All Questions
Tagged with galois-representations rt.representation-theory
64 questions
31
votes
1
answer
5k
views
Modern proof of Serre's open image theorem?
Let $E$ be an elliptic curve defined over a number field $K$ without complex multiplication. Serre's open image theorem (which appears in his book 'Abelian $l$-Adic Representations and Elliptic Curves'...
- 1,905
28
votes
1
answer
3k
views
Are there Maass forms where the expected Galois representation is $\ell$-adic?
Recall that by theorems of Deligne and Deligne--Serre, there is the following dichotomy:
Modular forms on the upper half plane of level $N$ and weight $k\geq 2$ correspond to representations $\rho:\...
- 18.7k
23
votes
3
answers
4k
views
Subgroups of GL(2,q)
I have been wondering about something for a while now, and the simplest incarnation of it is the following question:
Find a finite group that is not a subgroup of any $GL_2(q)$.
Here, $GL_2(q)$ is ...
- 295
14
votes
2
answers
2k
views
"Purely local" proof of local Langlands
As from this website
http://math.uchicago.edu/~lxiao/workshop_site/
My question is: What does it mean by "purely local"?
Also, I heard about this phrase "purely local" in other problems as well, ...
- 1,503
14
votes
1
answer
506
views
Local-global principle for split extensions of Galois representations
I guess the following is well-known (and probably follows from Chebotarev's density theorem, but I'm not very comfortable with it):
Define some notation:
$K$ a global field,
$G$ the absolute Galois ...
- 5,504
9
votes
1
answer
507
views
A question about Galois representations
Let $K$ be a number field and $(\rho,V)$, $(\rho',V')$ be two Galois representations of $\mathrm{Gal}(\overline{\mathbb{Q}}/K)$. Suppose that for some positive integer $n$ we have $\mathrm{Sym}^n\rho\...
9
votes
2
answers
759
views
Do two dimensional representations with the same adjoint representation differ by a character?
Let $K$ be a field of characteristic not equal to $2$. Let $\text{ad} : \text{GL}_2(K) \to \text{GL}_3(K)$ be the adjoint representation, obtained by $\text{GL}_2(K)$ acting on $2 \times 2$ matrices ...
- 93
8
votes
2
answers
1k
views
number of irreducible representations over general fields
For a finite group, there are finitely many irreducible representations of complex numbers.
What if the field is changed to some other fields? Like real numbers, p-adic field, finite field?
In ...
- 1,503
8
votes
2
answers
655
views
Misunderstanding in the hypotheses of Schlessinger's criterion
In studying deformation theory of Galois representations, I've come surely to an error, relating Schlessinger's criterion.
Let's fix a representation $\bar{\rho}$ of a group $G$ and let $D_{\bar{\...
8
votes
1
answer
567
views
Symmetric power lift of modular forms
Let $f_1$ and $f_2$ be two cuspforms of weights $k_1$ and $k_2$ and nebentypus $\epsilon_1$ and $\epsilon_2$ respectively such that $f_1 \neq f_2 \otimes \chi$ for some Dirichlet character $\chi$ of ...
- 424
7
votes
2
answers
1k
views
Classify 2-dim p-adic galois representations
Recently I have known how to classify 1-dim p adic Galois representations $\phi$. The p-adic Galois representations mean that a representation $G_K$ on a p-adic field $E$, where $K$ is also a p-adic ...
user141691
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
7
votes
0
answers
570
views
Which de Rham representations are trianguline?
Let $K/\mathbf{Q}_p$ be a finite extension, and let $V$ be an $n$-dimensional $\overline{\mathbf{Q}_p}$-vector space with a continuous action of $G_K$. Suppose $V$ is de Rham, so potentially ...
- 13.1k
6
votes
2
answers
846
views
Serre's conjecture for mod-p^n representations?
I think this may be a silly question, but here goes. Let $\rho:\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\to \mathrm{GL}_2(\overline{\mathbf{F}_p})$ be a representation; say $\rho$ is of S-type ...
- 13.1k
6
votes
1
answer
397
views
A question on the Hecke L-function
For a Hecke L-function, if all of the local eigenvalues are roots of unity, is it an Artin L-function?
- 63
6
votes
1
answer
350
views
If $G$ is compact, $H \leq G$ open, $V$ an irreducible $H$-rep, is $\text{Ind}_H^G$ semisimple?
Let $G$ be a compact group, $H$ a normal open subgroup, and $K$ a $p$-adic field (so that not all $G$-reps with coefficients in $K$ are semisimple). Let $V$ be a finite-dimensional topological $K$-...
- 1,453
6
votes
2
answers
985
views
Irreducible representations of compact groups
Let G be a compact group (or even profinite - Galois group). Let $V$ be a vector space over the field ${\mathbb F}_p$ with $p$ elements, $p$ a finite prime, such that $V$ is a contable product of ${\...
- 11.3k
5
votes
2
answers
237
views
Irreducibility of the $n$th symetric power of the reduction of the Galois representation of a non-CM newform
In "On $\ell$-adic representations attached to modular forms II", Ribet proved that the $\ell$-adic representation $\rho_{f,\ell}$ attached to a non-CM newform form $f$ satisfies
$${\rm SL}...
