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5 votes
2 answers
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Hodge-Tate weights of etale cohomology

Let $K/\mathbb Q_p$ be a local field, $X/K$ a proper scheme with semi-stable reduction. Question: What is the possible range of Hodge-Tate weights of the etale cohomology $H^i(X_{\overline K}, \...
natura's user avatar
  • 1,503
5 votes
0 answers
278 views

Tate's conjecture and symmetry of Hodge-Tate weights

I'm reading Bellaiche's notes on the Block-Kato conjecture (Hawaii summer school). Here is the link http://people.brandeis.edu/~jbellaic/BKHawaii5.pdf At page 10 he claims that an indirect ...
Bear's user avatar
  • 231
4 votes
1 answer
152 views

Semistability of the $\ell$-adic representation of variety with semistable reduction

The question is in the title, but here's some quick background. It's easy to show (assuming smooth-proper base change) that the $\ell$-adic cohomology of a variety over the fraction field of a DVR ...
Nico's user avatar
  • 141
3 votes
0 answers
232 views

$l$-adic Galois representations factor through a common finite quotient

Let $X$ be a smooth projective geometrically connected variety over $\mathbb{Q}$. Assume that for some $m>0$ we have $h^{i, 2m-i}(X)=0$ unless $i=m$. Does there exist a number field $E$ such that ...
user avatar
2 votes
0 answers
232 views

Berthelot’s comparison theorem and functoriality

Let $A$ be a noetherian $p$-adically complete ring with an ideal $I$ equipped with a PD structure and such that $p$ is nilpotent on $A/I$. Let $S = \text{Spec}(A)$, $S_0 = \text{Spec}(A/I)$, $Y\to S$ ...
Ari's user avatar
  • 181
2 votes
0 answers
141 views

Hodge-Tate weights of etale cohomology groups

Given a smooth algebraic variety $X$ over a number field $F$, its $p$-adic cohomology groups $H^i(X \times_F \bar F, \mathbb Q_p)$ carries an action of $\mathrm{Gal}(\bar F/F)$, which gives a ...
Shawn's user avatar
  • 453
1 vote
0 answers
125 views

Is the Frobenius semisimple on the de-Rham cohomology?

Suppose $K$ is a unramified finite extension of $\mathbb Q_p$, and $X$ is a projective smooth curve defined over $K$. By $p$-adic Hodge theory we know $D_{cris}(H_{et}^i(X,\mathbb Q_p))=H_{dR}^i(X)$. ...
Richard's user avatar
  • 775
1 vote
0 answers
272 views

$p$-adic Galois representation and Étale homology

Let $X$ be a smooth proper scheme over some $p$-adic field $K$. The "usual" way to get a Galois representation out of this is to consider the étale cohomology (either $p$ or $\ell$-adic). ...
curious math guy's user avatar