All Questions
43 questions
1
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0
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133
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Automorphy of the twisted representation
The Artin reciprocity says that if
$$
\chi: \operatorname{Gal}(K/\mathbb Q) \to \mathbb C
$$
is a 1-dimensional representation of a finite Galois extension $K/ \mathbb Q$, then it corresponds to a ...
- 663
3
votes
1
answer
220
views
Why locally algebraic characters of $\text{Gal}(\overline{\mathbb Q}/\mathbb Q)$ are associated to $A_0$ Grossencharacters/algebraic Hecke characters?
$\DeclareMathOperator\Res{Res}\DeclareMathOperator\Gal{Gal}$I am trying to understand lemma 3.1 of "Abelian Varieties over \mathbb Q and modular forms" of Ribet. ArXiv link
Just so everyone ...
- 31
4
votes
0
answers
100
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Local units in a family of $S_4$-extensions
Let $a \in \mathbb{Z}$ and consider the polynomial $f(X)=X^4+aX+1$; we assume that $a$ is chosen such that $f$ is irreducible and that the discriminant $4^4-27a^4=-p$ for some prime $p$ (for example $...
- 492
4
votes
0
answers
116
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The criterion for dimensional conjecture for universal Galois deformation rings
I’m writing to ask a question about Mazur’s dimensional conjecture in Lemma 7.5 of the paper [Galatius S, Venkatesh A. Derived Galois deformation rings. Advances in Mathematics. 2018 Mar 17;327:470-...
- 863
1
vote
1
answer
245
views
Existence of odd mod $p$ Galois representations whose image is $p'$-group
Let $K$ be a number field and let $G_K$ be the absolute Galois group of $K$. Let $p$ be an odd prime and $\mathbb{F}_p$ be a finite field of order $p$. Can we always find a continuous representation $\...
- 863
1
vote
0
answers
124
views
A question related to Kirillov model
I am reading Jackson - The theory of admissible representations of $\operatorname{GL}(2, F)$ and am not able to understand the following map related to Kirillov model. This result appears on p. 54:
I ...
- 424
3
votes
1
answer
206
views
Discrepancy between $\dim H^2(G, \mathrm{ad}(\bar \rho))$ and the number of relations in a minimal presentation of the universal deformation ring $R$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\ad{ad}\DeclareMathOperator\gen{gen}$Let $p$ be a prime and $G$ be a profinite group such that the pro-$p$ completion of every open subgroup is ...
- 863
4
votes
0
answers
175
views
A computation of nearby cycles
I'm currently reading P.Scholze's paper "THE LANGLANDS-KOTTWITZ APPROACH FOR THE MODULAR
CURVE". In Lemma 7.7, he showed a (maybe simple) nearby cycle computation, which I can't follow.
Now ...
2
votes
1
answer
355
views
Families of Galois representations over disks
Edit on Nov. 20, 2023. This question is answered below in the case that $0<r_i<1$. And indeed it is shown in the answers to not be an interesting question in that case. So please take all $r_i=1$...
anon
5
votes
1
answer
360
views
Conductor of determinant of a 2-dimensional Galois representation divides conductor of representation
I am studying Serre's paper "Sur les représentations modulaires de degré 2 de $\mathrm{Gal(\overline{\mathbb Q}/\mathbb Q)}$" and I'm stuck trying to prove that $N(\det\rho)$ divides $N(\rho)...
3
votes
0
answers
122
views
Description of $\operatorname{Gal}(K(E[n])/K)$ as a subgroup of $\operatorname{GL}_2(\mathbb{Z}/n\mathbb{Z})$ for a CM elliptic curve $E$
I am looking for a specific description of the Galois groups $\operatorname{Gal}(K(E[n])/K)$ as a subgroup of $\operatorname{GL}_2(\mathbb{Z}/n\mathbb{Z})$ for an elliptic curve $E$ with complex ...
- 309
4
votes
0
answers
154
views
A variant of the inverse Galois problem
In Theorem I of Construction of maximal unramified p-extensions with prescribed Galois groups, it's proved that
for any prime $p$ and any given finite $p$-group $G$, there exists a number field $F$ ...
