Questions tagged [p-adic-hodge-theory]
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218 questions
6
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Quasi-syntomic descent and prismatic F-crystals
I am reading Bhatt and Scholze's paper on F-crystals, and they seem to be using the following result in the proof of Theorem 5.6:
let $X \to Y$ be a quasisyntomic cover of formal schemes over $\...
- 461
2
votes
0
answers
187
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$G_K$-fixed points of sections of affinoids on the Fargues-Fontaine curve
Let $K$ be a finite extension of $\mathbb{Q}_p$ and let $G_K=\mathrm{Gal}(\overline{K}/K)$ be its absolute Galois group. There are the Fargues-Fontaine analytic curves $Y_{FF}$ and $X_{FF}$ associated ...
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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 ...
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1
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1
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300
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Exact sequence, de Rham representation
Let $k$ be a $p$-adic field and $G_k$ its absolute Galois group. Let $B_\text{dR}$ be the de Rham period ring with the usual filtration given by powers of $t$. For $i < j$ integers we have an exact ...
- 455
1
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1
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147
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Triangularizability of induced $(\phi, \Gamma)$-modules
Let $K$ be a finite extension of $\mathbb{Q}_p$ and $L/K$ a finite unramified extension.
Let $M$ be a $(\phi, \Gamma_L)$-module over the Robba ring of $L$ (with coefficients in some other $p$-adic ...
- 10.5k
3
votes
0
answers
173
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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
12
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3
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3k
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Witt vectors, the cotangent complex, and a solid construction of $B_{dR}^+$
In a remarkable lecture delivered on October 29th: New Foundations for functional analysis, Dustin Clausen suggests at the 40 minute mark a remarkable new construction interpretation of Fontaine's ...
- 858
4
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0
answers
205
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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
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0
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131
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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
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187
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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 ...
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8
votes
2
answers
631
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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
3
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1
answer
390
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The kernel from $A_\mathrm{inf}$ to $\mathcal{O}_{\mathbb{C}_K}$
I tried to understand this paper on page 31.
Let $K$ be an finite extension of $\mathbb Q_p$ and $\overline{K}$ be its algebraic closure; $\mathcal{O}_{\overline{K}}$ is the ring of integers of $\...
- 275
3
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1
answer
411
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Why does $\mathbb C_p$ not contain the periods?
I am reading the following article of Berger, p8 and I don't understand the idea:
$C_p:=\widehat{\overline{\mathbb Q_p}}$ does not contain the periods
The text seem to reason as follows
(under some ...
- 661
4
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1
answer
347
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A Tate-Sen theorem mod $p$
Let $p$ be a prime number, $G=\textrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$, and $\chi:G\rightarrow\mathbb{Z}_p^\times$ the cyclotomic character. Let $\mathbb{C}_p$ denote the completion of the ...
- 2,834
1
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1
answer
170
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Is the completion of the field generated by torsion points of a 1-dimensional formal group perfectoid?
Let $K$ be a finite extension of $\mathbb{Q}_p$ and let $G$ be a 1-dimensional formal group defined over $\mathcal{O}_K$. Consider the field $K_\infty$ obtained by adjoining to $K$ all the solutions ...
- 643
2
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0
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209
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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
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0
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148
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$p$-adic Hodge theoretic properties of global Galois representations via $\ell$-Frobenii
Let $G_{\mathbb{Q},S} = \mathrm{Gal}(\mathbb{Q}_S/\mathbb Q)$ where $\mathbb Q_S$ is the largest algebraic extension of $\mathbb Q$ unramified outside a finite set of places $S$. Then the union over $\...
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0
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336
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A question on the Robba ring
Notation is as in the question:
https://math.stackexchange.com/questions/4090045/some-questions-about-the-robba-ring.
We define a new operator over the Robba ring as follows. Put $$c=\frac{pE(u)}{E(0)}...
- 166
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2
answers
1k
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Vector bundles on adic spaces
Let $X = \mathrm{Spa}(A,A^+)$ be an analytic sheafy adic space. Let $\mathcal{E}$ be a locally finite free $\mathcal{O}_X$ sheaf. Does $\mathcal{E}$ correspond to a geometric vector bundle over $X$? ...
