p-adic Hodge Theory

Editors: Bhargav Bhatt, Martin Olsson

Publisher: Springer International Publishing

Book Series : Simons Symposia

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

About this book

This proceedings volume contains articles related to the research presented at the 2017 Simons Symposium on p-adic Hodge theory. This symposium was focused on recent developments in p-adic Hodge theory, especially those concerning integral questions and their connections to notions in algebraic topology. This volume features original research articles as well as articles that contain new research and survey some of these recent developments. It is the first of three volumes dedicated to p-adic Hodge theory.

Frontmatter

Notes on the -Cohomology of Integral p-Adic Hodge Theory

Abstract

We present a detailed overview of the construction of the \(\mathbb A_{{\mathrm {inf}}}\)-cohomology theory from the preprint Integral p-adic Hodge theory, joint with Bhatt and Scholze. We focus particularly on the p-adic analogue of the Cartier isomorphism via relative de Rham–Witt complexes.

Matthew Morrow

On the Cohomology of the Affine Space

Abstract

We compute the p-adic geometric pro-étale cohomology of the rigid analytic affine space (in any dimension). This cohomology is non-zero, contrary to the étale cohomology, and can be described by means of differential forms.

Pierre Colmez, Wiesława Nizioł

Arithmetic Chern–Simons Theory II

Abstract

In this paper, we apply ideas of Dijkgraaf and Witten [6, 32] on 3 dimensional topological quantum field theory to arithmetic curves, that is, the spectra of rings of integers in algebraic number fields. In the first three sections, we define classical Chern–Simons actions on spaces of Galois representations. In the subsequent sections, we give formulas for computation in a small class of cases and point towards some arithmetic applications.

Hee-Joong Chung, Dohyeong Kim, Minhyong Kim, Jeehoon Park, Hwajong Yoo

Some Ring-Theoretic Properties of

Abstract

The ring of Witt vectors over a perfect valuation ring of characteristic p, often denoted \(\mathbf {A}_{{{\mathrm{inf}\,}}}\), plays a pivotal role in p-adic Hodge theory; for instance, Bhatt–Morrow–Scholze have recently reinterpreted and refined the crystalline comparison isomorphism by relating it to a certain \(\mathbf {A}_{{{\mathrm{inf}\,}}}\)-valued cohomology theory. We address some basic ring-theoretic questions about \(\mathbf {A}_{{{\mathrm{inf}\,}}}\), motivated by analogies with two-dimensional regular local rings. For example, we show that in most cases \(\mathbf {A}_{{{\mathrm{inf}\,}}}\), which is manifestly not noetherian, is also not coherent. On the other hand, it does have the property that vector bundles over the complement of the closed point in \({{\,\mathrm{Spec}\,}}\mathbf {A}_{{{\mathrm{inf}\,}}}\) do extend uniquely over the puncture; moreover, a similar statement holds in Huber’s category of adic spaces.

Kiran S. Kedlaya

Sur une q-déformation locale de la théorie de Hodge non-abélienne en caractéristique positive

Abstract

Pour p un nombre premier et q une racine p-ième non triviale de 1, nous présentons les principales étapes de la construction d’une q-déformation locale de la “correspondance de Simpson en caractéristique p” dégagée par Ogus et Vologodsky en 2005. La construction est basée sur l’équivalence de Morita entre un anneau d’opérateurs différentiels q-déformés et son centre. Nous expliquons aussi les liens espérés entre cette construction et celles introduites récemment par Bhatt et Scholze. Pour alléger l’exposition, nous nous limitons au cas de la dimension 1. For p a prime number and q a non trivial pth root of 1, we present the main steps of the construction of a local q-deformation of the “Simpson correspondence in characteristic p” found by Ogus and Vologodsky in 2005. The construction is based on the Morita-equivalence between a ring of q-twisted differential operators and its center. We also explain the expected relations between this construction and those recently done by Bhatt and Scholze. For the sake of readability, we limit ourselves to the case of dimension 1.

Michel Gros

Crystalline -Representations and -Representations with Frobenius

Abstract

In the late ’80s, Faltings established an integral p-adic Hodge theory with coefficients, in which he generalized Fontaine–Laffaille theory of crystalline \(\mathbb Z_p\)-representations of the absolute Galois group of a p-adic field to the fundamental group of a non-singular algebraic variety over a p-adic field with good reduction. In this paper, we study the theory of coefficients above in the framework of integral p-adic Hodge theory via \(A_{\text {inf}}\)-cohomology recently introduced by Bhatt, Morrow, and Scholze. We give a local theory (i.e. a theory on an affine open) of \(A_{\inf }\)-cohomology for a p-torsion free crystalline \(\mathbb Z_p\)-representation of the fundamental group by constructing the associated \(A_{\inf }\)-representation with Frobenius, which is a variant of the construction by N. Wach of the \((\varphi ,\varGamma )\)-module associated to a crystalline \(\mathbb Z_p\)-representation of the absolute Galois group.

Takeshi Tsuji

Title: p-adic Hodge Theory
Editors: Bhargav Bhatt
Martin Olsson
Publisher: Springer International Publishing
Electronic ISBN: 978-3-030-43844-9
Print ISBN: 978-3-030-43843-2
DOI: https://doi.org/10.1007/978-3-030-43844-9

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p-adic Hodge Theory

About this book

Table of Contents

Frontmatter

Notes on the -Cohomology of Integral p-Adic Hodge Theory

On the Cohomology of the Affine Space

Arithmetic Chern–Simons Theory II

Some Ring-Theoretic Properties of

Sur une q-déformation locale de la théorie de Hodge non-abélienne en caractéristique positive

Crystalline -Representations and -Representations with Frobenius

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