All Questions
Tagged with p-adic-hodge-theory arithmetic-geometry
75 questions
2
votes
0
answers
161
views
What is the Galois representation structure of $B_{\text{cris}}^+/(t)$?
In $p$-adic Hodge theory, there is a nice exact sequence for quotients of $B_{\text{dr}}^+$. Denote by $t$ the typical uniformizer of $B_{\text{dr}}^+$ (the cyclotomic character), then there is a $G_{\...
- 2,907
3
votes
0
answers
185
views
Algebraic properties of Witt vectors $W(K^{\flat\circ})$, $K$ a characteristic 0 perfectoid field
Let $K$ be as in the title with tilt $K^\flat$. $W = W(K^{\circ\flat})$ satisfies a universal property: it is the unique $p$-adically complete $p$-torsion free $\mathbb{Z}_p$-algebra $A$ with $A / pA \...
- 658
1
vote
0
answers
227
views
Deformations over $A_{\inf}$
Setup:
Let $K$ be a perfectoid field of characteristic $0$ with tilt $K^{\flat}$.
Let $A_{\inf}=W(\mathcal{O}_{K^{\flat}})$ be the infinitesimal period ring.
Let $\mathcal{X}$ be a flat, projective $\...
- 282
3
votes
1
answer
469
views
Adic generic fiber of a small formal scheme in the sense of Faltings
$\DeclareMathOperator{\Spf}{Spf}\DeclareMathOperator{\Spa}{Spa}$In the Definition 8.5 of the paper "integral $p$ adic Hodge theory" by Bhatt-Morrow-Scholze, they define the adic generic ...
- 31
5
votes
0
answers
192
views
Image of $\gamma-1$ on etale $(\varphi,\Gamma)$-modules
Let $p\geq 3$ be a prime, $D$ be an etale $(\varphi,\Gamma)$-modules over the classical period ring $A_{\mathbb{Q}_p}=\mathbb{Z}_p[\![T]\!][1/T]^{\widehat{\phantom{xx}}}_p$ and $\gamma$ be a ...
- 549
1
vote
0
answers
215
views
$p$-adic étale cohomology group of open smooth varieties
Let $K$ be a finite extension of $\mathbb{Q}_p$ and let $X$ be a smooth variety over $K$.
Dr. Yamashita announced that he had proved the Galois representation of $p$-adic étale cohomology group $H^*_{\...
- 349
5
votes
0
answers
387
views
Calculating étale fundamental groups from the usual fundamental group
$\newcommand{Spec}{\operatorname{Spec}}$Let $X$ be a connected affine smooth variety over $\mathbb{Q}$, with a point $x\in X(\Spec(\mathbb{Q})$.
For any algebraically closed field $K$ of ...
- 541
2
votes
1
answer
324
views
Rank of $\mathbb{Z}_{p}$-module $H_{et}^{i}(X,\mathbb{Z}_{p}(r))$
I want to ask the following question.
Let $X$ be a smooth projective variety of dimension $d$ over $p$-adic field $k$ ( i.e. finite extension of $\mathbb{Q}_{p}$). Is it true that etale cohomology $H_{...
- 629
6
votes
0
answers
651
views
Are crystalline cohomology obsolete?
I'm interested in $p$-adic cohomology theories now. I have learned that since de Rham cohomology behaves badly in char $p$, people invented crystalline cohomology in smooth cases and later rigid ...
- 375
8
votes
1
answer
2k
views
Some questions from the paper by Scholze-Weinstein
The following is from the paper by Scholze-Weinstein on moduli of $p$ divisible groups.
My question is from a part of Lemma 4.1.7: If $R$ is a semiperfect ring, then the canonical map $W(R^{\flat}) \...
- 81
3
votes
0
answers
173
views
Smooth proper varieties over the integers that are not toric
Does there exist a smooth proper variety $X$ over $\operatorname{Spec} \mathbb Z$ that is not toric?
By Fontaine, we know that there is no Abelian scheme over $\operatorname{Spec} \mathbb Z$. Also by ...
- 7,746
4
votes
0
answers
205
views
Grothendieck group of admissible $p$-adic representations
Let $K$ be a $p$-adic local field; $G = \mathop{\mathrm{Gal}}(\overline K | K)$; $\tau \in \{\text{HT}, \text{dR}, \text{crys}\}$, $B_\tau$ the corresponding period ring; $\mathop{\mathrm{Rep}}_{\...
- 988
2
votes
0
answers
131
views
Base change of Hodge-Witt cohomology
Let $k$ be a perfect field of characteristic $p$, and $L$ be a finite extension of $k$.
For a smooth projective variety $X$ defined over $k$, we denote the base change $X \times_k L$ by $X_L$. In this ...
- 349
2
votes
0
answers
187
views
Does the map $\theta[1/p]: A_{\mathrm{inf}} \otimes \mathbb Q_p \to \mathbb C_p$ split?
This question might be very elementary to someone who knows p-adic hodge theory/perfectoid stuff etc.
Recall that $\mathbb C_p = \hat{\overline{\mathbb Q_p}}$ and $\mathbb C_p^\flat$ is it's tilt. We ...