- 667
5
votes
1
answer
574
views
Existence of even mod $ p $ Galois representations with full image
My question is that:
For any prime $p>7$, is there a mod $ p $ representation $ \bar{\rho}:G_{\mathbb{Q}}\to \operatorname{GL}_{2}(\mathbb{F}_{q}) $ of the absolute Galois group $ G_{\mathbb{Q}} $ ...
- 863
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
2
votes
0
answers
165
views
Is the cohomology of rigid varieties semisimple?
Let $X$ be a smooth projective geometrically connected scheme over $\mathbb{Q}_p$. Assume that $H^1(X, T_{X/\mathbb{Q}_p})=0$.
Is the Galois representation $H^*(X_{\overline{\mathbb{Q}_p}}, \mathbb{Q}...
- 21
5
votes
1
answer
251
views
$\mathrm{mod}\:p$ Galois representation with respect to Zariski topology
Let $G$ be the absolute Galois group of some number field. Can there be a semisimple continuous representation $G\to GL_n(\overline{\mathbb{F}_p})$ (the latter has Zariski topology) with infinite ...
user145520
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
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
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\...
2
votes
0
answers
138
views
Local polynomials of Frobenius-semisimple Weil representations which are tensor products of an Artin representation and an unramified character
Let $K$ be a local field and $\rho: W_K \to \operatorname{GL}(V)$ be a Weil representation. The for any finite extension $F/K$, we define the local polynomial
$$
P(\rho|_F,T) = \det{(1 - \operatorname{...
- 103
5
votes
0
answers
195
views
Is there some computational evidence of the $pq$ analog of Serre's conjecture?
The $pq$ analog of Serre's conjecture (see "Mod pq Galois representations and Serre's Conjecture"- Khare, Kiming) states that if $\bar{\rho}_1:G_{\mathbb{Q}}\rightarrow \text{GL}_2(\mathbb{F}_p)$ is a ...
user130124
5
votes
0
answers
174
views
Is there a conjectural analog of Ribet's theorem (Converse to Herbrand's Theorem) for Imaginary Quadratic fields?
For $p$ a prime, let $Cl(\mathbb{Q}(\mu_p))$ denote the class group of the extension of $\mathbb{Q}$ obtained by adjoining a primitive $p$th root of unity. Associated to an eigenform of weight 2 and ...
user130124
5
votes
1
answer
360
views
Is the following variant of Shafarevich's theorem known?
Let $Q$ be a finite simple group which may be realized as the Galois group of some extension of $\mathbb{Q}$ (like for instance $PSL_2(\mathbb{F}_p)$ for $p\geq 5$, or the monster group) and let $G$ ...
user130124
1
vote
0
answers
105
views
Local factors determine Weil representations - proof of the Artin representation case
This post can be seen as a continuation of this post I created on MathOverflow.
I want to understand the proof of the following Theorem from "Euler Factors determine Weil Representations" by Tim and ...
- 103
2
votes
1
answer
317
views
Local factors determine Weil representations - proof of the cyclic case
I already created this post on Math Stack Exchange but I was not so sure if this question fits better here. If it is not, I want to apologize in advance and feel free to delete my post.
I want to ...
- 103
5
votes
1
answer
306
views
Galois representation associated to CM-newforms
Let $f(z)=\sum_{n\ge 1}a(n)e(nz)$, be a newform of CM-type, and let $\psi_f$ be the associated Hecke character, so that,
$$
f(z)=\sum_{\mathfrak{a}}\psi_f(\mathfrak{a})e(N(\mathfrak{a})z),
$$
and let ...
- 400
10
votes
2
answers
4k
views
Reference book for Galois Representations
I am an undergrad. I have taken courses in algebraic number theory and have a basic idea about $p$-adic numbers. I have also read a little bit of infinite Galois theory. But I have no idea about ...
- 239
0
votes
0
answers
82
views
Normalizing factor in Reciprocity of traces of Frobenius with solutions of equations mod p
One kind of Reciprocity tells us that we can count solutions to polynomial equations over finite fields and relate them to traces of Galois Representations with some normalization factor.
