# Questions tagged [p-adic-hodge-theory]

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

129
questions

**12**

votes

**1**answer

284 views

### Can a covering space of the $p$-adic disc split over the circle?

Let $D = {\rm Sp}\, \mathbb{C}_p\langle x\rangle$ be the affinoid unit disc over $\mathbb{C}_p$.
Is there an example of a connected finite etale cover of $D$ whose restriction to the "unit circle" ${\...

**1**

vote

**0**answers

166 views

### Čech-Alexander complex in computing (crystalline/prismatic) cohomology

I have a naive question about Čech-Alexander complexes in prismatic cohomology (although I suspect that the situation is similar for crystalline cohomology).
They seemed to be introduced as a method ...

**1**

vote

**0**answers

89 views

### galois deformation ring with type is union of irreducible components

Notation:
$K$ finite extension of $\mathbb{Q}_p$, $G_K$ absolute Galois group of $K$,
$E$ is finite extension of $\mathbb{Q}_p$ (coefficient field), $O_E$ is ring of integer in $E$.
In this paper of ...

**4**

votes

**0**answers

205 views

### Equivalent definitions of the ring $B_{\mathrm{cris}}$

I'm reading Laurie's note about Fargues-Fontaine Curve and I think he uses a different definition of $B_{\mathrm{cris}}$. Usually when $R$ is a perfect ring of characteristic $p$, $A_{\mathrm{cris}}(R)...

**4**

votes

**0**answers

171 views

### Independence of $p$ of Hodge-Tate weights

Let $X$ be a smooth and proper variety over $\mathbb{Q}$. Then for each prime $p$ we have the representation $R_p=H^i_{et}(X\times \overline{\mathbb{Q}_p}, \mathbb{Q}_p)$ of $\mathrm{Gal}(\overline{\...

**4**

votes

**1**answer

185 views

### Irreducible local Galois representation with arbitrary Hodge-Tate weights

Let $p$ be a prime and $M$ be a finite multiset of non-negative integers. Does there exist a continuous irreducible de Rham representation $\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)\to GL_n(\...

**7**

votes

**1**answer

317 views

### Irreducible global Galois representation with weights 0, 1, 3?

Fix a prime number $p$. Can there exist a continuous irreducible representation $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \mathrm{GL}_3(\mathbb{Q}_p)$ that is unramified at almost all primes, ...

**3**

votes

**1**answer

114 views

### How does an analytic space correspond to a $p$-adic Banach space

Let $K$ be a finite extension of $\mathbb{Q}_p$, and $V$ be a Banach algebra over $K$, then what is the $K$-analytic space corresponding to $V$? What is the definition of $K$-analytic space? This is ...

**4**

votes

**0**answers

256 views

### Is the Fargues–Fontaine curve proper?

It is well known that Fontaine's curve $X=\bigoplus_{k\geq0}B_{\text{cris}}^{+,\varphi=p^k}$ is a Noetherian irreducible complete scheme of dimension $1$. And completeness means that the degree ...

**12**

votes

**2**answers

744 views

### Is there a version of algebraic de Rham cohomology that can be used to calculate torsion classes?

Much work has gone into the construction of cohomology theories which are defined on algebraic varieties (étale, crystalline, etc.) and comparison isomorphisms between them.
Say $X$ is an algebraic ...

**2**

votes

**1**answer

113 views

### An example that a $p$ adic Galois representation is crystalline but not $B_e$ admissible

$B_e=B_{\text{cris}}^{\phi=1}$, so if a $p$-adic Galois representation $V$ is $B_e$ admissible, then it is crystalline, so I want to know an example that $V$ is crystalline but not $B_e$ admissible.
...

**6**

votes

**1**answer

210 views

### $B_{\mathrm{dR}}=B_{\mathrm{cris}}+{B_{\mathrm{dR}}^+}$?

$B_{\mathrm{cris}}\subseteq B_{\mathrm{dR}}$ and $B_{\mathrm{dR}}^+$ are well-known period rings in $p$-adic Hodge. I know $B_{\mathrm{dR}}=B_{\mathrm{dR}}^+[\frac{1}{t}]$ and $\frac{1}{t}\in B_{\...