- 303
5
votes
2
answers
356
views
Is there a semisimple $\mathbf{Q}_\ell$-representation of $G_F$ ramified at an infinite set of places?
See http://math.uni.lu/~wiese/galois/Boeckle-Luxemburg-Notes.pdf, Theorem 1.4(a): Is there an example of a semisimple $\mathbf{Q}_\ell$-representation $V$ of $G_F$ ($F$ a global field) ramified at a ...
user19475
5
votes
1
answer
486
views
Generalized Hodge-Tate weights of an arbitrary p-adic Galois representation
Let $V$ be a continuous representation of the absolute Galois group of $\mathbb{Q}_p$ with coefficients in $\mathbb{Q}_p$. The theory of Sen attaches to $V$ generalized Hodge-Tate weights which are ...
- 643
5
votes
1
answer
254
views
Endomorphism algebras of restricted representations
Let $G$ be a group, and
$$\rho:G\to \mathrm{GL}(V)$$
be an absolutely irreducible, finite-dimensional representation over a characteristic $0$ field $k$. For each finite index subgroup $H\le G$, let
$...
- 874
5
votes
1
answer
289
views
Irrelevant parabolics and inner forms of GSp(4)
In Ralf Schmidt's appendix to "Jacquet-Langlands-Shimizu correspondence for theta lifts to $\mathrm{GSp}(2)$ and its inner forms" by Narita and Okazaki , he computes the representations of $\mathrm{...
- 2,429
5
votes
0
answers
169
views
Question About Page 11 of Cornell, Silverman, Stevens, "Modular Forms and Fermat's Last Theorem"
Looking at page 11 of the text, consider a Galois representation $\rho: G_{\mathbb Q} \to \operatorname{GL}_2(A)$, where $A$ is a coefficient ring (i.e. complete Noetherian local ring with finite ...
- 283
5
votes
0
answers
192
views
Globalizable Galois representations
Let $\rho$ be a $p$-adic representation of $G=\text{Gal}(\bar{\mathbb{Q}_p}/\mathbb{Q}_p)$.
When does $\rho$ extend to a representation of the global galois group? What can be said about the locus ...
5
votes
0
answers
132
views
Field of definition of compatible system of Galois representations
Let $K,F$ be number fields and suppose that there is a compatible system of Galois representations
$$(\rho_{\lambda} : \text{Gal}(\overline{K}/K) \longrightarrow \text{GL}_n(\overline{F}_\lambda) )$$
...
- 173
4
votes
2
answers
1k
views
Galois theory of endomorphism rings of irreducible representations
Let $k$ be a field which I don't suppose to be algebraically closed. Then, the endomorphism ring of any irreducible representation of a finite group over $k$ is a division ring.
What is known about ...
- 33.8k
4
votes
1
answer
202
views
Pro-$l$ Sylow action in a primitive representation of inertia over $\overline{\mathbb{F}}_l$
Let $K$ be a nonarchimedean local field of residue characteristic $p \neq l$ and let $I_K$ be the inertia subgroup of its absolute Galois group. Let $V$ an irreducible representation of $I_K$ over $\...
- 5,009
4
votes
1
answer
270
views
Are there Galois representations associated with any regular algebraic cuspidal automorphic representation?
Let $K$ be a number field and $\pi$ be a regular algebraic cuspidal automorphic representation on $\mathrm{GL}_2(\mathbb{A}_K)$. Let $\lambda$ be a prime of the field of Fourier coefficients of $\pi$ ...
user130124
4
votes
1
answer
239
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
4
votes
1
answer
275
views
Regular representations of Galois groups
Suppose $\mathcal{G}_k$ is the absolute Galois group of a number field $k$.
$\mathcal{G}_k$ is a topological group, with profinite topology. How does the theory of harmonic analysis of regular ...
- 221
4
votes
0
answers
173
views
Are there methods to compute the induced action of Frobenius map on the Neron-Severi group of a supersingular abelian surface over a finite field?
Let $A$ be a supersingular abelian surface over a finite field $\mathbb{F}_q$. In that case the Neron-Severi group $NS(A\otimes\overline{\mathbb{F}_q})$ is the lattice of rank $6$. Are there methods ...
- 1,833
4
votes
0
answers
380
views
Which properties of p-adic representations can be recovered from (\varphi,\Gamma)-modules?
In Berger's "An introduction to the Theory of $p$-adic Representations", he mentions that due the the equivalence of categories between etale $(\varphi,\Gamma)$-modules and $p$-adic representations, ...
- 41
3
votes
1
answer
324
views
Faithful representations of free pro-p groups
Let $p$ be a prime number, $m,n \in \mathbb{N}$, $F = F(p,m)$ be the free pro-$p$ group on $m$ generators. For which $(m,n)$ there is a continuous faithful representation (embedding) $\rho : F \...
