All Questions
Tagged with p-adic-hodge-theory ag.algebraic-geometry
49 questions with no upvoted or accepted answers
14
votes
0
answers
884
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, ...
- 141
11
votes
0
answers
749
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 ...
- 3,539
11
votes
0
answers
807
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 \...
10
votes
0
answers
2k
views
Roadmap for p-adic Hodge theory
I'd like to be able to start studying p-adic Hodge theory and hope that by posing this question, I can be better prepared to work towards it. I ask for a roadmap because I understand that I have a lot ...
- 544
10
votes
0
answers
834
views
Elementary aspects of The Fargues-Fontaine curve
To any pair $(E,F)$, where $E$ is a local field and $F$ is a perfectoid field, one can associate a curve $X^{\text{FF}}_{E,F}$, the so-called Fargues-Fontaine curve, which is unique up to Frobenius ...
- 989
9
votes
0
answers
910
views
Grothendieck's motivation of crystalline cohomology
Here Illusie mentions Grothendieck's observation that using Gauss-Manin connection one can give a non-canonical isomorphism between de Rham cohomology of smooth schemes over $W(k)$ with isomorphic ...
- 7,377
8
votes
0
answers
333
views
Triple comparison of cohomology in algebraic geometry
Let $X$ be a smooth proper variety over $\mathbb{Q}$ and $p$ a prime number. For an integer $k$, we have:
a finitely-generated abelian group $H^k(X^{\mathrm{an}}(\mathbb{C});\mathbb{Z})$
a finitely-...
- 15.4k
8
votes
0
answers
433
views
Intuition for de Rham comparison theorem in $p$-adic Hodge theory
The de Rham comparison theorem from $p$-adic Hodge theory compares the etale cohomology of a variety with the de Rham cohomology of that variety. It says the following:
Let $K/\mathbf{Q}_p$ be a ...
- 1,949
8
votes
0
answers
550
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 ...
- 2,923
8
votes
0
answers
581
views
On Topological Hochschild Homology
Nowdays, I hear talking about Topological Hochschild Homology more and more often, and I was wondering if someone could point out references to explain why it's important and interesting, and what ...
user125208
7
votes
0
answers
432
views
Failure of integral comparison between crystalline and de Rham cohomology over a highly ramified base
Let $K$ be a finite extension of $\mathbb{Q}_p$ with the ring of integers $\mathcal{O}_K$ and the residue field $k$.
By a theorem of Berthelot and Ogus(https://link.springer.com/article/10.1007%...
- 7,377
6
votes
0
answers
312
views
Geometry of syntomic cohomology
Deligne cohomology has a geometric interpretation. For example, $H^{2}_{\mathcal{D}}(X,\mathbb{Z}(1))$ is identified with the group $H^{1}(X,\mathcal{O}_{X}^{\ast})$ of isomorphism classes of line ...
- 1,404
6
votes
0
answers
630
views
On the Erratum to P. Scholze's "$p$-adic Hodge theory for rigid-analytic varieties"
I am trying to understand section (3) of the Erratum to P. Scholze's "$p$-adic Hodge theory for rigid-analytic varieties" in detail. In particular, there is the following sentence on page ...
- 507
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
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
6
votes
0
answers
267
views
Universal property of $A_{\mathrm{cris}}/p^n$
It is well known that the ring $A_{\mathrm{cris}}$ of Fontaine is the universal $p$-adically complete divided power thickening of $\mathcal{O}_{\mathbb{C}_p}$ over $\mathbb{Z}_p$; in fact, this is one ...
- 825
6
votes
0
answers
590
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 ...
5
votes
0
answers
556
views
Theorem 7.11 in Scholze's $p$-adic Hodge Theory
I was trying to understand the statement and proof of Theorem 7.11 in Scholze's paper "$p$-adic Hodge Theory for Rigid-Analytic Varieties". I'll reproduce part of the statement below:
Let $...
- 51
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
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
5
votes
0
answers
659
views
Comparison for cycle class maps for de Rham and etale cohomology via p-adic Hodge theory
Let $K$ be a p-adic local field, $X$ a smooth projective variety over $Spec(K)$, $CH^k(X)$ the Chow group of pure codimension $k$. Then there are cycle maps
$cl^X_{DR}:CH^k(X)\to H^{2k}_{DR}(X/K)$ ...
- 806
5
votes
0
answers
677
views
Basic question on p-adic Hodge theory
I am starting to study the rudiments of p-adic Hodge theory and I have the following basic question. Let $\chi$ be the unramified quadratic character of $G_p = \mathrm{Gal}(\bar{\mathbb{Q}}_p/\mathbb{...
