Questions tagged [langlands-conjectures]
Higher reciprocity laws
367
questions
10
votes
1
answer
451
views
History of points of view on Eisenstein series
What is the history of Eisenstein series? Did the mathematician Eisenstein actually encounter them?
There are, as far as I know, two major perspectives on what Eisenstein series are. The first is in ...
4
votes
1
answer
130
views
The meaning of $L_{\chi}^2(G(\mathbb Q) \backslash G(\mathbb A)^1)$
I'm reading James Arthur's notes on the trace formula and am confused on a point on pages 65 and 66. For $G$ a reductive group over $\mathbb Q$ we are going over the decomposition of the space $L^2(G(...
15
votes
2
answers
2k
views
Relationship between the TQFTs in Kapustin-Witten and Ben-Zvi-Sakellaridis-Venkatesh
In upcoming work of Ben-Zvi-Sakellaridis-Venkatesh, (see for instance these notes or this lecture) some important aspects of the Langlands correspondence are stated in the language of topological ...
6
votes
0
answers
541
views
Affine GIT quotients and the excursion algebra in Fargues–Scholze
Some background:
Let us fix a non-archimedean local field $E$ with residue characteristic $p$, and let $G$ be some connected reductive group over $E$. In [FS, §VIII.1.1] the authors define a moduli ...
4
votes
0
answers
255
views
The link between Satake parameter and Godement-Jacquet L-function of an automorphic representation of $GL_{n}$
Origin of the question: I'm reading the following survey of K. Martin, more generally I'm looking for the "best way" to define L-function associated to an automorphic representation of a ...
5
votes
0
answers
440
views
Do we expect the Langlands correspondence to be a functor?
In the literature I've read, it is often said that to a Hecke eigenclass, one would like (and sometimes succeeds) to "associate" or show the existence of a Galois representation such that ...
2
votes
1
answer
128
views
Do Artin L functions have polynomial growth in in the critical strip?
Given an irreducible representation $\rho$ of the Galois group $G$ of a number field $K$ over $\mathbb{Q}$, we have the associated Artin $L$ function which we denote by $L(s, \rho)$. By Brauer ...
4
votes
0
answers
158
views
From $\mathrm{GL}_{1}/K$ to $\mathrm{GL}_{2}/\mathbb{Q}$, where $K$ is a cyclic cubic extension
(Migrated from MSE)
Let $K$ be a cyclic cubic extension of $\mathbb{Q}$.
For example, one can take simplest cubic fields. By automorphic induction (which is known for cyclic extension of prime degree ...
2
votes
0
answers
102
views
Converse of Gelbart-Jacquet lift for $\mathrm{GL}(3)$ Maass forms
(This question is migrated from MSE)
Gelbart-Jacquet lift gives functoriality from $\mathrm{GL}(2)$ to $\mathrm{GL}(3)$ that corresponds to a symmetric square map $\mathrm{Sym}^{2}: \mathrm{GL}(2, \...
2
votes
0
answers
87
views
Local A-packet is singleton for unramified place?
Let $\pi$ be a generic $A$-parameter, that is an isobaric automorphic representation of linear group.
Decompose $\pi= \otimes \pi_v$ as a restricted tensor product. Then by the local Langlands ...
1
vote
0
answers
64
views
Generic Arthur-parameter of symplectic group
For a irreducible cuspidal automorphic representation $\pi$ of $Sp(2n)$, we can attach its generic $A$-parameter, that is isobaric sum automorphic representation of $GL_{2n+1}$.
I know it is of the ...
4
votes
0
answers
140
views
Global Vogan A-packet is infinite set?
For an cuspidal automorphic representation of general linear group, we can attach its global Vogan A-packet.
Though I thought that it is finite set, in some paper, it is written that there are ...
2
votes
0
answers
160
views
Representation of locally profinite group
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\GL{GL}$In Bushnell and Henniart's "The local Langlands conjecture for $\GL_2$", Chapter 1, 3.5, proof of the duality theorem it says
For ...
9
votes
1
answer
293
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}(...
15
votes
0
answers
2k
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 ...
21
votes
1
answer
2k
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
vote
0
answers
529
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 ...
5
votes
1
answer
709
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
votes
0
answers
71
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
votes
0
answers
146
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
votes
0
answers
130
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
vote
1
answer
260
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
1
answer
153
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
votes
0
answers
74
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
votes
1
answer
215
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 ...
7
votes
1
answer
1k
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
votes
1
answer
473
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 ...
12
votes
1
answer
477
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 ...
11
votes
1
answer
585
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 ...
6
votes
2
answers
911
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 ...
15
votes
0
answers
546
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 ...
13
votes
0
answers
313
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
votes
0
answers
588
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 ...
6
votes
1
answer
593
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
votes
0
answers
397
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
votes
0
answers
226
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
1
answer
671
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 ...
4
votes
0
answers
534
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
votes
0
answers
111
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
votes
1
answer
409
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
votes
0
answers
234
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
1
answer
522
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
votes
0
answers
79
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
0
answers
213
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
0
answers
299
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
votes
0
answers
161
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
votes
0
answers
152
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
votes
0
answers
118
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
votes
0
answers
89
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 $\Theta_{\pi}$ ...
23
votes
2
answers
2k
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 ...