Questions tagged [langlands-conjectures]

Higher reciprocity laws

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3
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0answers
120 views

Motive to motive via Langlands

I have been tying myself in knots trying to straighten out this speculative circle of connections, no doubt because I am just a novice in these matters, so I thought I'd pop the question here and ...
2
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1answer
149 views

Understanding moduli of shtukas of non-minuscule cocharacter

I have kind of a soft question. I've studied the basics of L. Lafforgue's proof of function field Langlands for GLn, and its use of the moduli of shtukas with two legs, and the cocharacters $[1,0,\...
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Residual and continuous spectra of $L^2( G(k) \backslash G(\mathbb A) ; \omega)$, and cuspidal automorphic data

Let $G$ be a connected, reductive group over a number field $k$. Let $\mathbb A$ be the ring of adeles of $k$, $\omega$ be unitary character of $Z_G(\mathbb A)/Z_G(k)$, and $V = L^2(G(k) \backslash G(...
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43 views

Globally generic automorphic representation is contained in some generic A-packet?

I am wondering whether every globally generic representation of special orthogonal group or unitary group is contained in some global generic A-packet. I think it would be not because the Ramanujan ...
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115 views

A complex analytic version of the eigencurve

I am very much a beginner to the theory of eigencurves so there might be many mistakes in what follows, especially since it is all very speculative. My understanding of the eigencurve $\mathcal C_{N,...
5
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104 views

reduction type of elliptic curves over $p$-adic fields and local Langlands correspondence for $GL_2(F)$

In an introductory note of local Langlands correspondence http://wwwf.imperial.ac.uk/~buzzard/maths/research/notes/old_introductory_notes_on_local_langlands.pdf, section $11$ describes a recipe to ...
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109 views

How does Langlands define Artin L-functions?

Let $\rho: \operatorname{Gal}(K/F) \rightarrow \operatorname{GL}_n(\mathbb C)$ be a representation for an unramified extension $K/F$ of $p$-adic fields. Let $\operatorname{Frob}_{K/F}$ be the (...
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79 views

Does Langlands use the geometric Frobenius or the classical Frobenius in his papers?

In several of Langlands' papers: Representations of Abelian Algebraic Groups, On Artin's L-functions, On the Functional Equation of Artin's L-functions, Langlands takes a finite Galois extension $K/F$ ...
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91 views

The definition of Langlands' L-function $L(s,\pi,r)$ in the case of $\operatorname{GL}_1$

Let $G$ be a split reductive group over a $p$-adic local field $k$. For $\pi$ an unramified representation of $G(k)$, and $r$ a finite dimensional representation of the L-group $^LG$, Langlands ...
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40 views

$p$-adic Harish-Chandra character of a stable virtual character

Let $F$ be a $p$-adic field and let $G$ be a reductive group over $F$. Associated to an irreducible admissible representation of $\pi$ of $G(F)$, we have a distribution character $\Theta_{\pi}$ ...
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2answers
653 views

How can I see the relation between shtukas and the Langlands conjecture?

The following bullet points represent the very peak of my understanding of the resolution of the Langlands program for function fields. Disclaimer: I don't know what I'm writing about. Drinfeld ...
4
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1answer
280 views

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 ...
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68 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 $...
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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 $M$ ...
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175 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 ...
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1answer
382 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 ...
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72 views

A question about Bushnell-Henniart book 'The Local Langlands Conjecture for Gl(2)'

This is a question that has vexed me for a long time. In the section 15.6 of the book "Local Langlands Conjecture for Gl2",written by Bushnell and Henniart, they investigate the structure of cuspidal ...
3
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1answer
218 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
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1answer
122 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
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1answer
184 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, ...
5
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1answer
469 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 ...
4
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140 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 $G$ ...
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46 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 ...
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102 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 ...
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68 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 ...
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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
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1answer
214 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_{\...
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59 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 ...
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543 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
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1answer
456 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 $\mathbb{C}$ ...
5
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2answers
216 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 ...
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202 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 ...
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74 views

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 ...
2
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1answer
400 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 ...
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0answers
387 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
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1answer
427 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 $\rho_{E, l}$ (...
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721 views

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 ...
25
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3answers
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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.....
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100 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
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0answers
112 views

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 ...
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1answer
435 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 ...
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119 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, ...
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917 views

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 ...
7
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1answer
211 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 ...
7
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1answer
239 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 ...
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186 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 ...
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159 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 ...
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192 views

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
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0answers
189 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 ...
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92 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}: \...

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