Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $X$ be a variety over a $p$-adic field $K$.

Is there a simple or intuitive explanation of why the $G_K$ representation $H^i(X_{├ęt},\mathbb{Q}_p)$ is Hodge-Tate? More precisely, why do the powers of the cyclotomic character appear and not other characters?

share|improve this question
    
For clarity, does "variety" assume smooth and proper as well? –  Matt Sep 8 '13 at 20:21
3  
Just to be precise, what is true is that, after tensoring over $\mathbb{C}_p$, the subspaces on which the (semi-linear!) Galois action is via powers of the cyclotomic character generate the whole space. As for whether there is an 'intuitive' explanation for this phenomenon, I don't know! –  Keerthi Madapusi Pera Sep 9 '13 at 3:07
2  
As Keerthi pointed out, it's after you tensor by Cp that the powers of the cyclo char appear. But a statement like that is true for any Galois repn V by Sen's theory. What is specific to the étale cohomology is that it's integer powers of the cyclo char that appear. This is where you use the geometric input. As to why this is so... –  Laurent Berger Sep 10 '13 at 8:02

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.