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I'm currently working through Tate's paper on $p$-divisible groups and have come to an impasse in his discussion of tangent spaces of $p$-divisible groups. I'd like to show that for an abelian variety ...
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Prismatic cohomology is a cohomology theory introduced by Bhatt and Scholze around 2019. It provides a unified framework for p-adic Hodge theory, bringing together various cohomology theories such as ...
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Let $X$ be an affinoid perfectoid space of characteristic either $0$ or $p$. It is well known that for a $p$-torsion sheaf $\mathcal F$, $H^i(X_{et}, \mathcal F)=0$ for $i>1$ (see Scholze's comment ...
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In Peter Scholze’s “Geometrization of Local Langlands, Motivically”, he discusses an object coined as the “inertia-Deligne group”, which is an object that lies in an extension of the inertia group by ...
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Definition 9.2.2 of Scholze and Weinstein's Berkeley Lectures on $p$-adic Geometry says "Consider the site Perf of perfectoid spaces of characteristic $p$ with the pro-ètale topology. A map $f:\...
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Illusie's article in volume II of the Grothendieck Festschrift cites a 1988 preprint: "Arithmetic of Fermat varieties I, Fermat motives and p-adic cohomologies", by Suwa-Yui. It does not ...
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Let $p$ be a prime, let $K$ be an unramified finite extension of $\mathbb{Q}_p$ with residue field $k$, and let $r$ be an integer in the range $[0, p - 2]$. We denote by $\mathrm{FL}^{\text{tors}} (\...
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Let $Y\to X$ denote a relative abelian scheme, with $X$ smooth proper over $k$ a finite field of characteristic $p$. Let $\mathcal{V} := R^1_{\mathrm{cr}}f_*\textbf{1}[1/p]$ denote the Gauss-Manin $F$-...
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Alexander Polishchuk proved that for $m=(m_1, \dots, m_n)$ a partition of $2g-2$ with also some negative entries, the closed subset of $\mathcal{M}_{g,n}$ parametrizing smooth marked curves $(C, q_1, \...
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Let $K/\mathbb Q_p$ be a finite extension, let $\pi$ be a uniformizer of $K$, let $\mathbb F_q$ be the residue field of $K$ and let $E/K$ be an elliptic curve such that $\tilde E$, the reduction $\...
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In Dwork's paper "Bessel functions as $p$-adic functions of the argument", a certain constant $\gamma$ arises as a matrix entry in his calculations of a Frobenius structure that he ...
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Fix a prime $p$, let $X$ be an open subspace of $\mathbb{P}^1_{\mathbb{Q}_p}$, and let $(E, \nabla)$ be a connection on $X$. In Definition 2.3.3 of https://arxiv.org/pdf/1912.13073, Kedlaya defines a ...
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My question concerns the compatibility of the complex and p-adic Hodge decompositions. To make this more precise, fix an isomorphism $\sigma: \mathbb{C}_{p} \simeq \mathbb{C}$, and let $K$ be a ...
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I am attempting to get a tourist's perspective on p-adic Hodge theory, and am confused about the precise relation between two very successful geometric interpretations of (different parts of) the ...
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Let $K$ be a finite extension of $\mathbb{Q}_p$ with ring of integers $A$. Let $F$ be a $p$-divisible group and $T$ be the Tate module. Consider the vector space $V=T \otimes_{\mathbb{Q}_p} C$, where $...
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Are there a polynomials $f_1,...f_n \in \mathbb{Z}_p[x_1,...x_n]$ with there coeficients $p$-adic integers s.t. A map $F:\mathbb{Z}_p^n\rightarrow \mathbb{Z}_p^n$ defined by $f_1,...f_n$ satisfy the ...
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The question is in the title, but here's some quick background. It's easy to show (assuming smooth-proper base change) that the $\ell$-adic cohomology of a variety over the fraction field of a DVR ...
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Following the notation in https://arxiv.org/abs/1011.3447 a representation $V$ is split trianguline iff $D(V)$ has a basis in which the matrices of $\varphi$ and of all the elements of $\Gamma$ are ...
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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_{\...
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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-...
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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 ...
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Suppose $K$ is a finite extension of $\mathbb Q_p$. Consider the one-point adic space $X=\operatorname{Spa}K$, and let $C=\hat {\bar K}$, $G=\operatorname{Gal}(\bar K/K)$. I heard that the category of ...
