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10
votes
0answers
389 views

Why does $H^i(X_{ét},\mathbb{Q}_p)$ have a Hodge-Tate structure?

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 ...
8
votes
0answers
337 views

Torelli-like theorem for K3 surfaces on terms of its étale cohomology

Is there a proof of a Torelli-like Theorem for a K3-surface over any field (non complex) in terms of its etale or crystalline cohomology? For example: If $K\ne \mathbb{C} $ and $X\rightarrow ...
7
votes
0answers
264 views

“Nontrivial” singular points on the eigencurve?

Let $\mathscr{C}$ be the Coleman-Mazur-Buzzard eigencurve of some fixed tame level $N$. Are there any known examples of a singular point $x\in \mathscr{C}$ which lies in a unique irreducible ...
6
votes
0answers
125 views

classifying reducible 2-dimensional mod-p Galois representations

I want to classify reducible $2$-dimensional mod-$p$ Galois representations of a field $E$ of characteristic $p > 0$ (i.e. representations $G_E = \mathrm{Gal}(E^{sep}/E) \to GL_n(\mathbf{F}_p)$) $$ ...
6
votes
0answers
137 views

simple proof of relation between H^1 crystalline and Dieudonne module?

Hi, Let $k$ be a perfect field of characteristic $p > 0$. Let $A/k$ be an abelian variety. Then the first crystalline cohomology group of $A$ with respect to $W(k)$ (= Witt vectors) is canonically ...
5
votes
0answers
231 views

Which de Rham representations are trianguline?

Let $K/\mathbf{Q}_p$ be a finite extension, and let $V$ be an $n$-dimensional $\overline{\mathbf{Q}_p}$-vector space with a continuous action of $G_K$. Suppose $V$ is de Rham, so potentially ...
3
votes
0answers
205 views

Semistable minimal model of a $K3$-surface and the special fibre

Suppose that $K$ is a $p$-adic field, that is a field of characteristic $0$ whose ring of integers is a complete discrete valuation ring $\mathcal O_K$ and with residue field $k$ (algebraic closed) of ...
3
votes
0answers
136 views

Hodge filtration over $\mathbb Z_p$

Let $p$ be a prime number. Let $X\to\operatorname{Spec}\mathbb Z_p$ be smooth and proper. Is it true that the map $H^i(X,\Omega^{\bullet\geq j}_{X/\mathbb Z_p})\to H^i(X,\Omega^\bullet_{X/\mathbb ...
2
votes
0answers
103 views

Bloch Kato Exponential as formal lie group exponential

Let $K$ be a $p$-adic field and $V$ a $p$-adic representation. In their paper on tamagawa numbers of motives, Bloch and Kato define an exponential map as the connecting homomorphism $$DR(V) ...
1
vote
0answers
97 views

$(\varphi, \Gamma)$-modules, geometric interpretation $D_{diff}$

Could anyone explain to me the first paragraph of page 29 (IV.4.1) of this course of L. Berger: http://perso.ens-lyon.fr/laurent.berger/articles/article05.pdf Specifically, I would like to ...
1
vote
0answers
166 views

lefschetz hyperplane theorem in positive characteristic

The proof of the Lefschetz Hyperplane theorem over $\mathbb{C}$ is well-known, and relies on Hodge theory in an important way. Does there exist an analogue of this theorem in positive characteristic, ...
0
votes
0answers
111 views

the definition of pro-infinitesimal thickenings

Let $R$ be a ring and $A$ and $R$-algebra. A pro-infinitesimal thickening of $R$ is a pair $(D, \theta)$ such that $\theta: D \rightarrow R$ is surjective and $D$ is separated and complete for the ...