The p-adic-hodge-theory tag has no usage guidance.

**11**

votes

**1**answer

564 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 $H^i(X_{ét},\mathbb{Q}_p)$ is Hodge-Tate? More precisely, why do the powers ...

**30**

votes

**0**answers

983 views

### Grothendieck's Period Conjecture and the missing p-adic Hodge Theories

Singular cohomology and algebraic de Rham cohomology are both functors from the category of smooth projective algebraic varieties over $\mathbb Q$ to $\mathbb Q$-vectors spaces. They come with the ...

**10**

votes

**0**answers

447 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

**0**answers

354 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

**0**answers

157 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

**0**answers

190 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

**0**answers

298 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 ...

**4**

votes

**0**answers

177 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 ...

**3**

votes

**0**answers

150 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

**0**answers

183 views

### Are there good properties of the divided power completion map?

Let $Y \to X$ be a closed immersion of smooth schemes over, say, the ${\rm Spec}(\mathbb{Z}_p)$. The completion map $$X_{/Y}\to X$$ is an ind-closed immersion (sometimes called pseudo-closed ...

**2**

votes

**0**answers

209 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) ...

**2**

votes

**0**answers

330 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 ...

**1**

vote

**0**answers

125 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

**0**answers

224 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

**0**answers

135 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 ...