Questions tagged [galois-representations]
The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local or global fields is an important tool in number theory.
627 questions
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}...
- 1,949
4
votes
3
answers
793
views
Quotients of Tate modules
Let $p$ be a prime number, let $K$ denote a finite extension $\mathbb{Q}_{p}$ and let
$\overline{K}$ be an algebraic closure of $K$. Let $A$ be an ellitpic curve over
$K$ and denote by $T_{p}A$ its ...
- 1,949
6
votes
0
answers
196
views
Status of Mazur's dimension conjecture on universal deformation rings
Fix a prime $ p $. Let $ \mathbb{F} $ be a finite field of characteristic $ p $ and $ W(\mathbb{F}) $ the ring of Witt vectors of $\mathbb{F} $ . Let $ \widehat{\mathfrak{A}}_{W(\mathbb{F})} $ be the ...
- 63.9k
1
vote
1
answer
282
views
Explicit Chebotarev density theorem for Galois representations associated to newforms
Let $f \in S_2(\Gamma_0(N))$ be a newform with associated residual Galois representation $\rho: \operatorname{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \to \operatorname{GL}_2(\mathbf{F})$, $\mathbf{F}$ ...
- 5,089
4
votes
1
answer
202
views
On presentations of universal rings of deformations
Let $k$ be a finite field of characteristic $p$, and $R$ a complete local noetherian algebra with residue field $k$. It is well known that $R$ has a natural structure of an algebra over the ring Witt ...
- 37k
3
votes
0
answers
300
views
Galois invariant of Tate module
Let $K$ be a finite extension of $\mathbb{Q}_p$. Let $V$ be a de Rham representation of $G_K=\operatorname{Gal}(\overline{K}/K)$. By Corollary 3.8.4 of Bloch and Kato - L-functions and Tamagawa ...
- 247
2
votes
0
answers
182
views
Image of the Kummer map for abelian varieties over $p$-adic local fields
The following statement might be well-known to the community: let $K$ be a finite extension of $\mathbb{Q}_p$ for some prime $p$. Let $A$ be an abelian variety over $K$. Then the image of the Kummer ...
- 1,428
5
votes
0
answers
197
views
On the pro-category of finite local artinian algebras
Let $\mathbb{F}$ be a finite field, and $W(\mathbb{F})$ its associated ring of Witt's vectors. On page 6 of the following lecture notes Deformations of Galois Representations, the category $\mathfrak{...
- 541
4
votes
1
answer
394
views
Reductive subgroups of $\mathrm{GL}_2$ over an algebraically closed field of characteristic zero
I am reading a very nice paper of Newton and Thorne, Symmetric power functoriality for holomorphic modular forms, and there is an argument concerning the (Zariski-closure of) image of certain $p$-adic ...
- 12.9k
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 ...
- 4,408
12
votes
3
answers
1k
views
Motivation for uniform surjectivity of mod l representations associated to elliptic curves
Background
Let $E$ be an elliptic curve over $\mathbb{Q}$ and let $G_{\mathbb{Q}}$ be the absolute Galois group $Aut(\overline{\mathbb{Q}})$. For any positive integer $n$ the $n$-torsion subgroup $E[...
- 1,283
5
votes
1
answer
625
views
What is the relation of the absolute Galois group and classical profinite groups?
Consider the absolute Galois group $G = \mathrm{Gal}(\overline{\mathbb{Q}}: \mathbb{Q})$ and $G_p = \mathrm{Gal}(\overline{\mathbb{Q}_p}: \mathbb{Q}_p)$.
Abelian class field theory gives us for the ...
- 35.5k
14
votes
2
answers
1k
views
Effective Chebotarev without Artin's conjecture
$\DeclareMathOperator\Frob{Frob}$Iwaniec and Kowalski, in their famous book Analytic Number Theory states a strong form
of the effective Chebotarev density theorem page 143, and prove it assuming both ...
- 63.9k
3
votes
1
answer
445
views
Galois cohomology of abelian varieties
Suppose $A$ is an abelian variety over a number field $K$ and call $M$ the maximal torsion free quotient of $A(\overline{K})$ equipped with its Galois action.
For the first Galois cohomology of $M$, ...
- 1,424
13
votes
2
answers
1k
views
Best bounds toward Serre's uniformity conjecture
If $E$ is a non-CM elliptic curve over $Q$, then it is a famous theorem of Serre
that there is some integer $M(E)$ such that for any prime $\ell > M(E)$, the image of the
Galois representations $\...
- 10.4k
4
votes
0
answers
421
views
Subgroup structure of certain maximal subgroups of PSL$(3,q)$
I have been reading the paper "On the images of modular and geometric three-dimensional Galois representations" (see also the earlier arXiv preprint). But there is a detail in section 6.2 ...
