Questions tagged [langlands-conjectures]

Higher reciprocity laws

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On Local Langlands correspondences

Both over global function fields and $p$-adic fields, we have a series of conjectures under the name of “geometric Langlands conjectures”. Over global function fields of char $p$, they are due to ...
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Langlands correspondence for higher local fields?

Let $F$ be a one-dimensional local field. Then Langlands conjectures for $GL_n(F)$ say (among other things) that there is a unique bijection between the set of equivalence classes of irreducible ...
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Diagonalizable pro-algebraic group in Kottwitz's 1985 Compositio paper

In Kottwitz's 1985 Compositio paper, Isocrystals with additional structure, first page, paragraph 4: Let $\mathbb{D}$ be the diagonalizable pro-algebraic group over $\mathbb{Q}_p$ with character ...
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How does the Bernstein-Zelevinsky construction of irreducibles from supercuspidals parallel the representations of the Weil-Deligne group?

In the Corvallis article Number Theoretic Background, here is what John Tate has to say on the local Langlands correspondence for a $p$-adic field $F$: So, granting a correspondence between ...
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L-functions for the Weil group over short exact sequences

Let $(\rho,V)$ be a continuous finite dimensional representation of the Weil group $W_F$ over a local field $F$. If $V$ decomposes as a direct sum $V_1 \oplus V_2$ of representations, then $$L(s,\...
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How to deduce Bernstein-Zelevinsky classification from the Langlands one

I am trying to understand the Langlands classification. To that end, I am trying to find how I could deduce the Bernstein-Zelevinsky classifcation from the second description of the Langlands ...
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What is known about the homomorphisms from local to global Weil groups?

I have been reading Tate's article Number Theoretic Background in the Corvallis proceedings about the Weil and Weil-Deligne groups. I understand that the global Weil group $W_K$ of a number field $K$ ...
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Examples of function fields Langlands for small genus (<= 2)

See Edward Frenkel's article "Lectures on the Langlands program and conformal field theory" for an exposition of the function fields Langlands correspondence (now a theorem of Drinfel'd, L.Lafforgue &...
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Supercuspidals and representations of $\operatorname{Gal}(\overline{F}/F)$

Let $F$ be a $p$-adic field, $G = \operatorname{Gal}(\overline{F}/F)$ and $W$ the Weil group of $F$. The inclusion map $W \subset G$ is continuous with dense image, so $\rho \mapsto \rho|_W$ defines ...
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Definition of Local L-function for a representation of a torus?

Let $G$ be a connected, reductive group over a $p$-adic field $k$. Let $\pi$ be an irreducible, admissible representation of $G(k)$, and $r$ a finite dimensional continuous representation of the $L$-...
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Arithmetic motivations for modularity in higher rank

The classical setting of modularity is that one can associate elliptic modular forms (or automorphic representations of GL(2)/$\mathbb Q$) to elliptic curves over $\mathbb Q$. This has far-reaching ...
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Symmetric powers of Ramanujan tau-function

Let $\Delta(z)$ be the modular form associated with Ramanujan $\tau$-function. For any $k=2,3,...$, $Sym^k\Delta$ is conjectured to be an automorphic form on $\mathrm{GL}(k+1)$ and $L(s, Sym^k\Delta)$...
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Connections between representations of $\operatorname{SL}_n$ and $\operatorname{GL}_n$

Let $G = \operatorname{GL}_n(F)$ for a $p$-adic field $F$, and let $G_D = \operatorname{SL}_n(F)$. I am wondering if there is a connection between irreducible, admissible representations of $G$ and ...
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Equivalence of formulations of Ihara's lemma

I'm wondering about the relationship between two formulations of Ihara's lemma for $\text{GL}_2$ I've seen: (1) the "concrete" version given in, for example, Darmon, Diamond, and Taylor, which says ...
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Orbits of arithmetic subgroups intersection a compact set

Let us suppose we have $G$ a connected reductive group over a number field $F$. Consider $G(\mathbb{A})$ the group over the adeles and $G(\mathbb{Q})$ embedded discretely. For $\gamma \in G(\mathbb{Q})...
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Shtukas for $\mathrm{Spec}\,\mathbf{Z}$

This is a very soft and speculative question. Please feel free to downvote, close or delete it. Studying the cohomology of moduli spaces of shtukas, Drinfeld proved the Langlands program for $\mathrm{...
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Bi-Whittaker functions and local Langlands compatibility

