Questions tagged [langlands-conjectures]
Higher reciprocity laws
370
questions
3
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0
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107
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Motivic $L$-functions came from automorphic representations
Langlands in his 1978 ICM talk made a conjecture that all motivic $L$-functions should arise as automorphic $L$-functions. A part of this conjecture, namely for some Hasse-Weil $\zeta$ functions is a ...
5
votes
0
answers
185
views
Applications of Langlands for GLn explicit reciprocity laws other than elliptic curves
Is there any explicit application of Langlands conjecture for $\mathrm{GL}(n)$
for $n\ge 3$, to get some reciprocity laws for higher dimensional varieties or higher genus curves?
I've never found such ...
1
vote
0
answers
136
views
Homotopical interpretation of Langlands correspondence
Recently I began learning about homotopy theory, I am very far from being familiar with all the basic notions and constructions, however I heard of the notion of topological modular forms. I also ...
6
votes
2
answers
245
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
167
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 ...
0
votes
1
answer
187
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 ...
1
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0
answers
68
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
167
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 ...
2
votes
0
answers
85
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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 ...
7
votes
0
answers
147
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
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0
answers
136
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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
181
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
488
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 ...
5
votes
1
answer
201
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
90
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
255
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
354
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 ...
1
vote
2
answers
249
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 $L$...
2
votes
1
answer
237
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 ...
4
votes
2
answers
280
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
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
176
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} $ ...
6
votes
0
answers
219
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 ...
3
votes
0
answers
323
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 ...
3
votes
1
answer
141
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
217
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
124
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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 ...
4
votes
1
answer
597
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 ...
3
votes
1
answer
473
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
1
answer
151
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 ...
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
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
503
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
169
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
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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
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0
answers
147
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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
184
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
627
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
807
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)...
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 $G=GL_n$. ...
4
votes
1
answer
165
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
318
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
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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 $$\pi_1(X)\to \mathrm{GL}_n\left(\overline{\mathbb{F}_p}\right)...
2
votes
0
answers
176
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Why are they called reductive groups? [duplicate]
The reductive groups play a central role in the Langlands correspondence. Why are these groups called reductive? Does this name suggest something conceptual about these groups?
1
vote
1
answer
336
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From Galois representations to automorphic forms for $\mathfrak{sl}_2$ (via Drinfeld's shtukas)
Drinfeld-Lafforgue have proven function fields Langlands conjectures in type A: see https://arxiv.org/pdf/math/0212417.pdf (Laumon's survey in English), https://arxiv.org/pdf/math/0212399.pdf (...
4
votes
0
answers
163
views
Understanding Shimura correspondence in context of Langlands functoriality
Recently, I started to read about automorphic forms and representations on covering groups, e.g. metaplectic groups. I set my first goal as understanding Shimura's correspondence in representation ...
7
votes
1
answer
420
views
Non-existence of "higher" Artin map
So rank $1$ local Langlands is special in as that it is given by the Artin map
$$\text{GL}_1(K)\to G_K^{ab},$$
whereas in the higher rank (to the best of my knowledge) there doesn't exist a map
$$\...