All Questions
Tagged with galois-theory galois-representations
12 questions with no upvoted or accepted answers
6
votes
0
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375
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How to construct this non-geometric mod $p$ Galois representation?
Let $ K $ be a totally real cubic field. I am concerned with the following kind of Inverse Galois Problem:
Is there a prime $ p>5 $ and a continuous representation $ \bar{\rho}:G_{K}\to {\rm
GL}_{...
- 863
4
votes
0
answers
154
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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
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
3
votes
0
answers
287
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Galois theory of ramified coverings vs classical Galois theory
That's an exact copy of my former MSE question I asked a couple of weeks ago and unfortunately not got the answer I was looking for.
The question adresses reuns' answer in this thread: Algebraic ...
- 6,018
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
147
views
Prime splitting in the division field of an elliptic curve
Let $E/\mathbb{Q}$ be an elliptic curve with good reduction at two distinct primes $p, \ell$. Suppose the mod $\ell$ Galois representation associated to $E$ is surjective. Let $K=\mathbb{Q}(E[\ell])$ ...
- 1,422
2
votes
0
answers
155
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Classification of mod p Galois Representations for l not equal to p
Let $l\neq p$ be primes and let $\text{G}_l:=\text{G}_{\mathbb{Q}_l}$. Let $k$ be a finite field of characteristic $p$ and $\bar{\rho}:\text{G}_l\rightarrow \text{GL}_2(k)$ a local Galois ...
user130124
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
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_{...
- 923
1
vote
0
answers
217
views
Is semi-simplicity of Galois representations local?
Let $\rho:G_{\mathbb{Q}}\rightarrow \text{Gl}(V)$ be a finite dimensional $\ell$-adic Galois representation. Then for each prime, by pre-composing $\rho$ with the natural inclusion $G_{\mathbb{Q}_p}\...
- 4,408
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
1
vote
0
answers
199
views
Class number of the cyclotomic tower
Let ${\Bbb Q}(\zeta_{\infty})$ be the field obtained by adjoining all roots of unity. We define
Cl(${\Bbb Q}(\zeta_{\infty})$)$\colon= \underset{m > 1}{\varinjlim}~{\mathrm{Cl}}({\Bbb Z}[\zeta_m])...
- 781