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Results tagged with galois-representations
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user 2481
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.
5
votes
Accepted
Robba ring and overconvergent (phi,Gamma)-modules
The theory of $(\varphi, \Gamma)$-modules works for $\mathbf{Z}_p$-linear representations, but one has to use a slightly different coefficient ring $\mathbf{A}_K$. This is explained very clearly in se …
- 37k
4
votes
Accepted
Rank one (phi,Gamma)-modules
Nice question! I remember doing this exercise myself once. This can be extracted from Fontaine's article in the Grothendieck Festschrift, but it takes a little bit of work. The key observation is that …
- 37k
4
votes
(phi, Gamma) module of ordinary elliptic curve
As Laurent has already pointed out, the representation is reducible and hence so is the phi-Gamma module, and writing down the two composition factors is easy; describing the extension class is harder …
- 37k
17
votes
Is every 3-dim self-dual Galois representation a symmetic square of 2-dim representation?
The correct statement is: any 3-dimensional selfdual Galois representation is isomorphic to a quadratic twist of the adjoint of some 2-dimensional representation. (The quadratic twist is really necess …
- 37k
8
votes
1
answer
860
views
Do Tamagawa numbers of Galois representations stabilise in the cyclotomic tower?
Suppose $T$ is a free finite rank $\mathbb{Z}_p$-module with a continuous action of $\operatorname{Gal}(\overline{K} / K)$, where $K$ is a number field. There is a definition of local Tamagawa numbers …
- 37k
10
votes
1
answer
1k
views
Tamagawa numbers of crystalline Galois representations
This is a followup to this question.
Let $p \ge 3$ be prime, and let $V$ be a crystalline 2-dimensional representation of $G_{\mathbb{Q}_p}$ and $T$ a lattice in $V$. I'm going to assume just about ev …
- 37k
11
votes
What geometric properties do properties of ell-adic Galois representations imply?
The converse is false. See the lecture notes by Chandan Singh Dalawat at http://arxiv.org/abs/math/0605326, which give some examples of varieties over finite extensions of $\mathbb{Q}_p$ whose $\ell$- …
- 37k
6
votes
Accepted
On the image of the residual representation attached to a CM form
No. For instance there are plenty of modular forms that are not CM, but are congruent mod p to CM forms or to Eisenstein series, and thus whose residual Galois representations have small image.
- 37k
3
votes
Accepted
Trianguline representation
No, triangulations are not in general unique.
A simple way of seeing this is to consider the case when $K = \mathbf{Q}_p$, $V$ is 2-dimensional and crystalline with distinct Hodge–Tate weights, say $\ …
- 37k
8
votes
Accepted
Crystalline when restricted to inertial subgroup
This is purely formal. If $V$ is crystalline, then $V \otimes \mathbf{B}_{\mathrm{cris}}$ has a basis as a $\mathbf{B}_{\mathrm{cris}}$-module in which the action of $G_K$ is trivial. Hence a fortiori …
- 37k
17
votes
Accepted
Does the p-adic Tate module of an elliptic curve with ordinary reduction decompose?
Serre has shown that there exists a complementary subspace invariant under the Lie algebra $\mathfrak{g}$ if and only if E has complex multiplication. Otherwise the image of Galois is open in the Bore …
- 37k
9
votes
Accepted
$(\varphi, \Gamma)$-modules of finite height
Proposition: If $D$ is not of finite height, then nor is $D \otimes \delta$ for any $\delta$ of rank 1.
Proof: It suffices to show the contrapositive: if $D$ is of finite height so is $D \otimes \de …
- 37k
5
votes
1
answer
622
views
Psi operator on Phi-Gamma modules
This is a question about the base-rings appearing in the the theory of $(\varphi, \Gamma)$-modules in $p$-adic Hodge theory.
Let $p$ be prime, $n \ge 1$, and let
$$ \mathbf{A}_{\mathbf{Q}_p}^{\dagger …
- 37k
8
votes
Accepted
Properties of representations attached to p-adic modular forms
There are two subtleties regarding how to formulate this question.
Firstly, there are several notions of "p-adic modular form". There's Hida's ordinary p-adic modular forms (a very small space); ther …
- 37k
8
votes
The existence of an elliptic curve with a specific Galois representation induced by a character
In this context, if $\rho$ is a mod $\ell$ representation of $Gal(\overline{F} / F)$, and $A$ is an elliptic curve over an extension $F' / F$, then the statement "$A[\ell] \cong \rho$" needs a little …
- 37k