Questions tagged [p-adic-hodge-theory]
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218 questions
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Galois invariant of Tate module
Let $K$ be a finite extension of $\mathbb{Q}_p$. Let $V$ be a de Rham representation of $G_K=\operatorname{Gal}(\overline{K}/K)$. By Corollary 3.8.4 of Bloch and Kato - L-functions and Tamagawa ...
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1
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Reductive subgroups of $\mathrm{GL}_2$ over an algebraically closed field of characteristic zero
I am reading a very nice paper of Newton and Thorne, Symmetric power functoriality for holomorphic modular forms, and there is an argument concerning the (Zariski-closure of) image of certain $p$-adic ...
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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 $\...
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$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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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 ...
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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 ...
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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 $\mathbb{Q}_p$...
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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 ...
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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 ...
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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}}_{\...
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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 ...
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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 ...
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Grothendieck's Period Conjecture and the missing p-adic Hodge Theories
Singular cohomology and algebraic de Rham cohomology are both functors from the category of smooth projective algebraic varieties over $\mathbb Q$ to $\mathbb Q$-vectors spaces. They come with the ...
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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 ...
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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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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 ...
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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 ...
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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 $\...
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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 ...
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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 ...
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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 ...
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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?
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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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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-...
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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 $$c=\frac{pE(u)}{E(0)}...
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475
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To identify $p$-adic Tate module $T_p(G)$ of $p$-divisible group $G$ in the category $\text{Rep}_{\mathbb{Q}_p}(G_{K_\infty})$
Let $k$ be a perfect field of characteristic $p>0$, $W=W(k)$ its ring of Witt vectors, $K_0=W(k)[\frac{1}{p}]$ and, $K/K_0$ be a totally ramified extension. Let $\pi \in K$ be an uniformizer.
...
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Vector bundles on adic spaces
Let $X = \mathrm{Spa}(A,A^+)$ be an analytic sheafy adic space. Let $\mathcal{E}$ be a locally finite free $\mathcal{O}_X$ sheaf. Does $\mathcal{E}$ correspond to a geometric vector bundle over $X$? ...
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Local to global for semistable $G_{\mathbb{Q}_p}$-representations
Let $\rho_p:G_{\mathbb{Q}_p} \to \text{Gl}_n(\mathbb{Q}_p)$ be semistable representation. In local to global Galois representation, it was asked if one can find a geometric global Galois ...
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periodic cyclic homology and tilting in the sense of Scholze
Suppose $R$ is a perfectoid ring in mixed characteristic, and $R'$ its characteristic-$p$ tilt. Scholze's results on tilting say that the étale theories over $R$ and $R'$ are equivalent in an almost ...
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Restriction of $(\varphi, N)$-modules
For any $p$-dic field $K$, we have an equivalence of categories
$$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-...
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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 ...
- 21.3k
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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 ...
CommunityBot
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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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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{\...
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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 ...
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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 $\mathrm{...
CommunityBot
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$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 ...
CommunityBot
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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 ...
CommunityBot
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272
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$p$-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). ...
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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 ...
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4
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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 $\delta$-...
- 63.9k
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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 ...
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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 ...
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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 $\mathcal{O}_\mathcal{E}...
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$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 $\text{Gal}(E/K)\cong G$? ...
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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" ${\...
- 7,377
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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
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0
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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 ...
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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)...
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