I'm trying to figure out the arithmetic analogue of a key conjecture in the geometric local Langlands correspondence. Briefly, one expects for $K=\mathbb{C}((t))$ an equivalence of dg categories $$\...
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Potential automorphy of abelian varieties

Let $A$ be an abelian variety over $\mathbb Q$. One could ask (1) is there a finite extension $K$ of $\mathbb Q$ such that the L-function $L(A/K,s)$ is the L-function of an automorphic form? or ...
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Uniqueness of class field theory map

Let $F$ be a local field of characteristic 0. The main theorems in local class field theory can be summarized by the existence of a group $W_F$ and a map $$ \phi_F:W_F\to W_F^\mathrm{ab}\simeq F^\...
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References for Langlands classification

I kindly ask about some references concerning the representation theory of the Langlands dual of a compact Lie group, and how it relates to things related to the original compact Lie group. My ...
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Idea of base change for Division algebras over local field

Let $F$ be a non-Archimedean local field of characteristic $0$ and $K/F$ be a finite extension. Let $D_F$ be the central division algebra of dimension $n^2$ over $F.$ Write $D_K=D_F\otimes_FK$, which ...
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Meaning of Ramanujan-Petersson conjecture? [closed]

I found it very hard to explain the Ramanujan-Petersson conjecture in a straightforward way. I can only say now: think about automorphic forms as sound waves, and then the conjecture predicts that ...
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Global Langlands function fields

Has V. Lafforgue proved the automorphic-to-Galois direction in the Global Langlands conjectures for general reductive groups over function fields? What is the current status, more generally? Related ...
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Where stands functoriality in 2017?

In 2002, R. Langlands put forward a new strategy to prove the general functoriality conjecture in the Beyond endoscopy paper. The main purpose of this strategy is to detecting the automorphic ...
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Alternative way to prove the functional equation for Eisenstein series?

Let $E(z,s):=\pi^{-s}\Gamma (s) \sum_{(m,n)=1}\frac{y^s}{|mz+n|^{2s}}$ be the real-analytic Eisenstein series. It satisfies the functional equation $E(z,s)=E(z,1-s)$ with two poles at $s=0,1$. The ...
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Critical values of L-functions and weights of Eisenstein Series

I have been reading Serre's paper on p-adic modular forms and there seems to be a connection between critical values of L-functions and weights of Eisenstein series in the following sense: For the ...
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Reference for shtuka and trace formula

I really want to learn the work of Laurent Lafforgue and the joint work of Zhiwei Yun and Wei Zhang. They both involve shtuka and trace formula, which I only know the basic idea. So I would like to ...
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Residue of Eisenstein Series on GL(n)

Reference: Mœglin, C.; Waldspurger, J.-L.: Le spectre residuel de GL(n) On GL($n$), the result of Waldspurger shows that if an automorphic representation $\pi$ is non-cuspidal and in the discrete ...
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What kind of non-cuspidal automorphic representation are not isobaric sums?

Let's say $\pi$ is an automorphic representation on $GL_3(A_{\mathbb Q})$ (or $GL_n(A_{\mathbb Q})$). If $\pi$ is not cuspidal, what $\pi$ can be other than isobaric sums? If there is such a thing, ...
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infinitesimal character of Langlands quotient for GL(n,R)

Let $G = GL(n,\mathbb{R})$. Consider a Langlands data $(Q_F, \sigma, \lambda)$ with $F \subset \Delta$ (the set of simple roots), $Q_F$ the associated standard parabolic subgroup, $\sigma$ an ...
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Is there a definition of supercupidal parameter in the Local Langland correspondence?

By the recent works of Mok, and Kaletha, Shin, White, James, I know that there is a notion of tempered $L$-parameter, square integrable $L$-parameter and generic $L$-parameter of unitary groups. ...
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Examples of conjectural functorial transfer which has $\times GL(1)$ functional equation?

I am look for some conjectural functorial transfer $X$ which (A)for any $GL(1)$ automorphic representation $\pi$, we have $L(s, X\times \pi)$ is holomorphic and satisfies certain functional ...
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Loss of cuspidality by Langlands tranfer

Given an $L$-homomorphism of Langlands dual groups $${}^LG \to {}^LG'$$ Langlands functoriality contectures predicts the existence of a tranfer map of automorphic representations $$Aut(G) \to Aut(G')$...
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Monstrous Langlands-McKay or what is bijection between conjugacy classes and irreducible representation for sporadic simple groups?

