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12
votes
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
2answers
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
2answers
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
1answer
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
2answers
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
2answers
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
3answers
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
1answer
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
0answers
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
0answers
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
2answers
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
0answers
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
0answers
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
0answers
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
0answers
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}(\...

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