**3**

votes

**1**answer

309 views

### Would it be a little but good exercise to construct or find out Breuil modules?

My question is about p-adic Hodge-Tate theory and p-adic Galois representation.
One of the important semi-linear object in p-adic Galois representation is the $\text{Breuil Module}$. There are ...

**9**

votes

**1**answer

483 views

### Tamagawa numbers

Let $K$ be a finite extension of $\mathbb{Q}_p$ with absolute Galois group $G_K$. Let $A$ be an abelian variety defined over $K$. The (geometric) Tamagawa number is defined as the order of the ...

**8**

votes

**1**answer

375 views

### Is the de Rham complex in characteristic $p$ a CDGA?

In the paper by Bhatt and Scholze on prismatic cohomology (https://arxiv.org/pdf/1905.08229.pdf), it is stated that the de Rham comparison theorem for prismatic cohomology can be lifted to an ...

**5**

votes

**1**answer

259 views

### Motivation behind Fontaine's Theory

I am reading Fontaine's theory of $p$-adic Galois representations. But I am not able find the motivation behind it. Please let me know some good reference where I can study the motivation behind ...

**4**

votes

**2**answers

275 views

### $p$-adic series bounded if and only if it has finitely many zeros

Let $L\subseteq\mathbb{C}_p$ be a finite extension of $\mathbb{Q}_p$, $r$ be a positive real number, and $f$ be a series $\sum_{n\in \mathbb{Z}} a_nz^n$ convergent in $D= \{x\in \mathbb{C}_p|0<v(x)\...

**5**

votes

**0**answers

236 views

### motivations of classifying $p$-divisible groups

Let $k$ be a perfect field of characteristic $p>0$ and $W:=W(k)$ is the witt ring. Let $K$ be a totally ramified extension of $K_0:=W(\frac{1}{p})$ and $\Lambda:=W[[u]]$ is the formal series ring ...

**5**

votes

**1**answer

203 views

### Generalized Hodge-Tate weights of an arbitrary p-adic Galois representation

Let $V$ be a continuous representation of the absolute Galois group of $\mathbb{Q}_p$ with coefficients in $\mathbb{Q}_p$. The theory of Sen attaches to $V$ generalized Hodge-Tate weights which are ...

**2**

votes

**1**answer

198 views

### Explicit semi-stable theorem for elliptic curves over $p$-adic fields

In this paper of Maja Volkov, the authur metions a number called "défaut de semi-stabilité" on page 9. It is defined as $\text{dst}(E)=\frac{12}{\text{pgcd} (12,v_p(\Delta_E))}$ where $E$ is an ...

**1**

vote

**1**answer

195 views

### References for the early history of Fontaine's tilting construction

Scholze attributes the tilting construction for perfectoid rings to Fontaine, who calls it "a classical construction in $p$-adic Hodge theory".
Would anyone happen to know an early reference where ...

**5**

votes

**0**answers

184 views

### Why does $\theta: \mathbb{B}^+_{dr} \rightarrow \mathbb{C}_p$ have no continuous or equivariant section?

Fix a $p$-adic field $K$ with perfect residue field $k.$ Let $\mathbb{C}_K$ be the completion of the algebraic closure of $K,$ and let $$R = \varprojlim \mathbb{C}_K/p,$$ where the transition maps in ...

**14**

votes

**0**answers

365 views

### Failure of local Fontaine Mazur

This question unfortunately has a very similar name to this one, but I what want to ask here is different.
Let $K$ be a finite extension of $\mathbb{Q}_p$. It seems to be well known that the local ...

**2**

votes

**1**answer

317 views

### Topology on $p$-adic period rings in an article by Fontaine, part II

This is a follow-up to this question. See that question for background and relevant notation.
In the answer to that question, it is claimed, if I understand the answer correctly, that a basis of ...