- 643
2
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1
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232
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Local to global for semistable $G_{\mathbb{Q}_p}$-representations
Let $\rho_p:G_{\mathbb{Q}_p} \to \text{Gl}_n(\mathbb{Q}_p)$ be semistable representation. In local to global Galois representation, it was asked if one can find a geometric global Galois ...
- 4,418
3
votes
1
answer
184
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Restriction of $(\varphi, N)$-modules
For any $p$-dic field $K$, we have an equivalence of categories
$$D_{st}:Rep_{\mathbb{Q}_p}^{st}(G_K)\rightarrow MF_K^{ad}(\varphi,N),\quad V\mapsto (B_{st}\otimes_{\mathbb{Q}_p} V)^{G_K}$$
with quasi-...
- 4,418
7
votes
1
answer
422
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Integral refinements of rigid cohomology
Disclaimer: I know absolutely nothing about p-adic cohomology, so it is possible that even the premises of this question are incorrect. But it turns out that I need to apply the theory of rigid ...
- 5,958
7
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1
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747
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Finite non-empty coproduct in the absolute prismatic site
Let $(R/A)_\Delta$ be the prismatic site over $R$ relative to a prism $(A, I)$, then it is known that $(R/A)_\Delta$ admits finite non-empty coproduct, for instance, by Cor. 5.2 in Bhatt's lecture ...
- 193
3
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1
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475
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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.
...
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2
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138
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Local deformation ring of representations with equal generalized Hodge-Tate weights
Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $\overline{\rho}:\mathrm{Gal}(\overline{K}/K)\rightarrow \mathrm{GL}_2(\mathbb{F})$ be a characteristic $p$ representation. According to a ...
- 643
24
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1
answer
3k
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Condensed criterion for sheafiness of adic spaces
Multiple times in talks about condensed mathematics (e.g. the Masterclass talks, Clausen's RAMpAGe talk), it is stated that the derived structure sheaf given by the condensed formalism "fixes&...
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1
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How many untilts?
I read the following passage in Endomorphisms of power series fields and residue fields of Fargues-Fontaine curves by Kedlaya-Temkin:
"One can construct many algebraic extensions of $\mathbb{Q}_p$...
- 282
1
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0
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270
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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
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0
answers
390
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Is there a period ring B_dif?
Let $K$ be a finite extension of $\mathbb{Q}_p$ and let $V$ be a p-adic representation of $\mathrm{Gal}(\overline{\mathbb{Q}}_p/K)$. Write $K_\infty=K(\mu_{p^{\infty}})$ for the cyclotomic extension ...
- 643
4
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0
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293
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Galois representation with infinite image but finite image everywhere locally
Fix a prime $l$. Let $\phi:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to GL_n(\mathbb{Q}_l)$ be a semisimple continuous representation. Assume $\phi$ has finite image when restricted to $\mathrm{...
user158636
3
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0
answers
232
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$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
1
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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
6
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0
answers
231
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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
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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
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0
answers
244
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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
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1
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405
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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
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1
answer
204
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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
3
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0
answers
272
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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 $\mathcal{O}_\mathcal{E}...
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0
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150
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$SL_2(\mathbb{Z}_p)$ extension of a local field
Let $G$ be an arbitrary open group of $SL_2(\mathbb{Z}_p)$ and $K$ be a finite extension of $\mathbb{Q}_p$. Can we construct a Galois extension field $E$ of $K$ such that $\text{Gal}(E/K)\cong G$? ...
user141691
12
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1
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510
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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" ${\...
- 16.2k
2
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0
answers
670
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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 ...
- 21
1
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0
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125
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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
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0
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361
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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
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0
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230
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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
4
votes
1
answer
278
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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(\...
user145520
7
votes
1
answer
402
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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
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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
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0
answers
419
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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
14
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2
answers
2k
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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 ...
- 1,347
2
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1
answer
281
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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