Questions tagged [langlands-conjectures]

Higher reciprocity laws

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6 votes
2 answers
230 views

Reference for Langlands dual homomorphisms

I am looking for a reference that explains in detail the existence of Langlands dual homomorphisms. It seems that in the literature two references are given most often. The first is Borel's article ...
2 votes
1 answer
164 views

Orbital integrals of $\operatorname{SL}_2$ and the fundamental lemma

$\DeclareMathOperator\SL{SL}$When I was checking some orbital integral computations of Sally-Shalika's The Plancherel Formula for $\SL_2$ over a Local Field, Proceedings of the National Academy of ...
19 votes
1 answer
2k views

What are "branes", and why do they form a category?

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 ...
0 votes
1 answer
181 views

Variants of the classical Satake classfication

Let us fix a connected split reductive group $G$ over a local non-archimedean field $K$ with maximal torus $T$, then most notes including that of [Gross] describes as a consequence of the Satake ...
4 votes
2 answers
267 views

Parabolic induction, and tensoring (Iwahori/affine) Hecke algebras

In several places, such as [1] and [2], it seems to be implicitly known that (normalized) parabolic induction corresponds to tensoring affine Hecke algebras. I say implicitly because both [1] and [2] ...
1 vote
0 answers
64 views

Automorphisms of Iwahori/affine Hecke algebras

Has there been any serious study of automorphisms of extended affine Hecke algebras? Has anyone determined the automorphism group of say, type A extended affine Hecke algebras? I ask because the ...
5 votes
0 answers
159 views

Question on the unramified local Langlands conjecture

I'm working on the unramified local Langlands conjecture and there is something that I don't understand if it is true or not. I want to start by saying that I don't care about endoscopic transfer or ...
3 votes
1 answer
432 views

$L$-parameters and parabolic induction

I apologize in advance if the answer to this question is well-known to experts. So let $F$ be a $p$-adic field, and $G$ a reductive group over $F$. Let $P$ be an $F$-parabolic subgroup of $G$ and $M$ ...
2 votes
0 answers
84 views

Local systems on $\mathbb P^1$ and on the formal punctured disc

Consider the projective curve $\mathbb P^1$ over a finite field $k$. Consider $\ell$-adic local systems $E$ on $\mathbb P^1\backslash \{0,\infty\}$ such that a) $E$ is tame at $\infty$ b) The ...
14 votes
2 answers
4k views

Consequences of the Langlands program

In the one-dimensional case the Langlands program is equivalent to the class field theory and the two-dimensional case implies the Taniyama-Shimura conjecture. I would like to know: are there any ...
7 votes
0 answers
136 views

Non-abelian ray class fields for local fields

Let $K$ be a non-Archimedean local field. Then, thanks to work of Koch (when $K$ has positive characteristic) and Jannsen-Wingberg (when $K$ has characteristic zero, and odd residual characteristic) ...
1 vote
0 answers
129 views

Langlands dual of torus

Let $T$ be a split torus over a field $k$. Then the dual torus $\hat{T}$ is defined to be the unique torus such that $$ X_*(T)=X^*(\hat{T}), $$ where the left hand side is the cocharacter lattice of $...
1 vote
1 answer
179 views

Non-split extension of representations of $\mathrm{GL}_2$ and $\mathrm{Hom}$

Let $0\to V_1\to V\to V_1\to0$ be a sequence of representations of $\mathrm{GL}_2(\mathbb{Q}_p)$ over $\overline{\mathbb{F}}_p$, where $V_1$ is irreducible, smooth and admissible. Assume that this ...
5 votes
2 answers
478 views

Non-trivial extension of representations have same central character

Let $\pi_1, \pi_2$ be two irreducible complex representations of $G=\mathrm{GL}_2(\mathbb{Q}_p)$ and assume that there exists a non-split extension $0\to\pi_1\to \pi\to\pi_2\to0$ of representations ...
3 votes
0 answers
273 views

Can you prove and/or generalize this formula involving the Möbius function at n = square free numbers for elliptic curve related sequence in the OEIS?

Let $g(n)$ be the Dirichlet inverse of the Euler totient function: $$g(n) = \sum\limits_{d|n} d \cdot \mu(d)$$ and let $f(x,y)$ be the elliptic equation: $$f(x,y)=x^3 - x^2 - y^2 - y$$ Show that the ...
4 votes
1 answer
596 views

Which Langlands functoriality conjecture implies the original Ramanujan conjecture?

I heard that the Langlands functoriality conjecture implies the Ramanujan conjecture for $GL(2)$. especially for the Maass form. There are various versions of the Langlands functoriality concerning to ...
2 votes
1 answer
728 views

Reference on Casselman-Shalika formula for GL(n) and PGL(n)?