- 4,713
1
vote
0
answers
129
views
Galois extension of $ C_{k} $ field and irreducible polynomial over $ C_{k} $ field
A field is called a $ C_{k}$- field if every form i.e. every homogeneous polynomial of degree $ d $ in $ n > d^{k} $ indeterminates has a nontrivial zero. We know that any finite field is a $ C_{...
- 63.9k
29
votes
2
answers
2k
views
Does the image of a p-adic Galois representation always lie in a finite extension?
I have been looking at Serre's conjecture and noticed that there are two conventions in the literature for a p-adic representation $\rho:\mbox{Gal}(\bar{\mathbb Q}/\mathbb Q)\to \mbox{GL}(n,V).$ In ...
- 50.6k
13
votes
1
answer
771
views
Abelian $\ell$-adic representations in $\widehat{\mathrm{SL}(2,\mathbb{Z})}$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Gal{Gal}\newcommand{\Z}{\mathbb{Z}}$In Grothendieck's Esquisse he claims that the action of
$$\Gal(\mathbb Q)\to\text{Out}(\pi_1(M_{1,1})=\text{Out}(\...
- 63.9k
4
votes
0
answers
166
views
Constructing motivic representations through extensions of $\mathrm{SL}(2, \mathbb{Z})$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$In the Esquisse, Grothendieck explains how to find the representations of $G_{\mathbb{Q}}$ on Tate modules of Jacobians through its action on $\...
- 63.9k
3
votes
0
answers
60
views
Family of residual representations compatible with a motive
Suppose $S$ is a set of primes. For each prime $p$ in $S$ we are given a Galois representation $\rho_p : Gal(\bar{\mathbb{Q}}/\mathbb{Q}) \rightarrow GL_2 (\mathbb{F}_p) $. What are the necessary or ...
- 31
2
votes
1
answer
247
views
Galois representations attached to a cusp form for different primes
If I have a cusp form $f$, I can consider the associated Galois representation $\rho_l(f)$ for any prime $l$. For two distinct primes $p$ and $q$, what is the relationship between $\rho_p(f)$ and $\...
- 6,039
3
votes
0
answers
529
views
The cycle class map with values in crystalline cohomology
Let $ k = \mathbb{F}_q $ be a finite field of characteristic $ p > 0 $.
Let $ X $ be a smooth proper scheme of dimension $ d $ over $ k $.
Consider the associated $ K $ - linear cycle class map ...
- 595
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}(...
- 37k
1
vote
0
answers
178
views
Compute dimension of space of modular forms by counting Galois representations
It is known that we can compute the dimension of the space $S_{k}^{\mathrm{new}}(N, \chi)$ of new forms of weight $k\geq 2$ and level $N$ and Nebentypus $\chi$ via Riemann-Roch theorem or using ...
- 2,215
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 ...
- 103
3
votes
0
answers
311
views
Eichler orders in a certain quaternion algebra
Let us consider a totally real number field $K$ such that $[K \colon {\Bbb Q}] = {\mathrm{odd}}$. We shall consider the quaternion algebra $D$ over $K$ such that $D$ splits everywhere at finite places ...
- 12.9k
8
votes
2
answers
629
views
Motivation of the construction of $p$-adic period rings
Let $B$ be either $B_{\text{dR}}$ or $B_{\text{crys}}$. For a $\mathbb{Q}_p$-representation $V$ of the absolute Galois group $\mathrm{Gal}(\overline{K}/K)$ of a $p$-adic field $K$ (a finite extension ...
- 10.9k
6
votes
2
answers
1k
views
Topology on $p$-adic period rings in an article by Fontaine, part II
This is a follow-up to this question. See that question for background and relevant notation.
In the answer to that question, it is claimed, if I understand the answer correctly, that a basis of ...
- 9,273
3
votes
0
answers
220
views
How the Galois group acts on a Néron–Severi group of a variety?
Let $K/k$ be a Galois extension with Galois group $\Gamma$ and let $X$ be a variety over $k$. Assume that either $X(k)\neq\varnothing$ or $\mathrm{Br}(k)=0$, the Brauer group of $k$. By the Hochschild-...
- 165
2
votes
0
answers
230
views
Determining existence of a $p$-isogeny from $p|E(\mathbb{F}_{\ell})$
In Siksek's notes The modular approach to Diophantine equations he uses the following result:
Let $p$ be an odd prime. For an elliptic curve $E$ over $\mathbb{Q}$ if $p|E(\mathbb{F}_{\ell})$ then for ...
- 131
3
votes
0
answers
106
views
A uniform version of Bashmakov's theorem for elliptic curves
Let $E/\mathbb Q$ be an elliptic curve. Serre's open image theorem is the statement that the image of the Galois group $G_{\mathbb Q}$ into $GL_2(\mathbb Z/n\mathbb Z)$ by it's action on the torsion ...
