# Questions tagged [langlands-conjectures]

Higher reciprocity laws

Higher reciprocity laws

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My field is far from the Langlands conjectures. I am just trying to understand some basic ideas.
At the moment I am interested in a global field $K$ of positive characteristic and the group $G=GL_n$. ...

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Let $G$ be a $p$-adic reductive group and $\pi$ an irreducible non-supercuspidal representation. Then there exist a parbaolic subgroup $P=MN$ and a supercuspidal representation of $M$ such that $\pi$ ...

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Suppose $G$ is a semisimple algebraic group over the rational numbers, $\pi$ is a cuspidal automorphic representation of $G$, and $r: \widehat{G}(\mathbf{C}) \rightarrow \mathrm{GL}_N(\mathbf{C})$ is ...

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In conjecture 6.1.14 of this article, Emerton-Gee-Hellmann formulate the p-adic local Langlands conjecture, which posits the existence of a fully faithful functor from (the appropriate derived ...

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Do we have a finite coefficients Langlands correspondence for function fields? By which I mean a bijection between Galois representations $$\pi_1(X)\to \mathrm{GL}_n\left(\overline{\mathbb{F}_p}\right)...

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The reductive groups play a central role in the Langlands correspondence. Why are these groups called reductive? Does this name suggest something conceptual about these groups?

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Drinfeld-Lafforgue have proven function fields Langlands conjectures in type A: see https://arxiv.org/pdf/math/0212417.pdf (Laumon's survey in English), https://arxiv.org/pdf/math/0212399.pdf (...

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Recently, I started to read about automorphic forms and representations on covering groups, e.g. metaplectic groups. I set my first goal as understanding Shimura's correspondence in representation ...

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So rank $1$ local Langlands is special in as that it is given by the Artin map
$$\text{GL}_1(K)\to G_K^{ab},$$
whereas in the higher rank (to the best of my knowledge) there doesn't exist a map
$$\...

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I've been trying to read Kapustin–Witten - Electric–Magnetic Duality And The Geometric Langlands Program recently, as someone whose mathematical interests are in the Langlands program. I have some ...

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I am reading David Geraghty's paper, 'Modularity lifting theorems for ordinary Galois representations'(https://link.springer.com/article/10.1007/s00208-018-1742-4) and I have a related question, which,...

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In the paper by L.Clozel in this book (a French text), there is this conjecture (conjecture 4.5 p139)
Conjecture: Given $\pi$ an algebraic cuspidal representation of $Gl(n)$ of weight $w$ and denote ...

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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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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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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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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 ...

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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 ...

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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 ...

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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 ...

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(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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(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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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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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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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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$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\GL{GL}$In Bushnell and Henniart's "The local Langlands conjecture for $\GL_2$", Chapter 1, 3.5, proof of the duality theorem it says
For ...

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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}(...

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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 ...

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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 ...

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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 ...

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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?...

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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}_{...

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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 $\...

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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 ...

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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 ...

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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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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 = ...

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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 ...

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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?...

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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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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 ...

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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 ...

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$\DeclareMathOperator\End{End}$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 ...

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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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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 ...

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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 ...

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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 ...

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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 ...

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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,...

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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 ...

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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 ...