Langlands.
Let $F$ be a totally real number field with $[F:\mathbb{Q}]=r$, let $B_1=M_2(F)$ be the split ...
1
vote
0
answers
82
views
Connection between global and local notions of a cuspidal representation
Let $k$ be a number field, and $G$ a connected, reductive group over $k$. Let $\omega$ be a unitary character of $Z_G(\mathbb A_k)/Z_G(k)$. An irreducible subspace $(\pi, V)$ of $L^2(G(k) \backslash ...
4
votes
0
answers
97
views
Siegel Levi in $\operatorname{GSpin}(2n+1)$ and image into $\operatorname{SO}(2n+1)$
Let $T$ be a maximal torus of split $\operatorname{SO}_{2n+1}$ with basis $e_1, ... , e_n$. Let $$\Delta = \{e_1 - e_2, ... , e_{n-1} - e_n, e_n\}$$ be a set of simple roots of $T$ corresponding to a ...
6
votes
1
answer
447
views
Cusp forms have an orthonormal basis of eigenfunctions for all Hecke operators
I am reading Langlands' pape Euler Products and have a few questions. Let $G$ be a split adjoint semisimple group over $\mathbb Q$. If $p$ is a place of $\mathbb Q$, finite or infinite, let $G_{\...
3
votes
2
answers
201
views
Definition of cusp form in $L^2$ and convergence over $N_{\mathbb Q} \backslash N_{\mathbb A}$
Let $G$ be an adjoint semisimple group over $\mathbb Q$ with parabolic subgroup $P = MN$ in good position relative to a compact subgroup $U= \prod\limits_v K_v$ of $G(\mathbb A)$. Let $L$ be the ...
11
votes
0
answers
720
views
What is known at $\ell = p$ about realizing Jacquet-Langlands & local Langlands as the cohomology of Lubin-Tate space with level structure?
Background:
(Mostly my paraphrased interpretation of the introduction of Strauch's Deformation spaces of one-dimensional formal
modules and their cohomology, with additional details from Carayol's ...
9
votes
1
answer
899
views
Why is the Langlands dual group always taken over $\mathbb{C}$?
Whenever I read a statement of the Langlands conjectures for a reductive group $G$, they are formulated in terms of the Langlands dual group, which is essentially the reductive group over $\mathbb{C}$ ...
6
votes
2
answers
438
views
When is compact induction cuspidal?
Let $G=GL_2(\mathbb{Q}_p)$, and let $K$ be a compact-modulo-center subgroup of $G$, $\rho$ an irreducible smooth representation of $K$.
Question 1: Is $\mathrm{ind}_K^G \rho$ cuspidal?
Here ...
5
votes
0
answers
221
views
The number of rational semisimple conjugacy class/the Arthur-Selberg trace formula
I was trying to understand a statement in Theorem 1.5 of this where the author seems to imply that if $G$ is a reductive group over $\mathbb{Q}$ such that $G/Z(G)$ is anisotropic, then for any ...
2
votes
0
answers
80
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Pseudocoefficients and Traces of Standard Representations
Let $G$ be a connected reductive group over $\mathbb{R}$ (you may assume that $G/Z(G)$ is anisotropic if necessary) and suppose $\pi$ is a discrete series representation of $G(\mathbb{R})$ with ...
4
votes
1
answer
622
views
Modifying the Langlands correspondence
I am trying to understand various ways in which one can modify the Langlands correspondence. Hopefully I will be able to learn something from you. First, one can categorify/decategorify.
It is my ...
27
votes
3
answers
5k
views
Why to believe the Fargues geometrization conjecture?
In the study of the arithmetic local Langlands correspondence, there is a conjecture that was recently (in this decade) formulated by Fargues.
I can't even concisely state the conjecture so I will ...
3
votes
1
answer
596
views
What does the Langlands philosophy have to say about the weight and the level?
I have recently attempted to read some number-theoretic texts. Here is an excerpt from a paper by Breuil-Conrad-Diamond-Taylor:
Now consider an elliptic curve $E/\mathbb{Q}$. Let $\rho_{E, l}$ (resp. ...
25
votes
0
answers
1k
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Caramello's theory: applications
In this text, the author says (well, he says it in French, but I am too lazy to fix all the accents, so here is a Google translation):
In any case, contemporary mathematics provides an example of ...
28
votes
3
answers
2k
views
What is a tamely-ramified Weil-Deligne representation?
Let $W_F$ denote the Weil group of a finite extension of $\mathbb{Q}_p$. Let $I$ denote the inertia subgroup and $I^{>0}$ the (pro-$p$) subgroup of wild inertia. (I hope I've got my notation right.....
2
votes
0
answers
174
views
Local-global compatibility and modular curves
I have been told by some people that local-global Langlands compatibility for $GL_2$ (the vanilla version, not the one being developed in this decade by Emerton and others) implies Shimura conjecture ...
3
votes
0
answers
146
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Representations of Galois groups of structured ring spectra and topological automorphic forms
Rognes and Mathew have defined Galois groups for certain structured ring spectra ($E_\infty$-ring spectra and axiomatic stable homotopy theories resp.)
Is a connection expected between ...
6
votes
1
answer
623
views
Is there an English translation of Laumon's proof of geometric Langlands for $\mathbb{G}_m$?
I'd like a detailed proof in English of Laumon's proof that the two Fourier-Mukai transforms taking the derived category of quasicoherent sheaves on $\mathbb{G}_m$-local systems of a curve $X$ to the ...
3
votes
0
answers
152
views
Decomposition of $L^2(\Gamma \backslash H)$ into irreducible representations using the spectral theorem
I'm reading the introduction of An Introduction to the Trace Formula by James Arthur and wanted to understand something in the introduction.
Let $H$ be a unimodular locally compact Hausdorff group, ...