- 7,746
2
votes
0
answers
208
views
Galois-action on spectral sequence
Let $X_\bullet\to S$ be a proper surjective hypercover of a $k$-scheme by smooth proper $k$-schemes. This gives a proper surjective hypercover $X'_\bullet\to S_{\bar{k}}$ where $X'_n:=X_n\times_k \bar{...
- 4,408
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
0
votes
0
answers
75
views
An invariant subspace under $G_Q$ action , and BSD-rank
Let $E/Q$ be an elliptic curve. $E(\bar{Q})$ is a complicated abelian group, which equals to all closed points of $\bar{E}$, and also, a $G_Q$-Galois module. Its torsion part, $E(\bar{Q})_{tor}$ is a ...
- 547
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
2
votes
0
answers
229
views
Reference Request - New proof of Ribet's level lowering by Khare and Wintenberger
I'm currently following the note of Sug Woo Shin's course at Berkeley with notes taken by Rong Zhou. In Section 24.3 (Page 86), Ribet's level lowering theorem is stated:
[Theorem 24.7] $E = E_{a^{\...
- 639
4
votes
0
answers
238
views
Diophantine consequences of the Buzzard–Diamond–Jarvis conjecture
Serre's modularity conjecture famously implies Fermat Last Theorem. More generally, Serre's conjecture implies that certain generalized Fermat equations have no non-trivial solutions (see Section 4.1 ...
- 595
3
votes
1
answer
380
views
Comparison of two $GL_N(\mathbb{Z}_\ell)$ Galois representations
I have a question about comparing two $\ell$-adic Galois representations. Suppose we have two irreducibible Galois representaions
$$
\rho_1,\rho_2: Gal(\overline{\mathbb{Q}}/\mathbb{Q})\to GL_N(\...
CommunityBot
- 1
3
votes
0
answers
335
views
A question on the Robba ring
Notation is as in the question:
https://math.stackexchange.com/questions/4090045/some-questions-about-the-robba-ring.
We define a new operator over the Robba ring as follows. Put $$c=\frac{pE(u)}{E(0)}...
- 166
3
votes
1
answer
474
views
To identify $p$-adic Tate module $T_p(G)$ of $p$-divisible group $G$ in the category $\text{Rep}_{\mathbb{Q}_p}(G_{K_\infty})$
Let $k$ be a perfect field of characteristic $p>0$, $W=W(k)$ its ring of Witt vectors, $K_0=W(k)[\frac{1}{p}]$ and, $K/K_0$ be a totally ramified extension. Let $\pi \in K$ be an uniformizer.
...
- 166
4
votes
0
answers
121
views
Looking for a crystalline analogue of , $\mathcal {Z}_{\sim}^* (X)_F \simeq \mathcal{Z}_{\sim}^* (X_{k ' })_F^{\mathrm{Gal} ( k ' / k )} $
Is there a crystalline analogue for the following formula, using the crystalline Frobenius $ F_v $ instead of the absolute Galois group $ \mathrm{Gal} (\overline{k} / k) $ ?
Here is the formula, which ...
- 595
3
votes
1
answer
776
views
On the coherence of a Néron-ring
Let $A:= \underset{\lambda \in \Lambda}{\varinjlim} \,A_{\lambda}$ be an inductive limit of geometric regular local ring $(A_{\lambda}, {\frak m}_{\lambda})$, whose transition map $\phi_{\mu\lambda} \...
CommunityBot
- 1
9
votes
0
answers
441
views
Commutative algebra details on patching when proving $R = \mathbb{T}$ theorem (Calegari-Geraghty Paper)
I have originally posted this on math.SE and been suggested to post this here. I'm merely an undergraduate student and it is the first time for me to ask questions here. I'm sincerely sorry if these ...
- 639
4
votes
1
answer
807
views
Meaning of Atkin-Lehner eigenvalues
Suppose I have $f\in S_2(\Gamma_0(N))$ a classical modular newform of level $N$. I want to understand what information (if any) is carried by its Atkin-Lehner eigenvalues for primes $p\mid N$, as ...
- 5,089
4
votes
1
answer
280
views
Semisimplicity of the étale cohomology mod $p$
Let $X$ be a smooth projective variety over a field $k$. Then if $\ell\neq \text{char} k$, $k$ is finite, and $X$ is an abelian variety it was shown by Weil that the $\ell$-adic cohomology of $X_{k^{...
- 86
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 ...
- 21.3k
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 ...
- 63.9k
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-...
- 37k
7
votes
0
answers
444
views
Status of the conjectured vanishing of Bloch-Kato H^2
There is a folklore conjecture that $\operatorname{Ext}^2$ vanishes in the category of geometric $p$-adic Galois representations (i.e. representations that are unramified almost everywhere and de Rham ...
- 35.5k