**1**

vote

**2**answers

356 views

### Topology on $p$-adic period ring in an article by Fontaine

Fix a $p$-adic field $K$ with perfect residue field $k.$ Let $\mathcal{C}$ be the completion of the algebraic closure of $K,$ and let $$R = \varprojlim \mathcal{C}/p,$$ where the transition maps in ...

**11**

votes

**0**answers

543 views

### What is known at $\ell = p$ about realizing Jacquet-Langlands & local Langlands as the cohomology of Lubin-Tate space with level structure?

Background:
(Mostly my paraphrased interpretation of the introduction of Strauch's Deformation spaces of one-dimensional formal
modules and their cohomology, with additional details from Carayol's ...

**2**

votes

**1**answer

144 views

### Jacobson radical of a derived $I$-complete ring

Let $A$ be a commutative ring and $I \subseteq A$ a finitely generated ideal (I am not assuming that $A$ is Noetherian).
Assume that $A$ is derived $I$-complete, meaning, let's say, that $\mathrm{...

**3**

votes

**1**answer

150 views

### maximal unramified extension of Breuil ring in $A_{cris}$

Here is the notation. Let $k$ be a perfect field of characteristic $p$, $W=W(k)$ the ring of Witt vector and $\mathfrak{S}:=W[[u]]$, $\mathcal{O}_{\mathcal{E}}$ is the $p$-adic completion of $W[1/u]$ ...

**4**

votes

**1**answer

426 views

### Fontaine-Fargues curve and period rings and untilt

When I read the paper "THE FARGUES–FONTAINE CURVE AND DIAMONDS" of Matthew Morrow, I have a question on page 11.
Question: The arthur said that the de Rham and crystalline period rings implicitly ...

**2**

votes

**1**answer

334 views

### How to prove the p-adic Galois representations atteched to the Tate module of an abelian variety is de Rham directly?

Recently I read a thesis p-adic Galois representations and elliptic curves. Using Tate's curve, the author proved the p-adic Galois representations atteched to the Tate module of an elliptic curve is ...

**7**

votes

**2**answers

682 views

### Classify 2-dim p-adic galois representations

Recently I have known how to classify 1-dim p adic Galois representations $\phi$. The p-adic Galois representations mean that a representation $G_K$ on a p-adic field $E$, where $K$ is also a p-adic ...

**4**

votes

**1**answer

541 views

### Prisms and Hodge-Tate comparisons

A few weeks ago, Bhatt and Scholze uploaded a preprint of their paper 'Prisms and Prismatic Cohomology' to arxiv.
In Theorem 6.3 they state their Hodge-Tate comparison. Recently, I started reading ...

**1**

vote

**1**answer

151 views

### An archimedean analogue of the non-canonicity of Hodge--Tate decomposition

For smooth proper schemes over $\mathbb{C}_p$, there is no canonical Hodge--Tate decomposition (but there is something close). Is there an analogue of this on the archimedean side? I thought about ...

**3**

votes

**0**answers

170 views

### Variations of $p$-adic Hodge structures

What is the analogue of variations of Hodge structures in $p$-adic Hodge theory? What does Griffiths transversality correspond to? Is there any reference explaining it in detail and containing ...

**8**

votes

**0**answers

332 views

### Foundational Questions on Adic Spaces

There are some foundational questions on adic spaces that I can't find in the literature. It seems that these questions are pretty natural, so I guess that an answer should be known to the experts in ...

**1**

vote

**0**answers

164 views

### Hodge--Tate weights of an abelian surface

Let $X$ be an abelian surface over a finite extension of $\mathbb{Q}_p$. When does $X$ have distinct Hodge--Tate weights (in étale cohomology)?

