Suppose K is a local field , G is its galois group, V a fine dimensional Vector space over F, which is a sub field of K, and totally ramified over $Q_p$. Consdider the linear action of G on V (V is not just a $Z_p$ representation ), are there similar theories dealing with such situation like Fontaine's theory, someting like filtered $\varphi$ module with srong p-divisible peoperties and maybe with some extra structure? Are there any reference? Thank you!
$\begingroup$
$\endgroup$
4
-
$\begingroup$ I have edited it to more easy case. $\endgroup$– TOMCommented Sep 6, 2012 at 7:09
-
$\begingroup$ You also changed the setting completely. Is $F$ an extension of $K$ or a subfield of $K$?? $\endgroup$– Laurent BergerCommented Sep 6, 2012 at 7:15
-
$\begingroup$ I am sorry, F should be a subfield of K. $\endgroup$– TOMCommented Sep 6, 2012 at 7:18
-
2$\begingroup$ If you're looking at linear representations of $G$ with coefficients, then everything works "the same". See for example 3.1 of Breuil-Mézard's 2002 Duke paper. $\endgroup$– Laurent BergerCommented Sep 6, 2012 at 11:50
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
0
If $F$ is not finite but rather equal to $C_p$ then this is really Sen's theory (see for instance Fontaine's course notes in Astérisque 295). If $F$ is merely a finite extension of $K$, then I'm not sure that you need to introduce a lot of machinery: restrict your representation to $G_F$ so that it's linear, do what you have to do, and then take in account the extra structure that you had. Alternatively, a semilinear representation is the same as an element of $H^1(G,GL_d(F))$, so you could use Galois-cohomological techniques, especially the inflation-restriction sequence with $G_F$ and $G_K$.
EDIT : this answered the question for $F$-semilinear representations of $G_K$ with $F$ an extension of $K$. Since then the OP has modified his question so my answer is not relevant anymore :(
answered Sep 6, 2012 at 7:06