- 111
5
votes
0
answers
278
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 ...
- 231
5
votes
0
answers
585
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) \...
- 3,555
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
4
votes
0
answers
244
views
Uniqueness of $\delta$-structure on a $p$-torsion ring
I was working through Bhargav's notes on $\delta$-rings and prismatic cohomology, specifically lecture 2, page 2, point 5 where he claims that the ring $\mathbb Z[x]/(px,x^p)$ has a unique $\delta$-...
- 7,746
4
votes
0
answers
244
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 ...
user138661
3
votes
0
answers
347
views
Modern integral $p$-adic Hodge theory and modularity lifting and Fontaine-Mazur
As a follow-up to a comment on this answer, I'm wondering if there are expected to be applications of the new point of view on integral $p$-adic Hodge theory, à la Bhatt-Morrow-Scholze and others, to ...
- 15.4k
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
3
votes
0
answers
272
views
Explanation for devissage argument
Let $K$ be a local field of characteristic $0$ with the ring of integers $\mathcal{O}_K$ and uniformizer $\pi$. Let $k$ be the residue field of $K$ with $\text{card}(k)=q$. Let $\mathcal{O}_\mathcal{E}...
- 101
3
votes
0
answers
518
views
Comparison theorem between étale and de Rham cohomology for local system
This question is based on Milne "canonical models of Shimura varieties and automorphic vector bundles"
Let $(G,X)$ be a Shimura datum, $(V,\xi)$ be a rational representation of $G$ (I guess it means ...
- 31
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
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
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
2
votes
0
answers
127
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}(\...
- 27k
2
votes
0
answers
357
views
Does the pro-étale local system defined over a p-adic period domains interpolate crystalline representations?
There is a Grothendieck-Messing period morphism of rigid-analytic spaces $\pi: \mathcal{M}_\eta^{rig}\to \mathcal{Fl}$ going from the generic fiber of an EL-type Rapoport-Zinks to a flag variety. The ...
- 435
2
votes
0
answers
389
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 immersion)...
- 33
1
vote
0
answers
138
views
Syntomic f-cohomology for open varieties
Syntomic cohomology $H^{i+j}_{\mathrm{syn}}(X,n)$ of a proper variety $X$ with good reduction over a $p$-adic field $K$ is computed via a spectral sequence in terms of $H^i_{\mathrm{f}}(G_K;H^j_{\...
- 15.4k
1
vote
0
answers
80
views
The bound for zeros of the composition of polynomials and analytic functions
Suppose $K$ is a number field, and $A\in M_n(K)$. $v$ is a place of $K$, and $f_1,\cdots,f_n$ are analytic functions (one variable) on $m_v\mathcal O_{K,v}$, satisfying: $\frac{\mathrm d \bf {f}}{\...
- 785
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
1
vote
0
answers
184
views
Calculation of de Rham cohomology of abelian varieties/ jacobian varieties
It's known that for a elliptic curve like $E:y^2=x(x-1)(x-t)$ we have a basis $\frac{dx}{y}, \frac{xdx}{y}$ for $H_{dR}^1(E)$. But find such a basis is not an easy thing. I wonder for a general ...
- 785
1
vote
0
answers
125
views
Is the Frobenius semisimple on the de-Rham cohomology?
Suppose $K$ is a unramified finite extension of $\mathbb Q_p$, and $X$ is a projective smooth curve defined over $K$. By $p$-adic Hodge theory we know $D_{cris}(H_{et}^i(X,\mathbb Q_p))=H_{dR}^i(X)$. ...
- 785
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
1
vote
0
answers
270
views
Almost ring theory and derivations
I don't understand the definition of $\boldsymbol{\Omega}_A$ in the context of almost rings. In Gabber and Ramero https://arxiv.org/pdf/math/0409584.pdf it is covered in 9.6.12. How is $\boldsymbol{\...
- 4,418
1
vote
0
answers
272
views
$p$-adic Galois representation and Étale homology
Let $X$ be a smooth proper scheme over some $p$-adic field $K$. The "usual" way to get a Galois representation out of this is to consider the étale cohomology (either $p$ or $\ell$-adic). ...
- 4,418
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 ...
1
vote
0
answers
298
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)?
user140654
1
vote
0
answers
610
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, ...
- 3,779
0
votes
0
answers
111
views
Prime to $p$ monodromy of local system on rigid variety
Suppose $F$ is a finite extension of $\mathbb Q_p$, and $X$ is a rigid variety over $F$. I saw in proposition 3.7 of Oswal, Shankar, Zhu, and Patel - A $p$-adic analogue of Borel's theorem: "Let $...
- 785