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Brian Conrad and Ofer Gabber have some results that were announced 9 years ago here: https://www.ihes.fr/~abbes/Gabber/OferGabber.pdf and there's a talk by Gabber about them here: https://www.youtube....
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Suppose $X$ is an abelian variety over a $p$-adic field $K$, and it's well known that $X$ has good reduction is equivalent to the étale cohomology of $X$ is crystalline, and $X$ has semistable ...
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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 $...
Richard's user avatar
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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_{\...
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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: $\frac{\mathrm d \bf {f}}{\...
Richard's user avatar
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I was trying to understand the statement and proof of Theorem 7.11 in Scholze's paper "$p$-adic Hodge Theory for Rigid-Analytic Varieties". I'll reproduce part of the statement below: Let $...
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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 \...
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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 $\...
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Does anyone have a copy of Kato's article Generalized explicit reciprocity laws in Advanced Studies in Contemp. Math which is used heavily in his paper constructing his eponymous Euler system? I used ...
xir's user avatar
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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 ...
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$\newcommand{\cris}{\mathrm{cris}}$In my setting, $K/\mathbb Q_p$ is finite and unramified, and $V$ is a $2$-dimensional crystalline representation of $G_K$. Then we have $D_{\cris}(V)$, which is $2$-...
Richard's user avatar
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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 ...
Richard's user avatar
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Suppose $K$ is an unramified extension of $\mathbb Q_p$ of degree $m$, and $\sigma$ is the $p$ power frobenius on $K$. Suppose $V$ is a $2$ dimensional admissible filtered $\phi$ module over $K$. I ...
Richard's user avatar
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2 votes
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Suppose $K$ is an finite unramified extension of $\mathbb Q_p$ with residue field $k$, and let $Y$ be an proper smooth variety defined over $k$. We know if $Y$ admits a proper smooth lifting $X/W(k)$ ...
Richard's user avatar
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This question is following the previous question. Definitions: Suppose $F$ is an unramified finite extension of $\mathbb Q_p$ and $E$ is an elliptic curve defined over $F$ with good reduction. Denote ...
Richard's user avatar
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I'm reading "An introduction to the theory of $p$-adic representations" by Laurent Berger. In the page 14 it says: Suppose $F$ is an unramified finite extension of $\mathbb Q_p$ and $E$ is ...
Richard's user avatar
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$\DeclareMathOperator{\Spf}{Spf}\DeclareMathOperator{\Spa}{Spa}$In the Definition 8.5 of the paper "integral $p$ adic Hodge theory" by Bhatt-Morrow-Scholze, they define the adic generic ...
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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 $D_{cris}(H_{et}^i(X,\mathbb Q_p))=H_{dR}^i(X)$. ...
Richard's user avatar
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Faltings proved the following: Fix integers $w, d \geqslant 0$, and fix a number field $K$ and a finite set $S$ of primes of $\mathcal{O}_K$. There are, up to conjugation, only finitely many ...
Richard's user avatar
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I am interested in learning about Fontaine-Messing theory. Besides the original papers, though, I don't know any good expository literature on this topic (crystalline representations, etc.). Can ...
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Is [Scholl, Motives for modular forms, Theorem 1.2.4 (ii)] proven for any $p$ independent of the weight? Concretely, let $f$ be a normalized eigenform of weight $w$. Let $p$ be a prime not dividing ...
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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 ...
Yijun Yuan's user avatar
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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 $H^*_{\...
OOOOOO's user avatar
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$\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 ...
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In Lawrence and Venkatesh's paper on the Mordell conjecture, they prove that there are finitely many $K$-rational points on a hyperbolic curve $X$, where $K$ is a number field, by showing that there ...
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Let $K$ be a discretely valued extension of $\mathbb{Q}_p$ with perfect residue field $k$, and $\mathcal{C}$ a completed algebraic closure of $K$ with the ring of integers $\mathcal{O}_{\mathcal{C}}$. ...
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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 ...
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2 answers
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Recently I've been rewatching some recordings of old talks on L-functions and explicit reciprocity laws (in particular, the series of talks by Loeffler and Zerbes given at this workshop at the CRM in ...
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