Questions tagged [langlands-conjectures]

Higher reciprocity laws

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6
votes
1answer
141 views

A question about mod $p$ local Langlands for $\mathrm{GL}_{2}(\mathbb{Q}_{p})$

In the mod $p$ local Langlands correspondence for $\mathrm{GL}_{2}(\mathbb{Q}_{p})$, the irreducible supercuspidal representation $\left(\mathrm{ind}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}_{\mathrm{GL}_{2}(...
12
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0answers
679 views

A question on Fargues-Scholze

As far as I understand it, the main goal of the recent work of Fargues and Scholze on the geometrization conjecture is to show that the local Langlands conjecture of a local field is equivalent to the ...
18
votes
1answer
1k views

Cohomology of Shimura varieties and coherent sheaves on the stack of Langlands parameters

In Zhu's Coherent sheaves on the stack of Langlands parameters theorem 4.7.1 relates the cohomology of the moduli stack of shtukas to global sections of a certain sheaf on the stack of global ...
1
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0answers
413 views

Langlands program and complexity theory

Back when I was an undergraduate, I spent some time reading the about the modularity conjecture, but the details are fuzzy now. One of the motivations I imagined for the Langlands program was for ...
4
votes
1answer
343 views

Categorical-geometric Langlands for tori

Fix a "nice" curve $X$ (smooth, projective, proper, geometrically connected, what-have-you) and an algebraic torus $G$, both over a field of characteristic $0$ (possibly algebraically closed?...
3
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0answers
47 views

Density of the Mellin transform inside the direct integral of induced representations

I'm trying to better understand the continuous spectrum of $G = \operatorname{GL}_2(\mathbb A_{\mathbb Q})$, which is the direct integral of induced representations $\mathbf H(s) = \operatorname{Ind}_{...
3
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0answers
123 views

$p$-adic Hodge theoretic properties of global Galois representations via $\ell$-Frobenii

Let $G_{\mathbb{Q},S} = \mathrm{Gal}(\mathbb{Q}_S/\mathbb Q)$ where $\mathbb Q_S$ is the largest algebraic extension of $\mathbb Q$ unramified outside a finite set of places $S$. Then the union over $\...
3
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0answers
98 views

The mystery of the jumps of functions with the prescribed jumps: Eisenstein series and hidden symmetries(?)

Say that a function $f(t)$ “changes only by jumps” if $f(t) + \text{const} = C ∑_k j_k θ(t-t_k)$ for a certain constant $C$. Here $θ(t)$ is the Heaviside step function which has a jump 1 at $t=0$ (it ...
1
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1answer
185 views

The norm of the principal series intertwining operator for $\operatorname{GL}_2$

Is there a known bound on the norm of the standard intertwining operator for the principal series of $G = \operatorname{GL}_2(\mathbb Q_p)$? Background: For a character $\chi = (\chi_1,\chi_2)$ of the ...
3
votes
1answer
71 views

Calculating the residue of Eisenstein series from the residue of the intertwining operator

I've been reading the article Forms of $\operatorname{GL}(2)$ from the analytic point of view by Gelbart and Jacquet in Corvallis and am confused on a particular claim (equation 5.17 on page 232). The ...
4
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0answers
49 views

How to see the surjectivity of $L^2_{\text{cont}}$ onto the direct integral of Hilbert space representations?

I've been reading the article Forms of $\operatorname{GL}(2)$ from the analytic point of view by Gelbart and Jacquet in Corvallis and am confused on one point. Let $G = \operatorname{GL}_2$, and $V = ...
9
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1answer
172 views

Why are characters orthogonal to cusp forms?

