All Questions
Tagged with galois-representations langlands-conjectures
42 questions
4
votes
1
answer
238
views
A specific $2$-dimensional Galois representation of $G_{\mathbb{Q}_2}$ and its Langlands correspondence
I am interested in understanding a situation in (classical, not $p$-adic) local Langlands for $\mathrm{GL}_p(\mathbb{Q}_p)$. An example of
it is as follows: Let $F=\mathbb{Q}_2$ and $E$ be the ...
- 1,856
5
votes
0
answers
232
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 ...
- 151
2
votes
0
answers
103
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 ...
- 7,037
5
votes
1
answer
220
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}...
- 833
5
votes
1
answer
833
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 ...
- 1,270
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 ...
- 3,309
5
votes
0
answers
481
views
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 ...
- 4,398
9
votes
1
answer
322
views
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}(...
- 311
21
votes
1
answer
2k
views
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 ...
- 3,309
3
votes
0
answers
148
views
$p$-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 $\...
- 503
12
votes
1
answer
541
views
Eigenvarieties and functoriality
In Langlands' review of Hida's book "$p$-adic automorphic forms on Shimura varieties", he discusses a nexus of 4 areas of modern number theory: automorphic representations, motives, spaces ...
user136098
11
votes
1
answer
646
views
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 ...
- 1,267
6
votes
0
answers
230
views
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,...
- 311
11
votes
1
answer
770
views
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 ...
- 643
3
votes
0
answers
184
views
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 (...
- 6,170
5
votes
0
answers
192
views
Globalizable Galois representations
Let $\rho$ be a $p$-adic representation of $G=\text{Gal}(\bar{\mathbb{Q}_p}/\mathbb{Q}_p)$.
When does $\rho$ extend to a representation of the global galois group? What can be said about the locus ...
2
votes
0
answers
151
views
Central character of automorphic representations of $Sp_{2n}$
Let $F$ be a CM field. Given a regular algebraic self-dual cuspidal automorphic representation $\Pi$ of $GL_n(\mathbb A_F)$ and a prime $l$, there is a continuous Galois representation $r_{\Pi}: \...
- 453
3
votes
1
answer
136
views
If $\Pi$ and $\Sigma$ agree at almost all places, then the central character of $\Pi$ corresponds to $\operatorname{Det} \Sigma$
Let $\Sigma$ be an $n$-dimensional representation of the global Weil group $W_F$ for a number field $F$, and $\Pi$ an automorphic representation of $\operatorname{GL}_n(\mathbb A_F)$. Suppose that at ...
- 6,170
3
votes
0
answers
191
views
Automorphy of families of motives
I have a couple of elementary questions regarding automorphy of Galois representations arising from geometric families.
Suppose we have an algebraic family of varieties over a number field, and ...
user127738
3
votes
0
answers
504
views
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 ...
user125985
28
votes
1
answer
2k
views
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_{...
- 3,058
36
votes
3
answers
2k
views
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 ...
- 461
5
votes
1
answer
289
views
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{...
- 2,429
13
votes
0
answers
284
views
Propagation of modularity and the Artin conjecture
The (still incomplete) solution of the Artin conjecture on dimension $\leq2$ has been a massive research effort that has spanned (knowingly or not) around a century.
A very natural question is, what ...
- 17.6k
18
votes
1
answer
564
views
To what extent are modular parametrizations expected to generalize?
By the Modularity Theorem (a.k.a. the Shimura--Taniyama--Weil Conjecture), if $E$ is an elliptic curve over $\textbf{Q}$ with conductor $N$, then there exists a “modular parametrization” $\psi: X_0(N) ...
- 281
15
votes
2
answers
994
views
Explicit Chebotarev and Langlands - irreducibility of X^5-X-1 mod primes
Is there an explicit infinite set of primes, modulo which $X^5 - X - 1$ is irreducible?
Since our polynomial's Galois group over $\mathbb{Q}$ is $S_5$, Chebotarev's density theorem implies that there ...
- 11.3k
5
votes
1
answer
451
views
The infinity-type of automorphic representations in the Langlands correspondence
Let $K$ be a number field, $\rho\colon \mathrm{Gal}_K\to \mathrm{GL}_n(\overline{\mathbf{Q}_p})$ a geometric (i.e.: unramified a.e., de Rham above $p$) irreducible Galois representation. One piece of ...
