Questions tagged [p-adic-hodge-theory]
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218 questions
6
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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 ...
7
votes
0
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570
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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 ...
- 13.1k
1
vote
0
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610
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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, ...
- 3,779
18
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1
answer
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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 $H^i(X_{ét},\mathbb{Q}_p)$ is Hodge-Tate? More precisely, why do the powers ...
- 527
15
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1
answer
1k
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What is the classification of characters in $p$-adic Hodge theory?
Let $K$ be a $p$-adic field and $\chi : Gal_K \rightarrow \mathbb{Q}_p^\times$ be a character. I know that $\chi$ is Hodge-Tate of weight $0$ iff $\chi(I_K)$ is finite (by Sen's theory), and that it ...
- 527
11
votes
0
answers
807
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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
0
answers
491
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"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 ...
- 13.1k
3
votes
0
answers
204
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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 Z_p})...
- 2,842
8
votes
0
answers
315
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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 ...
- 2,842
19
votes
1
answer
2k
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Analogue of Tate or Hodge conjecture for varieties over $\mathbb Q_p$
I've been learning about p-adic Hodge theory recently (I'm a beginner), and I've been wondering about the following question the past couple of weeks. Sorry for the long setup, it's mainly background;...
- 317
2
votes
1
answer
466
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Minimal semistable model for K3-surfaces.
I wonder if a semistalbe K3 surface over a $p$-adic field has a minimal semistable model. I guess yes but I do not find any reference.
Also, if we have a semistable K3 surface with a log structure, ...
7
votes
1
answer
425
views
showing that abelian varieties are de Rham *without* showing that they are crystalline
If $X$ is a smooth projective variety over a $p$-adic field $K$, then Faltings' Theorem says that the etale cohomology of $X_{\overline{K}}$ is crystalline.
There have been various steps towards this ...
- 287
1
vote
1
answer
459
views
Submodule of a Kisin module
By M. Kisin, let $k$ be an algebraically closed field of characteristic $p$, and $K$ be a totally ramified extension of $B(k)$, the fraction field of the Witt vector ring $W(k)$, the category of ...
- 51
5
votes
2
answers
2k
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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}, \...
- 1,503
0
votes
0
answers
391
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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 $I:=...
- 287
8
votes
1
answer
333
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Describing the ratio of uniformizers in B_dR
In Conrad and Brinon's notes http://math.stanford.edu/~conrad/papers/notes.pdf, two uniformizers of $B_{dR}$ are produced: one is $\xi := [\tilde{p}]-p$ (bottom of p.58), where $\tilde{p} = (p, p^{1/p}...
- 287
11
votes
1
answer
648
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Are D_dR and D_st "potentially comparable"?
Suppose we have a de Rham Galois representation $G_K\rightarrow GL(V)$ for some $p$-adic field $K$ and some finite dimensional vector space $V$ over $\mathbf{Q}_p$. Then it is a theorem that there is ...
- 1,988
13
votes
1
answer
3k
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p-adic Hodge theory for varieties defined over \C _p ?
I have a question on p-adic Hodge theory:
When e.g. $X$ is a smooth proper scheme over a finite extension $K$ of $\mathbf{Q}_{p}$ then e.g. one variant of $p$-adic Hodge theory says that there is a ...
- 131