**6**

votes

**1**answer

371 views

### Integral $p$-adic Hodge theory and the space of comparisons of cohomology theories

Weil cohomology theories can be considered as fibre functors from the category of motives. Given two such functors, we have an affine scheme of invertible natural transformations between them, and ...

**6**

votes

**1**answer

384 views

### Applications of $h$-topology and $h$-descent

This is a technical problem about applications of Grothendieck topologies. In some recent works, the technique of $h$-topology and $h$-descent is very useful, for an introduction see https://stacks....

**1**

vote

**1**answer

94 views

### 2-dimensional absolutely irreducible $p$-adic Galois reps

Here the following is stated:
It's a basic fact in $p$-adic Hodge theory that any 2-dim. absolutely
irreducible $G_{\mathbb Q_p}$-representation with distinct Hodge-Tate weights is uniquely ...

**12**

votes

**0**answers

711 views

### On mixed $p$-adic Hodge theory

Does mixed $p$-adic Hodge theory exist? Can we extend the scope of comparison theorems using simplicial resolutions a la Deligne? Do we get 3 opposite filtrations as in classical mixed Hodge theory, ...

**4**

votes

**1**answer

208 views

### Finite image but not crystalline

What is an example of a $p$-adic representation of the absolute Galois group of a $p$-adic field that has finite image on the inertia subgroup, but is not crystalline?

**14**

votes

**3**answers

1k views

### Tower of moduli spaces in Scholze's theory

My question is related to another one I read here in Overflow. I am reading Scholze's papers about moduli spaces of $p$-divisible groups and elliptic curves, and I am very interested in the formal ...

**14**

votes

**1**answer

468 views

### P-adic Volume Conjecture

Let $M$ be a closed hyperbolic 3-manifold. One can use hyperbolic structure on $M$ to define hyperbolic volume $Vol(M)$. Thanks to Mostow's rigidity theorem the volume depends only on the topology of ...

**2**

votes

**0**answers

161 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 $S = \text{Spec}(A)$, $S_0 = \text{Spec}(A/I)$, $Y\to S$ ...

**3**

votes

**0**answers

275 views

### Condition on a Fontaine Laffaille module which prescribes the image of the associated Galois representation

The Setup:
Let $m\geq 1$ be an integer, $\mathbb{F}$ be a finite field of characteristic $p$ and $W(\mathbb{F})$ the ring of Witt-vectors with residue field $\mathbb{F}$ and $\sigma:W(\mathbb{F})\...

**52**

votes

**2**answers

4k views

### What is Prismatic Cohomology?

Prismatic cohomology is a new theory developed by Bhatt and Scholze; see, for instance, these course notes. For the sake of the community, it would be great if the following question is discussed in ...

**5**

votes

**0**answers

220 views

### Fargues's Theorem for $Spa(C,C^+)$ (rather than $Spa(C,O_C)$

$\DeclareMathOperator\Spa{Spa}$Fargues's Theorem for $\Spa(C,O_C)$ states that the category of (mixed characteristic) shtukas with one paw at $x_C$ is equivalent to the category of Breuil-Kisin-...

**1**

vote

**0**answers

152 views

### Reference to a particular result of Scholl and Faltings

Let $f=\sum_{n\geq 1} a_n q^n$ be a normalized eigenform which is supersingular and crystalline at a prime $p$ and let $V_f$ be the associated crystalline representation, then it follows from the work ...

**1**

vote

**0**answers

135 views

### A family of crystalline representations

Let $K$ be a number field and let $v$ be a finite place of $K$. Further, let $g \geq 1$ be a positive integer. Consider the family $F(K,v,g)$ consisting of abelian varieties $A$ of dimension $2g$, ...

**2**

votes

**0**answers

90 views

### Generalizing characterizing crystalline representations of dimension 2 to certain special classes of crystalline representations of higher dimension

Let $A$ be an abelian variety defined over a number field $K$, and let $v$ be a prime of $K$ such that $A$ has good reduction modulo $v$. Let $\rho$ be the representation of $G_K = \text{Gal}(\...