I am looking for reference on Casselman-Shalika formula for GL(n) and PGL(n) at finite place p.
1 vote
2 answers
230 views

How to assign the $L$- and $A$-parameters for the trivial representation

Let $G$ be a (split) reductive group over a $p$-adic field $F$, and $\mathbf{1}$ the trivial representation of $G(F)$. Under the (conjectural) Langlands correspondence, this should correspond to an $L$...
5 votes
1 answer
197 views

Two different local Langlands parameters for quadratic extension

Let $E/F$ be a quadratic extension of nonarchimedean local field, and let $G$ be a reductive group over $E$, and $G(E)$ the $E$-points of $G$. (For my question, one may just assume $G=\operatorname{GL}...
3 votes
0 answers
89 views

Langlands parameters and Weyl group actions

Let $F$ be a $p$-adic field and $\mathbf{G}$ a connected reductive group over $F$, assumed to be quasi-split. Let $\mathbf{T}$ be a maximal split torus of $\mathbf{G}$ and $\mathbf{P}=\mathbf{M}\...
6 votes
0 answers
248 views

What, if anything, do we hope and expect to understand about (classical) Galois groups?

I was reading Franz Lemmermeyer's introduction to Fermat's Last and Wiles' Theorem, where he states Galois representations $\rho_p : G_\mathbb Q\rightarrow GL_2(\mathbb Z_p)$ are used for studying ...
6 votes
0 answers
338 views

Can Langlands correpondence be restated using topos?

Langlands correspondence describes an equivalence between Galois representations and automorphic representations under some conditions. Laurent Lafforgue applying Olivia Caramello thesis described in ...
2 votes
1 answer
234 views

Non-continuous group homomorphism from p-adic field to C*

Let $F$ be a p-adic field. A character on $F^\times$ is defined as a continuous group homomorphism $F^\times\longrightarrow\mathbb{C}^\times$. But is there any way to construct a non-continuous group ...
2 votes
1 answer
82 views

Image of the intertwining operator for GL(2) is $K$-invariant at the "pole" $s=1$

I am taking a look at the residues of Eisenstein series and have a question about a local computation. Let $k$ be a local field, $G = \operatorname{GL}_2(k)$, and $P$ (resp. $K$) the standard ...
0 votes
1 answer
805 views

Euler product of Asai L-function?

Let $\pi$ be an automorphic form of GL(n)/$\mathbb{Q}$ with standard $L$-function $$L(s,\pi)=\prod_p \prod_{i=1}^n(1-\frac{\alpha_{p,i}}{p^s})^{-1},$$ where $\{\alpha_{p,i}:i=1,\dots,n\}$ are the ...
3 votes
0 answers
317 views

References on $p$-adic Langlands

As a student with a background in Algebraic Geometry (up to chapter 3 of Hartshorne) and basic Algebraic Number Theory, where should I begin learning about the $p$-adic Langlands program? What are the ...
1 vote
1 answer
103 views

Continuity of central character [closed]

Let $G$ be a $p$-adic reductive group and $Z\subseteq G$ the center. Let $\pi$ be an irreducible admissible representation of $G$. By Schur's lemma, it is easy to show that there is a group ...
4 votes
0 answers
174 views

Several L-functions but one Galois representation: How to choose

Let $\mathbf{G}$ be a reductive group which enjoys all the nice properties a reducive group can dream of. Fix $(\mathbf{G},X)$ a Shimura datum associated with it and assume that if $K\leq\mathbf{G} $ ...
4 votes
0 answers
208 views

Correspondence between motives and automorphic representations

What I know: I understand motives via its realization; in Coates' and Perrin-Riou's paper On $p$-adic L-functions Attached to Motives over $\mathbb{Q}$ (see http://doi.org/10.2969/aspm/01710023), the ...
2 votes
0 answers
143 views

Meaning of the meromorphic continuation of intertwining operators

I am trying to make sure the meaning of the meromorphic continuation of the intertwining operators. Assume we deal with a non-Archimedean field $F$ and just consider $G= SL_2$, for simplicity. We fix ...
3 votes
1 answer
139 views

Residue of a local $\gamma$-factor and its relation with adjoint $\gamma$-factor

I met the following relation (if my understanding is correct) of local $\gamma$-factors when I was reading Hiraga-Ichino-Ikeda's paper "Formal Degrees and Adjoint $\gamma$-Factors": Let $F$ ...
2 votes
0 answers
216 views

Compatibility of Lefschetz formula and categorical local Langlands correspondence

Lefschetz-Verdied formula is formulated for etale sheaves and coherent sheaves in SGA 5 Ⅲ Theorem 4.4 for noetherian schemes. My questions are that 1.Do we formulate the formula for objects and ...
3 votes
1 answer
122 views

