Questions tagged [p-adic-hodge-theory]
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218 questions
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A question about Hasse Invariant and Modular curve
Let $N\geq4$ be a positive integer and p be a prime such that $(p,N)=1$, and $X=X_1(N)$ be the modular curve parameterizing (generalized) elliptic curves with $\Gamma_1(N)$-level structure. Base ...
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Complete characteristic p perfect Tate rings are uniform?
In Lemma 7.1.6 of his lecture notes on perfectoid spaces, Bhatt states that every complete characteristic p perfect Tate ring $A$ is uniform. In the proof he uses the Banach open mapping theorem on ...
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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 ...
CommunityBot
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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
$cl^X_{DR}:CH^k(X)\to H^{2k}_{DR}(X/K)$ ...
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1
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Equivalence of vector bundles over $Spec(A_{\inf})$ and the punctured spectrum
I'm trying to understand the Lemma 4.6 of Bhatt-Morrow-Scholze's paper Integral $p$-adic Hodge Theory.
In the proof, for proving the restriction functor is fully faithful, it used a affine open cover $...
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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%...
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1
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A question about Kato's explicit reciprocity law
In the paper Iwasawa Theory and F-analytic Lubin-tate $(\phi,\Gamma)$-modules
Prop 3.4.2 says that for any $x\in{S}$, there exists (not uniuqely) $f(T)\in{B_{rig,F}^+}$ such that
$f(u_n)=\log_{LT}(...
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Why does $H^i(X_{ét},\mathbb{Q}_p)$ have a Hodge-Tate structure?
Let $X$ be a variety over a $p$-adic field $K$.
Is there a simple or intuitive explanation of why the $G_K$ representation $H^i(X_{ét},\mathbb{Q}_p)$ is Hodge-Tate? More precisely, why do the powers ...
CommunityBot
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1
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Injectivity of Frobenius on $A_{cris}$
I am reading Brinon, Conrad "Notes on $p$-adic Hodge theory" and I can't find any reference for the proof of Theorem 9.1.8, namely the injectivity of the Frobenius endomorphism of $A_{cris}$. Does ...
CommunityBot
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The Breuil-Mezard Conjecture and Generalizations (Survey)
What's the current state of the Breuil-Mezard conjecture? Has the original version (from the 2002 paper) been solved in its entirety? What are some of the new directions being explored?
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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 ...
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$p$-adic Hodge Theory for rigid spaces, after P. Scholze
I was going over P. Scholze's paper on $p$-adic Hodge Theory for rigid analytic varieties.
This question is around the "Poincaré Lemma" in the paper.
Throughout, let $X$ be a proper smooth rigid ...
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6
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Two Definitions of Barsotti-Tate Representations
In different articles I have seen different definitions of Barsotti-Tate representations. I am wondering if and how these definitions are equivalent.
In Section 1.1 of Conrad-Diamond-Taylor they say ...
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Smooth intertwining operators
Let $V$ be a crystalline irreducible representation of the absolute Galois group of $\mathbb{Q}_p$ with distinct Hodge Tate weights $(0,k-1), k \in \mathbb{Z}_{\geq 2}$.
Then $V$ is uniquely ...
CommunityBot
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Locally analytic vectors of a quotient space
My question here is in connection with one of my previous question
"A definition of a (amalgamated) direct sum"
Following the notations there, my question is:
Why the locally analytic vectors of $B(...
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How large is Dcris of certain twists of modular forms?
I want to determine $\mathrm D_{\mathrm{cris}}$ of certain twists of the Galois representations attached to modular forms. For one particular twist it is not clear to me how $\mathrm D_{\mathrm{cris}}$...
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1
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Reference on a result on local Galois representation associated to classic modular form in p-adic Hodge theory
At the end of Fontaine’s rings and p-adic L-functions, P. Colmez states a Theorem 8.4.8 (click here) of Faltings-Tusji-Saito without references.
So I am wondering is there any references for this ...
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A definition of a (amalgamated) direct sum
I am wondering about a definition of a direct sum in page $31$ of this paper by R. Liu.
I am following the notations in page $31$ of the above paper. Let $V$ be a crystalline irreducible ...
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267
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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 ...
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1
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p-adic representations of $GL_2(\mathbb{Q}_p)$
Let $L$ be a finite extension of $\mathbb{Q}_p$. Colmez defines here
the trainguline representations which are extensions of Robba rings of dimension $1$. Then, in this paper he contructs the ...
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1
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Describing the ratio of uniformizers in B_dR
In Conrad and Brinon's notes http://math.stanford.edu/~conrad/papers/notes.pdf, two uniformizers of $B_{dR}$ are produced: one is $\xi := [\tilde{p}]-p$ (bottom of p.58), where $\tilde{p} = (p, p^{1/p}...
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1
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Uniqueness of finite flat models over bases of low ramification via Breuil-Kisin modules
Let $R$ be a complete DVR of mixed characteristic $(0, p)$, let $K$ be its fraction field, and assume that the absolute ramification index $e$ of $R$ satisfies $e < p - 1$ and that the residue ...
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Ramification for subgroups of Lubin-Tate formal group
Let $K/\mathbb{Q}_p$ be a finite field extension and $E/\mathcal{O}_K$ be the local N\'eron model of a CM elliptic curve with CM by $\mathcal{O}_F$ and let $G\subseteq E[p^n]$ be a subgroup over $\...
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1
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282
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Serre tensor construction on finite flat group schemes
Let $K/\mathbb{Q}_p$ be a finite field extension and $\mathcal{O}_K\subseteq K$ be its ring of integral elements. Let also $G/\mathcal{O}_K$ be a finite flat $p$-group scheme that is also an $\mathcal{...