Let $G = \operatorname{GL}_2$, and let $V = L^2(Z(\mathbb A)G(\mathbb Q) \backslash G(\mathbb A),\omega)$, for $\omega$ a character of the ideles $\mathbb A^{\ast}$, identified with a central ...
6
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1answer
340 views

Contemporary introduction to Godement-Jacquet "Zeta functions of simple algebras"

The question is in the title: The book Godement-Jacquet "Zeta functions of simple algebras" is from 1971. Has there ever been a textbook introduction to this material, or at least part of it?...
7
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1answer
389 views

Dictionary of arithmetic symmetries and Langlands

To a number theorist automorphic forms appear to be adelic point-counting generating functions for arithmetic schemes. This is what the conjectured equality of their $L$ functions tells us. The fact ...
11
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1answer
284 views

Eigenvarieties and functoriality

In Langlands' review of Hida's book "$p$-adic automorphic forms on Shimura varieties", he discusses a nexus of 4 areas of modern number theory: automorphic representations, motives, spaces ...
9
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1answer
385 views

Modularity of higher genus curves

The modularity conjecture for elliptic curves over number fields is well known, and indeed, is a theorem for all elliptic curves over $\mathbb{Q}$, and at least potentially, over any CM field. What ...
5
votes
2answers
500 views

What is the theorem of the highest weight used for?

$\DeclareMathOperator\End{End}$Over the past few months, I have taught myself the classification of reductive groups, and continued to non-abelian (as well as a small venture to non-compact) Harmonic ...
11
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0answers
243 views

Exposition of Drinfeld's proof of function field Langlands for GL(2)

I know, or think I know, the vague outline of the proof: the Galois-to-automorphic direction is "classical," i.e. follows from converse theorems due to Grothendieck et al., and for the ...
9
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0answers
227 views

How do we deduce the Jacquet-Langlands correspondence from Fargues' two towers?

In trying to understand the geometric proof of the local-Langlands and Jacquet-Langlands correspondence which uses Fargues's two tower theorem, I am having trouble finding a nice source on this, and I ...
25
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0answers
497 views

Galois representations attached to Shimura varieties - after a decade

In an answer to the question Tools for the Langlands Program?, Emerton, in his usual illuminating manner, remarks on the reciprocity aspect of Langlands Program: "...As to constructing Galois ...
5
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1answer
303 views

What is a map for the representation theory of reductive groups?

I have finished learning about linear algebraic groups (minus their representation theory) and the associated algebraic structures (root data, root systems, etc.), and will next attempt to summarize ...
8
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0answers
274 views

On a remark of Langlands

I'm been wondering about this for a while and hope someone can enlighten me. In this interview of Robert Langlands's from 2010, on pg 21 (Question 8) he states "At one point, when fairly young, I ...
6
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0answers
196 views

Modularity switching for primes $p>7$

In Freitas, Le Hung, and Siksek's 2014 paper proving that elliptic curves over totally real quadratic fields are modular they prove (and use) the following result (Theorems 3 and 4): let $\bar \rho_{E,...
11
votes
1answer
384 views

Roadmap for studying Galois deformation theory/modularity theorems from a modern perspective

I am a graduate student with some background in Galois deformation theory. I am familiar with the basics (the existence of a universal deformation space with prescribed conditions) and with some ...
3
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0answers
458 views

Modern example of a reciprocity law and intuition behind it

I'm very new to the Langlands program and I was going through the Gauss reciprocity law, Hilbert's 9th problem, Artin's reciprocity law which allowed him to identify the Artin's L-functions with the ...
2
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0answers
75 views

Parabolic inductions for p-adic reductive groups

So I wish to ask for articles/comments surveying conjectures and theorems about parabolic induction for p-adic (non-archimedean case) reductive groups, and how local Langlands behaves under such. That ...
9
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1answer
291 views

Langlands dual group in math vs. Goddard-Nyuts-Olive dual group in physics

Given a group $G$, there is a so-called Langlands dual group $G^{∨}$. Given a group $G$, there is also a so-called Goddard-Nyuts-Olive dual group $G^{'}$ that relates to the magnetic charge. Are ...
3
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0answers
169 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 ...
3
votes
1answer
307 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,\...
3
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0answers
71 views

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(...
5
votes
0answers
142 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
votes
0answers
200 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 ...
3
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0answers
121 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 (...
2
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0answers
109 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 $K/F$ ...
2
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0answers
103 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 ...
2
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0answers
51 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 $\Theta_{\pi}$ ...
19
votes
2answers
1k 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 ...
3
votes
1answer
294 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 ...
3
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0answers
73 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
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0answers
77 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
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0answers
179 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
1answer
502 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
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1answer
251 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
1answer
136 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
218 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
1answer
586 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
votes
0answers
151 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 $G$ ...
3
votes
0answers
55 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
0answers
119 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
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0answers
78 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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