- 5,759
6
votes
1
answer
489
views
Extending a representation from the Weil group to the Galois group
Let $F$ be a nonarchimedian local field. Since the Weil group $W_F$ is a dense subgroup of $G_F=Gal(\bar{F}/F)$, it's clear that restriction gives an injection $Irr(G_F)\rightarrow Irr(W_F)$ of ...
- 1,062
7
votes
1
answer
914
views
Explicit calculation of Weil Deligne representations
According to Grothendieck monodromy theorem, l-adic galois representations of a local field corresponds to Weil-Deligne representations.
However, given a galois representation, it is usually difficult ...
- 945
9
votes
0
answers
656
views
Langlands program beyond CM fields?
I apologize since this is a quite vague question. And I am personally at an expert in these fields at all.
It seems to me that there are two main directions of the Langlands program, namely, ...
- 91
0
votes
0
answers
585
views
Local Langlands conjecture for GL(2)
Let $F_v$ be a local field. Let $\sigma_v$ be a two-dimensional representation of $Gal(\overline{F}_v:F_v) \rtimes W_{F_v}$. Now, there exists an infinite-dimensional representation $\pi_v$ of $GL_2(...
- 11.2k
3
votes
0
answers
739
views
The operator \boxtimes and \boxplus in automorphic representations
Given two automorphic representations $\pi_1, \pi_2$ of $GL_2(\mathbb A_Q)$ and $GL_3(\mathbb A_Q)$ respectively. Let $\pi_i =\otimes_v \pi_{i, v}$.
Now, for each $v$, let $\pi_{1, v}\boxtimes \pi_{...
- 31
13
votes
0
answers
2k
views
Why doesn't functoriality immediately imply the modularity theorem?
Let $E/\mathbb{Q}$ be an elliptic curve. By the modularity theorem, the prime indexed coefficients of its $L$-function agree with those of a weight $2$ cusp eigenform $f$ with integer coefficients. ...
- 7,062
3
votes
0
answers
519
views
local deformation rings and Hecke algebras
Let $\bar{\rho}_p$ be a two-dimensional irreducible local Galois representation of $Gal(\bar{\mathbb{Q}}_p / \mathbb{Q}_p)$ on a $k$-vector space, where $k$ is a finite extension of $\mathbb{F}_p$. ...
4
votes
1
answer
394
views
semisimplicity of automorphic Galois representations
Is it known that the Galois representation constructed by Harris and Taylor in their book is semisimple? I can't see this proven in the book, but on the other hand, everywhere else the representation ...
- 2,842
18
votes
1
answer
2k
views
Galois representations attached to Maass form
So, how does one construct a galois representation from a Maass form?
For a modular cusp eigenform, I am familiar with the work of Eichler-Shimura, Deligne, Deligne-Serre, and realize these are ...
- 4,593
2
votes
1
answer
977
views
Rapoport-Zink proof of purity of monodromy
Hi,
Does anyone know if the article
"Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindene Zyklen in ungleicher Charakteristik", INvent. Math, 68 (1980)
by ...
- 647
10
votes
1
answer
1k
views
Carayol via the trace formula
Hi,
Is there a proof of the result that Carayol proves in "Sur les representations l-adiques..."
using the Langlands-Kottwitz method of comparing the Lefschetz trace formula and the Selberg trace ...
- 2,842
7
votes
1
answer
968
views
Semisimple Weil-Deligne representations
I've just realized that I don't understand something important and basic about the Weil-Deligne group and its representations. (I'm not very surprised by this).
Following Deligne's article, Section ...
- 13.3k
10
votes
1
answer
1k
views
P-adic local Langlands for non-unitary representations?
In Colmez's work on the p-adic local Langlands correspondence for ${\rm GL}_2(\mathbb{Q}_p)$, he works with ${\rm GL}_2(\mathbb{Q}_p)$-representations on $p$-adic Banach spaces which admit an ...
- 37k
31
votes
2
answers
3k
views
Elementary Aspects of Galois Deformation
Galois deformations are an important tool in Wiles' arsenal
for proving FLT. Are there any more elementary aspects (I'm
thinking of 1-dimensional Galois representations attached to
number fields) ...
- 32.6k
86
votes
8
answers
13k
views
What are the local Langlands conjectures nowadays, for connected reductive groups over a $p$-adic field?
Let me stress that I am only interested in $p$-adic fields in this question, for reasons that will become clear later. Let me also stress that in some sense I am basically assuming that the reader ...
- 41.4k