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Conjecture on classification of $p$-divisible over the ring of integers of $\widehat{\bar{\mathbb{Q}}_p}$
I am reading the paper of Fargues Quelques résultats et conjectures concernant la courbe. In the end of this paper, there is a conjecture on the classification of $p$-divisible groups over $\...
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Kisin module for CM elliptic curve
Let $E$ be a CM elliptic curve with CM by the field $K$ and assume that $p$ is ramified in $K$ so that $\pi^2 = p \in \mathcal{O}_K$. In particular, then $E$ has supersingular reduction at $p$ and by ...
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1
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Crystalline extension the $p$-adic cyclotomic character
Let $\epsilon_p$ be the $p$-adic cyclotomic character, $F$ be a real quadratic extension of $\mathbb{Q}$ in which $p$ splits, $\psi$ be an odd character of $G_\mathbb{Q}$ of finite image and with ...
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2
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1
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extension of the universal cyclotomic character
Let $p$ be a prime number, $\psi:G_\mathbb{Q} \rightarrow \bar{\mathbb{Q}}_p$ be an odd character of conductor $N$ prime to $p$, with finite image and such that $\psi(p)=1$. Let $\mathcal{W}$ be the ...
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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{...
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Morphisms for good reduction are maps respecting filtration
Please see edits below!
So, let $A,A'/K$ be abelian varieties where $K$ is a $p$-adic local field with residue field $k$. Suppose further that they have good reduction with models $\mathscr{A},\...
CommunityBot
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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 ...
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$p$-divisible groups and Breuil-Kisin modules with coefficients
Let $K$ be a finite extension of $\mathbb{Q}_p$ with ring of integers $\mathcal{O}_K$ and residue field $k$. Choose a uniformizer $\pi \in \mathcal{O}_K$ and $E(u)$ be the minimal (Eisenstein) ...
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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 ...
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1
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Semistability of local Siegel Galois rep:
When are the $l$-local $p$-adic Galois representations of Siegel modular forms semistable? By this I mean $\rho_{f}: G_{\mathbb{Q}}\to \operatorname{GSpin}_{2n+1}(\overline{\mathbb{Q}}_p)$ restricted ...
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congruent number problem [closed]
I am studying the congruent number problem
and I heard that there is a paper by Kazuma Morita
which claims to solve this problem from my colleague.
I saw the paper on his homepage but it is very ...
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4
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1
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508
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Katz $p$-adic L function and ordinary condition
Let $H$ be a CM field and $F$ be the maximal totally real subfield of $H$. Can we construct a Katz $p$-adic L-functions of Hecke characters without the ordinary condition (i.e every prime of $F$ above ...
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Is Faltings' $p$-adic Eichler-Shimura isomorphism the $p$-adic comparison isomorphism?
This is a question about Faltings' $p$-adic Eichler-Shimura isomorphism from his 1987 article "Hodge-Tate structures and Modular Forms".
Let $N\ge5$, $k\ge2$ be integers. Denote by $X(N)$ the proper ...
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Hodge-Tate weights of etale cohomology
Let $K/\mathbb Q_p$ be a local field, $X/K$ a proper scheme with semi-stable reduction.
Question: What is the possible range of Hodge-Tate weights of the etale cohomology $H^i(X_{\overline K}, \...
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Examples of p-adic representations
When reading the books or papers on p-adic Hodge theory, non trivial example of p-adic representation seems to be only the example of Tate curves. To be sure, I had read the very readable introduction ...
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Hodge–Tate structures of modular forms
The title refers to the paper of Faltings:
Hodge-Tate structures and modular forms.
Math. Ann. 278 (1987), no. 1-4, 133–149.
The main theorem in the paper says that the associated Galois rep to a ...
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1
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Hodge-Tate representations
I know that the p-adic representaions from geometries are de Rham representations and hence they are Hodge-Tate
representations. Then, are there (more than 2-dimensional) Hodge-Tate representations ...
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2
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Describing the crystalline extension of $\mathbb{Q}_p$ by $\mathbb{Q}_p$
Let $K$ be a finite extension of $\mathbb{Q_p}$. The group $\ker H^1(G_K, \mathbb{Q}_p) \rightarrow H^1(G_K, B_{crys})$ is one-dimensional, which tells us that among all extensions of Galois modules
...
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Status of Fontaine-Mazur conjecture
In the language of Richard Taylor's 2004 (extended) ICM article (''Galois Representations'', Annales de la faculté des sciences de Toulouse (2004) Tome XIII, no. 1, 73-119), the conjecture is the ...
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Relationship between the syntomic cohomology of Kato and of Fontaine-Messing
Fix a prime $p$ and let $X$ be a $\mathbb{Z}_{p}$-scheme. Write $X_{n}:=X\otimes\mathbb{Z}/p^{n}$ and $\phi:X_{1}\rightarrow X_{1}$ for the absolute Frobenius. Let $X\hookrightarrow Z$ be a (suitable) ...
CommunityBot
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Applications of $p$-adic Hodge theory
I am trying to learn $p$-adic Hodge theory. I found some materials explaining main theorems (or aspects) of the theory. However, I could not find references which explaining how to use the theory. ...
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1
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notion of $\mathrm{Gal}(\overline{\mathbb{Q}}_p / \mathbb{Q}_p)$ representation with complex multiplication
In usual Hodge theory, there is the notion of Hodge structure $H$ with complex multiplication, that can be defined in several ways, i.e. asking that there exists a CM number field $E$ such that $\dim ...
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Analogue of Tate or Hodge conjecture for varieties over $\mathbb Q_p$
I've been learning about p-adic Hodge theory recently (I'm a beginner), and I've been wondering about the following question the past couple of weeks. Sorry for the long setup, it's mainly background;...
CommunityBot
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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)...
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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 ...
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0
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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) \...
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