- 7,746
8
votes
2
answers
631
views
Motivation of the construction of $p$-adic period rings
Let $B$ be either $B_{\text{dR}}$ or $B_{\text{crys}}$. For a $\mathbb{Q}_p$-representation $V$ of the absolute Galois group $\mathrm{Gal}(\overline{K}/K)$ of a $p$-adic field $K$ (a finite extension ...
- 1,428
2
votes
0
answers
209
views
Is there a smooth proper family whose fibers are not Mazur-Ogus?
Set $K$ to be a number field, denote by $\mathcal{O}_K$ the integer ring of $K$. My question is the following:
Is there a smooth proper family $X \to \mathcal{O}_K$ whose fibers are not Mazur-Ogus?
- 519
3
votes
1
answer
475
views
To identify $p$-adic Tate module $T_p(G)$ of $p$-divisible group $G$ in the category $\text{Rep}_{\mathbb{Q}_p}(G_{K_\infty})$
Let $k$ be a perfect field of characteristic $p>0$, $W=W(k)$ its ring of Witt vectors, $K_0=W(k)[\frac{1}{p}]$ and, $K/K_0$ be a totally ramified extension. Let $\pi \in K$ be an uniformizer.
...
- 930
3
votes
0
answers
232
views
$l$-adic Galois representations factor through a common finite quotient
Let $X$ be a smooth projective geometrically connected variety over $\mathbb{Q}$. Assume that for some $m>0$ we have $h^{i, 2m-i}(X)=0$ unless $i=m$.
Does there exist a number field $E$ such that ...
user158636
6
votes
0
answers
231
views
Variety over $\mathbb{F}_p$ that does not embed into flat scheme over $\mathbb{Z}/p^2\mathbb{Z}$
Let $X\to\mathrm{Spec}\:\mathbb{F}_p$ be a smooth proper morphism. Is there a closed immersion $X\to Y$ where $Y$ is flat of finite type over $\mathbb{Z}/p^2\mathbb{Z}$?
As mentioned in the comments ...
user158636
9
votes
1
answer
471
views
Hodge numbers rule out good reduction
A theorem of Fontaine says that if a geometrically connected smooth proper variety $X$ over $\mathbb{Q}$ has good reduction everywhere then $h^{i, j}(X)=0$ for $i\neq j$, $i+j\leq 3$.
This means that ...
user158636
4
votes
1
answer
405
views
Can Hodge symmetry fail if there is a lift to $W_2$ and the crystalline cohomology is torsion-free?
Let $f:X\to \mathrm{Spec}\:\mathbb{F}_p$ be a smooth proper morphism with $p>\mathrm{dim}\:X$. Assume that $H^i_{\mathrm{crys}}(X/\mathbb{Z}_p)$ is torsion-free for all $i\geq 0$ and that there is ...
user158636
0
votes
1
answer
204
views
Can $h^{1, 0}$ and $h^{1, 1}$ jump for smooth projective surfaces over $\mathbb{Z}[1/N]$?
Let $N$ be a positive integer. Let $f:X\to S=\mathrm{Spec}\:\mathbb{Z}[1/N]$ be a smooth projective morphism of relative dimension 2 such that $R^1f_*\mathcal{O}_X$ and $R^2f_*\mathcal{O}_X$ are both ...
user158636
1
vote
0
answers
125
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 ...
5
votes
0
answers
361
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)...
- 1,093
3
votes
0
answers
230
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{\...
user145520
7
votes
1
answer
402
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, ...
user145520
5
votes
1
answer
312
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 ...
user141691
4
votes
0
answers
419
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 ...
user141691
2
votes
1
answer
281
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.
...
user141691
6
votes
1
answer
297
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_{\...
user141691
4
votes
2
answers
336
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)\...
user141691
5
votes
0
answers
315
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 ...
user141691
2
votes
1
answer
435
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 ...
user141691
3
votes
1
answer
455
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 ...
- 4,164
4
votes
0
answers
234
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 ...
- 217
6
votes
2
answers
1k
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 ...
- 217
3
votes
3
answers
1k
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 ...
4
votes
1
answer
888
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 ...
user141691
2
votes
1
answer
882
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 ...
user141691
7
votes
2
answers
1k
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 ...
user141691
5
votes
1
answer
1k
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 ...
- 309
6
votes
1
answer
563
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 ...
user138661
7
votes
1
answer
978
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....
- 6,622
16
votes
3
answers
2k
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 ...
- 161
2
votes
0
answers
232
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 $S = \text{Spec}(A)$, $S_0 = \text{Spec}(A/I)$, $Y\to S$ ...
- 181
56
votes
2
answers
10k
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 ...
- 2,751
10
votes
1
answer
564
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-...
- 435
1
vote
0
answers
157
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 $2g$, ...
- 27k
2
votes
0
answers
180
views
Is there a Hodge structure for smooth proper varieties over $\mathbb{C_p}$? [duplicate]
For smooth proper varieties over $\mathbb{Q_p}$, we have several comparison theorems in p-adic Hodge theory, in particular a p-adic Hodge structure.
Now for $\mathbb{C_p}$, is there any such results ...
- 806
9
votes
0
answers
1k
views
Moduli interpretation of Fargues-Fontaine curve
The Fargues-Fontaine curve is, in his schematic version, a noetherian regular scheme $X$ of dimension 1 associated to a pair $(E,F)$, where $E$ is a local field (i.e. complete w.r.t. a discrete ...
- 1,386