Modular forms on central division algebra of degree $\ge 3$

I just learned from some online sources including Buzzard's note and Emerton's answer on this MO question about quaternionic modular forms and explicit version of Jacquet-Langlands correspondence in ...
27 votes
2 answers
4k views

Status of (global) Langlands conjecture for $\mathrm{GL}_2$ over $\mathbb{Q}$

$\DeclareMathOperator\GL{GL}$Apologies if this question has already been dealt with on MO. I am wondering about the status of the global Langlands conjectures for $\GL_2$ over the rational numbers. ...
2 votes
1 answer
149 views

Basic results concerning the intertwining operator in the $\mathrm{SL}_2$ case

I am reading [Ikeda, Tamotsu, On the location of poles of the triple L-functions]. On page 194, the author recalled some known results concerning $\operatorname{SL}_2$. I would like to know any ...
3 votes
0 answers
113 views

$L^2$-spectrum versus automorphic discrete spectrum

Let $G$ be a classical group defined over a number field $F$. In his monumental book, (https://www.ams.org/books/coll/061/coll061-endmatter.pdf) Arthur described a spectral decomposition of $L_{disc}^...
6 votes
1 answer
499 views

Langlands-Shahidi method in classical language

The Langlands-Shahidi method says that the $L$-functions of automorphic representations appear in the constant terms of Eisenstein series. Since those Eisenstein series have analytic continuation and ...
3 votes
1 answer
166 views

Frobenius-Schur indicator of a self-dual L-parameter

Let $F$ be a non-archimedean field and let $\pi$ be a self-dual supercuspidal representation of $\mathrm{GL}_n(F)$ (which, by a result of Adler exists only when $n=1$ or $n$ is even). Then, under LLC ...
1 vote
0 answers
86 views

Pure, residual, 2-dimensional, semisimple $G_{\mathbb{Q}}$ representations

Let $G_{\mathbb{Q}}$ be the absolute Galois group of the rationals. I want to know how many isomorphism classes of 2-dimensional, irreducible, pure Galois representations are there over a finite field ...
4 votes
0 answers
146 views

On decomposition of the space of automorphic forms (via central characters)

Let $F$ be a number field and $\mathbb{A}$ be its adele. For simplicity, we assume $G$ is a connected reductive group. Given a unitary central character $\chi: Z_{G}(F)\backslash Z_{G}(\mathbb{A}) \to ...
4 votes
0 answers
179 views

What are the modularity conjectures for Artin motives?

Classically, singular cohomology is an important tool for studying topological spaces, in particular, complex varieties. In the mid-twentieth century it was realized that there are many analogues of ...
5 votes
1 answer
600 views

Understand the $p$-adic local Langlands correspondence with examples

Let $\rho:G_{\mathbb{Q}}\rightarrow \mathbf{Gl}_{n}(\mathbb{Q}_{p})$. I would like to understand in depth why the local Langlands correspondence for $\rho_{|\mathbb{Q}_{p}}$ must consider $p$-adic ...
5 votes
1 answer
799 views

Can Taniyama-Shimura conjecture be generalized to curves of higher genus (within Langlands framework)?

The Shimura-Taniyama-Weil conjecture asserts that if E is an elliptic curve over Q, then there is an integer N ≥ 1 and a weight-two cusp form f of level N, normalized so that a1(f) = 1, such that ap(E)...
22 votes
1 answer
2k views

What do Hecke eigensheaves actually look like?

Let $\mathbb F_q$ be a finite field, $C$ a curve over $\mathbb F_q$ of genus $g\geq 2$, $\rho: \pi_1(C) \to GL_2(\overline{\mathbb Q}_\ell)$ an irreducible local system. The geometric Langlands ...
3 votes
0 answers
223 views

Basic question on the Langlands conjectures for $GL_n$ over global field of positive characteristic

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 $G=GL_n$. ...
8 votes
1 answer
624 views

Absolute convergence of Rankin–Selberg series

Let $\pi$ and $\pi'$ be two general automorphic representations on $\operatorname{GL}(n)$ and $\operatorname{GL}(n')$ over $\mathbb{Q}$. I heard that the Rankin-Selberg $L$-function $L(s,\pi\times\pi')...
4 votes
1 answer
164 views

On a theorem of Bernstein-Zelevinsky regarding supercuspidal resentations

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 $\pi$ ...
8 votes
1 answer
316 views

Is the nonvanishing of Langlands L-functions at $s=1$ conjectured?

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 ...
17 votes
1 answer
2k views

How does the cohomology of the Lubin-Tate/Drinfeld tower fit into categorical p-adic local Langlands?

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 ...
5 votes
0 answers
183 views

Finite coefficients Langlands for function fields

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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