- 11.3k
3
votes
1
answer
131
views
Depth of the filtration of higher ramification groups in the ramified case in Serre's modularity conjecture
I am studying Serre's paper "Sur les représentations modulaires de degré 2 de $\mathrm{Gal(\overline{\mathbb Q}/\mathbb Q)}$" and I have some questions about Serre's definition of "peu ...
3
votes
1
answer
184
views
Restriction of $(\varphi, N)$-modules
For any $p$-dic field $K$, we have an equivalence of categories
$$D_{st}:Rep_{\mathbb{Q}_p}^{st}(G_K)\rightarrow MF_K^{ad}(\varphi,N),\quad V\mapsto (B_{st}\otimes_{\mathbb{Q}_p} V)^{G_K}$$
with quasi-...
- 4,408
3
votes
0
answers
183
views
Representability of $\operatorname{Hom}(G_{\mathbb{Q}}, \operatorname{GL}_2)$
Let $G_{\mathbb{Q}}$ be the absolute Galois group of the rationals, and let $F: \mathrm{Aff}/\textbf{Q}_p\longrightarrow \mathrm{Sets}$ be the functor which associates to every affine $\mathbb{Q}_p$ ...
- 2,907
3
votes
0
answers
86
views
Freeness of completed homology over universal deformation ring
In Theorem 7.4 of the paper "patching and the $p$-adic Langlands program for $\mathrm{GL}2(\mathbf{Q}_p)$" (arXiv link), it is proved that in the minimally ramified case, the completed homology of ...
- 497
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
3
votes
0
answers
213
views
Galois action on local deformation ring
Let ${\Bbb Q}_p$ be a local field. For a prime $q \not= p$, we consider an irreducible residual Galois representation
$\overline{\rho} \colon {\mathrm{Gal}}(\overline{{\Bbb Q}}_p/{{\Bbb Q}_p}) \to \...
- 781
3
votes
0
answers
199
views
Decompositions of representations of pro-p groups
Let $P$ be a pro-p group. Assume that there is a filtration of $P$ by normal subgroups $P_i$ such that $P_0=P$ and $P_{i+1} < P_i(i\in\mathbb N)$. Let $V$ be an $l$-adic representation of $P$, ...
- 93
3
votes
0
answers
740
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
2
votes
2
answers
584
views
Lifting projective Galois representation with condition
Let $\bar{\rho}: G_K\to PGL_n(\mathbb{C})$ be projective representation of the absolute Galois group of a number field $K$ and $\varphi\in Aut(G_K)$.
A theorem of Tate tells us that we can always ...
- 845
2
votes
1
answer
449
views
$l$-adic representations from Shimura curves
This question may be kind of vague. And we use the same notations as in Carayol's papers:
H. Carayol, Sur les représentations l-adiques associées aux formes modulaires de Hilbert;
H. Carayol, Sur la ...
- 73
2
votes
1
answer
232
views
Local to global for semistable $G_{\mathbb{Q}_p}$-representations
Let $\rho_p:G_{\mathbb{Q}_p} \to \text{Gl}_n(\mathbb{Q}_p)$ be semistable representation. In local to global Galois representation, it was asked if one can find a geometric global Galois ...
- 4,408
2
votes
1
answer
317
views
Local factors determine Weil representations - proof of the cyclic case
I already created this post on Math Stack Exchange but I was not so sure if this question fits better here. If it is not, I want to apologize in advance and feel free to delete my post.
I want to ...
- 103
2
votes
1
answer
881
views
How to prove the p-adic Galois representations atteched to the Tate module of an abelian variety is de Rham directly?
Recently I read a thesis p-adic Galois representations and elliptic curves. Using Tate's curve, the author proved the p-adic Galois representations atteched to the Tate module of an elliptic curve is ...
user141691
2
votes
1
answer
173
views
Norm 1 elements of an unramified quadratic extension of a local field
Let $E$ be an unramified quadratic extension of a local field $F$, with $p$ odd. Let $E^1$ denote the set of norm $1$ elements of $E$. What can be said about the following index:
$$
{\rm 1.}\ \ \ \ [ ...
- 63
2
votes
0
answers
125
views
Semisimplicity of induced representation of a irreducible representation
This question occurs when I read this one.
Suppose $k$ is a field of char $0$ (not necessarily complex numbers, in my case it's $\mathbb Q_p$), and $G$ is a profinite group (in my case, $\operatorname{...
- 775
2
votes
0
answers
101
views
On the irreducible submodules of adjoint representations $\text{ad}^{0}$
Let $k$ be a finite field of characteristic $p$. Let $H$ be a subgroup of $\rm{GL}_{n}(k)$ of order prime to $p$ where $n\geq2$. Assume that the representation $H\hookrightarrow \rm{GL}_{n}(k)$ is ...
- 667
2
votes
0
answers
76
views
Question on GSpin-Valued L-parameters
Let $\Gamma$ be a topological group, $n \geq 1$ an integer, $\ell$ a prime number, and $\overline{\mathbb{Q}}_{\ell}$ the algebraic closure of the $\ell$-adic integers. We set $\Phi(GSpin_{2n + 1})$ ...
- 21