For ...
- 1,629
23
votes
2
answers
2k
views
Even Galois representations "mod p"
Consider an irreducible $\mathrm{mod}$ $p$ representation:
$$\rho: \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_2(\bar{\mathbb{F}}_p)$$
If $\rho$ is odd, it was conjectured by Serre in ...
- 17.6k
2
votes
0
answers
223
views
Residual Representation of a Motive
Suppose we have $M$ a hypergeometric motive, and $\rho$ its associated Galois rep over $\mathbb{Q}_{l}$. Is there any easy/concrete way to find $\bar{\rho}$, the residual representation at a prime (in ...
- 2,429
0
votes
0
answers
459
views
Relation between cyclotomic character and fundamental character of level
The question I have is the following: is it true that the $p+1$ exponentiation of the fundamental character of level $2$ gives us the reduction (mod $p$) of the cyclotomic character?
For a review of ...
- 85
6
votes
1
answer
658
views
Connection of Galois representation and arithmetic geometry
There are lots of Galois representations which arise naturally from geometric objects, for example, Galois representations attached to elliptic curves. I know that people are interested in studying ...
- 601
3
votes
1
answer
510
views
$\ell$-conductor of a two-dimensional $\ell$-adic Galois representation
Let $\ell$ be a prime number, denote by $K_\ell$ the maximal algebraic extension of $\Bbb{Q}$ ramified only at $\ell$. Let $f = \sum a_n q^n$ be a Hecke eigenform of level $1$ with integer ...
- 31
8
votes
1
answer
466
views
Extensions of Galois representations
Let $G=Gal(\bar{\mathbb Q}/{\mathbb Q})$ be the absolute Galois group of the rationals. Fix two continuous group homomorphisms $\alpha,\beta: G\to {\mathbb Q}_l^\times$, where $l$ is a prime and ${\...
user1688
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
1
vote
0
answers
330
views
Extending systems of l-adic representations to other l
I'm asking this not because I have an idea how one might approach it, but because it seems natural and inherently interesting.
Let $K$ be a number field, $G_K$ its absolute Galois group, and $\ell\...
- 15.4k
5
votes
0
answers
288
views
Do infinite and ramified local factors of the Dedekind zeta function of a tame number field characterize its local root numbers?
Let say you have two number fields, that are tamely ramified, and suppose that the $p$-part of their Dedekind zeta functions coincide for all prime $p$ which is ramified in either field. Suppose ...
- 2,396
9
votes
0
answers
596
views
Tameness criterion in the reducible case
Dear MO,
This is a follow up to a previous question here in MO, but I will make this question self-contained for convenience. Those already familiar with the following paper [G] by Gross can safely ...
- 1,694
2
votes
2
answers
1k
views
Decomposition of Artin L functions
The Dedekind zeta function of an abelian extension $E$ of $\mathbb{Q}$ factors as a product of Artin L function $L(s, \chi)$, where the product runs over all irreducible representations $\chi$ of $Gal(...
- 11.2k
9
votes
3
answers
2k
views
Crystalline Characters
Let $K$, $L$ be finite extensions of the $p$-adic numbers. Suppose $\chi:G_K\rightarrow L^{\times}$ is crystalline. It is my understanding that if either $K$ or $L=\mathbb{Q}_p$, then $\chi$ must be a ...
- 4,790
9
votes
1
answer
1k
views
On the determinant of an odd, continuous Galois representation
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Frob{Frob}$In his paper, Duke paper, Serre consider continuous, odd Galois representation
$\rho: G_{\mathbb{Q}}\longrightarrow \GL_{n}(\overline{\...
- 337
17
votes
0
answers
1k
views
Special values of Artin L-functions
This question might be naive and might carry the heuristic that we are living in the best possible world a little too far. If so, I appreciate being told so.
Background: Stark's conjecture interprets ...
- 13k
19
votes
2
answers
2k
views
Applications of Artin's holomorphy conjecture
I wonder why the Artin conjecture is so important. The only reason I could figure out is that one could use the holomorphy of Artin L-series and Weil's converse theorem to show modularity of two-...
user19475