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Results tagged with galois-representations
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user 5513
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.
6
votes
Irreducibility of the $n$th symetric power of the reduction of the Galois representation of ...
For (i), if $\overline{\rho}_{f,\ell|G_{\mathbb{Q}(\zeta_\ell)}}$ was absolutely reducible, then in particular the image of $G_{\mathbb{Q}(\zeta_\ell)}$ under $\overline{\rho}$ would be solvable. Howe …
- 4,790
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 …
CommunityBot
- 1
12
votes
Can one ignore primes lying over $l$ in the Fontaine-Mazur conjecture? Counterexamples?
In fact there are one-dimensional counterexamples: if $\chi$ is the $l$-adic cyclotomic character, and $k\in \mathbb{Z}_l \backslash \mathbb{Z}$, then $\chi^{(l-1)k}$ is unramified outside $l$, but do …
- 4,790
3
votes
Example of a diophantine application of an open image theorem
Well, this isn't explicitly diophantine, but here goes:
If $f$ is a level one weight $k$ eigenform with rational coefficients, the image of the attached Galois representation
$\rho_f:G_{\mathbb{Q}} …
- 4,790
3
votes
The significance of modularity for all Galois representations
Proving modularity of finite image Galois representations seems to be the most feasible way of proving the Artin conjecture. In fact, this was one of Langlands' original motivations.
- 4,790
5
votes
1
answer
434
views
Can a p-adic representation and its twist by a non-crystalline character both have nontrivia...
For a continuous irreducible representation
$\rho: G_{\mathbb{Q}_p}\rightarrow GL_n(\overline{\mathbb{Q}_p})$,
is it possible for both $D_{cris}(\rho)$ and $D_{cris}(\chi\otimes\rho)$ to be nonzero, …
- 6,924