18
votes
0
answers
1k
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Number Theory and Gravity
Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by Robert Langlands at IAS (1967, 1970), it seeks to relate Galois ...
8
votes
1
answer
396
views
L-packets in the local Langlands correspondence: why finite sets?
Let $G$ be a connected, reductive group over a local field $k$, and let $^LG$ be the Langlands dual group. As explained by Borel in his article in the Corvallis proceedings, the general local ...
8
votes
1
answer
417
views
How should the local Langlands correspondence for general reductive groups take into account different inner forms?
Let $G$ be a connected, reductive group over a local field $k$, and let $^LG$ be the Langlands dual group. As explained by Borel in his article in the Corvallis proceedings, the general local ...
2
votes
0
answers
283
views
Question about the Fourier expansion of adelic Eisenstein series for $\operatorname{GL}_2$
My reference is Daniel Bump's book, Automorphic Forms and Representations, Chapter 3.7. Let $k$ be a number field, $G = \operatorname{GL}_2$, $B$ and $T$ the usual Borel subgroup and maximal torus ...
7
votes
0
answers
283
views
What does it mean for a complex valued function on $G(\mathbb A)$ to be smooth (or smooth of compact support)?
Let $G$ be a linear algebraic group over a number field $k$. Let $\mathbb A$ denote the adeles of $k$, $\mathbb A_f$ the finite adeles, and $k_{\infty} = \prod\limits_{v \mid \infty} k_v$. Here are ...
5
votes
0
answers
978
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Some questions about cuspidal representations and automorphic representations
My reference is Daniel Bump's book, Automorphic Forms and Representations. $G$ is a connected reductive group over a number field $k$ (in Bump's book he takes $G = \operatorname{GL}_n$). Let $K = K_{...
4
votes
0
answers
228
views
Carayol's "ramified Eichler-Shimura relation" and its applications
In his paper "Sur la mauvaise reduction des courbes de Shimura" from '86 H. Carayol shows the following congruence relation:
Let $M$ be the tower of Shimura curves over a totally real $F$, associated ...
2
votes
0
answers
146
views
Central character of automorphic representations of $Sp_{2n}$
Let $F$ be a CM field. Given a regular algebraic self-dual cuspidal automorphic representation $\Pi$ of $GL_n(\mathbb A_F)$ and a prime $l$, there is a continuous Galois representation $r_{\Pi}: \...
7
votes
3
answers
584
views
Physical Applications of Locally Symmetric Spaces
Locally Symmetric Spaces are the basis of the Langlands program—a set of ambitious and interconnected conjectures connecting representation theory to number theory, firstly proposed in 1967 by Robert ...
3
votes
1
answer
130
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If $\Pi$ and $\Sigma$ agree at almost all places, then the central character of $\Pi$ corresponds to $\operatorname{Det} \Sigma$
Let $\Sigma$ be an $n$-dimensional representation of the global Weil group $W_F$ for a number field $F$, and $\Pi$ an automorphic representation of $\operatorname{GL}_n(\mathbb A_F)$. Suppose that at ...
16
votes
1
answer
601
views
What is the automorphic interpretation of the Weil conjectures over finite fields
I am very much a beginner in the theory of automorphic forms and I might (will?) make mistakes in what follows. Please correct me.
A loose interpretation of the Langland's philosophy is that to any ...
8
votes
0
answers
338
views
Results conditional on Langland's conjectures?
I know in number theorythere are loads of conditional results, dependant on RH for instance. On the other hand, Langland's programme is supposed to provide some understanding of the absolute Galois ...
16
votes
0
answers
731
views
What would be the simplest analog of Langlands in algebraic topology?
It is oversimplified, I know, but just as a superficial analogy, one may think of the fact that abelianization of the fundamental group is the first homology group, as some remote relative of class ...
4
votes
1
answer
200
views
Local L-function $L(s,\pi_p\times \chi_p)=1$
Let $\pi_p$ be a ramified representation of $GL(n,\mathbb{Q}_p)$.
Let $\chi_p$ be a ramified representation of $GL(1,\mathbb{Q}_p)$.
Is it generally known that
$L(s,\pi_p\times \chi_p)=1$ if $\...
17
votes
0
answers
956
views
Why arithmetic Langlands?
In trying to understand the import of Akshay Venkatesh' most recent work I found myself wondering anew about that old gnawing mystery: why Langlands? Why should arithmetic of polynomial equations over ...
3
votes
0
answers
191
views
Automorphy of families of motives
I have a couple of elementary questions regarding automorphy of Galois representations arising from geometric families.
Suppose we have an algebraic family of varieties over a number field, and ...
1
vote
0
answers
164
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Some simple question of the base change of the unitary group to general linear group
Let $E/F$ be a quadratic extension of number fields and $\chi$ is a unitary automorphic character of $E^{\times}$.
Let $\pi$ be an automorphic representation of $U(n)(F)$ associated to $E/F$, which ...
12
votes
0
answers
444
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The trace formula over function fields
There are many examples in number theory where an "arithmetic" problem (i.e. for number fields) has an easier analogue for function fields over finite fields. This is also true for questions ...
1
vote
0
answers
53
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A normalized embedding $\mathbb C \rightarrow \mathfrak a_M^{\ast}$ via $\tilde{\alpha}$
Let $G$ be a connected, reductive group over a field $k$. Let $S$ be a maximal $k$-split torus of $G$ with Weyl group $W$, $\Delta$ a set of simple roots of $S$ in $G$, and $P = MN$ a maximal ...
3
votes
0
answers
487
views
On Local Langlands correspondences
Both over global function fields and $p$-adic fields, we have a series of conjectures under the name of “geometric Langlands conjectures”.
Over global function fields of char $p$, they are due to ...