Questions tagged [langlands-conjectures]

Higher reciprocity laws

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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 ...
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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(...
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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 ...
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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 ...
Alex Youcis's user avatar
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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 ...
Marsault Chabat's user avatar
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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 ...
curious math guy's user avatar
2 votes
1 answer
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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 ...
Krishnarjun's user avatar
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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 ...
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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, \...
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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 ...
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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 ...
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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 ...
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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 ...
user837898's user avatar
9 votes
1 answer
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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}(...
Mihir Sheth's user avatar
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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 ...
curious math guy's user avatar
21 votes
1 answer
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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 ...
Anton Hilado's user avatar
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1 vote
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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 ...
user321680's user avatar
5 votes
1 answer
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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?...
Dat Minh Ha's user avatar
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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 $\mathbf H(s) = \operatorname{Ind}_{...
D_S's user avatar
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$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 $\...
Ashwin Iyengar's user avatar
3 votes
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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 ...
Ilya Zakharevich's user avatar
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1 answer
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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 ...
D_S's user avatar
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3 votes
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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 ...
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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 = ...
D_S's user avatar
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9 votes
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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 ...
D_S's user avatar
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7 votes
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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?...
user175904's user avatar
7 votes
1 answer
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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 ...
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12 votes
1 answer
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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 ...
George R's user avatar
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11 votes
1 answer
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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 ...
Nimas's user avatar
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6 votes
2 answers
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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 ...
Andrew NC's user avatar
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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 ...
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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 ...
Catherine Ray's user avatar
25 votes
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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 ...
Nimas's user avatar
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6 votes
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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 ...
Andrew NC's user avatar
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8 votes
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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 ...
Waleed Qaisar's user avatar
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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,...
Avi's user avatar
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1 answer
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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 ...
xlord's user avatar
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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 ...
Rachid Atmai's user avatar
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2 votes
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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 ...
Cheng-Chiang Tsai's user avatar
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 ...
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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 ...
nvc's user avatar
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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,\...
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3 votes
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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(...
D_S's user avatar
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5 votes
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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,...
Asvin's user avatar
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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 ...
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3 votes
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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 (...
D_S's user avatar
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2 votes
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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 $K/F$ ...
D_S's user avatar
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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 ...
D_S's user avatar
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2 votes
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$p$-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 $\Theta_{\pi}$ ...
Alexander's user avatar
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23 votes
2 answers
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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 ...
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