Context: The number of conjugacy classes equals to the number of irreducuble representations (over C) for any finite group. Moreover for the symmetric group and some other groups there is "good ...
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Bijection between conjugacy classes and irreducible representation of Weyl group = Langlands correspondence over "field with one element"

Context: The number of conjugacy classes equals to the number of irreducuble representations (over C) for any finite group. Moreover for the symmetric group there is well-known "natural bijection" ...
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Endoscopic group that is not a subgroup

The question is a very little more than what's in the title. It is easy (for some values of ‘easy’) to produce examples of endoscopic groups that are not subgroups. When I asked a colleague, he ...
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Reference for the proof of Langlands conjecture for $GL_n$ over function fields

Is there any reference written in English for the proof of Langlands conjecture for $GL_n$ over function fields?
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Intuition behind centralizers of Langlands parameters

In the description of the Langlands correspondence for $\mathbb{Q}_p$, we consider admissible representations of $G(\mathbb{Q}_p)$ for $G$ a reductive group defined over $\mathbb{Q}_p$, and admissible ...
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The role of Honda-Tate theory in (Scholze's refinement of) the Langlands-Kottwitz method?

I was wondering if somebody could give me a quick primer on Honda-Tate theory, specifically the number of isogeny classes of elliptic curves, and how specifically it's utilized in (Scholze's ...
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Is there a "Langlands philosophy" reason for the fact that the L-function of the Jacobi theta function is (almost) the Riemann zeta function?

The Jacobi theta function $\theta(z) = 1 + 2 \sum_{n = 1}^\infty q^{n^2}$, with $q = e^{\pi i z}$ is a (twisted) modular form with weight $1/2$. It has an associated $L$-function $L(\theta, s) = \sum_{...
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What is a common name for these automorphic objects?

I am looking for a name which includes these objects: 1. automorphic forms, cusp forms and non-cusp forms 2. Rankin-Selberg convolution between automorphic forms (which is conjectured to be ...
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nonabelian reciprocity law

I heard the following relation in a talk by Peter Scholze. Could someone explain "in a simple way" what is the precise relation between the polynomial $x^4-7x^2-3x+1 $ and the integral homology of the ...
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The proof of Kazhdan's density theorem (And does it hold over positive characteristic?)

When proving identities about traces of functions on representations of $p$-adic groups, Kazhdan's density theorem indicates one only has to check equalities of traces on tempered representations. ...
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Selberg trace formula, quadratic L-values, and generalization

It is known that the geometric side of the Selberg trace formula on GL(2) is related to values of quadratic L-functions (due to Sarnak, Zagier, etc). Are there any conjectures or results about its ...
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Relationship among class field theory, modularity theorem, and the langlands program

Here is some background: I have taken an introductory algebraic number theory course up to valuation-theoretic approach, learned a bit of local class field theory, representation theory, and modular ...
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Irrelevant parabolics and inner forms of GSp(4)

In Ralf Schmidt's appendix to "Jacquet-Langlands-Shimizu correspondence for theta lifts to $\mathrm{GSp}(2)$ and its inner forms" by Narita and Okazaki , he computes the representations of $\mathrm{...
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Does local Langlands functoriality preserve genericity?

Let $G=Sp_4$ over a p-adic field $F$. After Gan-Takeda's work, the local Langlands correspondence for $G$ is known. Thus we have the local functoriality from $G$ to $GL_5$. Do we know that this local ...
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local root number calculation

For discrete series L-parameter $\phi:W_\mathbb{R}\rightarrow Sp(2n)(\mathbb{C})$ given by Harish-Chandra parameter $(a_1,\cdots, a_n)$ with $a_1>\cdots>a_n>0$ and $a_i\in \mathbb{Z}+1/2$. ...
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Relationship between motivic Galois groups and Langlands program [duplicate]

I would like to know if there is any relationship between the motivic Galois groups and the Langlands program. Many thanks.
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Bernstein–Zelevinsky classification for classical groups

Bernstein and Zelevinsky classifies the irreducible complex smooth representations of a general linear group over a local field in terms of cuspidal representations. The irreducible modules are ...
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