A theory of ordinary \(p\)-adic curves
type: Item
instance of: scholarly article
title: A theory of ordinary \(p\)-adic curves
zbMATH Open document ID: 0879.14009
author: Shinichi Mochizuki
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1998-01-29T00:00:00Z
review text: For elliptic curves in characteristic \(p\) there is a difference between ordinary curves (the generic case) and supersingular curves. The paper under review gives a generalization to genus \(g\geq 2\), different from the obvious look at the Jacobian. A curve is called ordinary if it admits a certain type of rank two bundle with filtration and Frobenius, or a certain family of \(p\)-divisible groups of dimension one and height two. The moduli space of ordinary curves is étale over the usual moduli space of stable curves, and covers the most degenerate curves (the Mumford curves). Its lift to a formal \(p\)-adic scheme (étale over the moduli space) admits a canonical Frobenius-lift which is also ``ordinary''. In turn this defines the notion of a canonical lift from characteristic \(p\) to characteristic zero, over any base-ring with Frobenius-lift. Finally an ordinary curve contains a dense open set of ordinary points, and on the canonical lift of this curve Frobenius also lifts on these ordinary points. There are also logarithmic variants, and results about local moduli problems. The proofs consist mostly of ingenious calculations of obstruction-classes.
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 14D10
Mathematics Subject Classification ID: 14G20
zbMATH DE Number: 1034053
zbMATH Keywords: characteristic \(p\)
zbMATH Keywords: elliptic curves
zbMATH Keywords: moduli space of ordinary curves
zbMATH Keywords: Mumford curves
zbMATH Keywords: Frobenius-lift
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195145686
OpenAlex ID: W2071618654
cites work: The real analytic theory of Teichmüller space
cites work: The Picard groups of the moduli spaces of curves
cites work: Théorie de Dieudonne cristalline. II
cites work: Uniformization by Beltrami equations
cites work: Q3972724
cites work: Q4202168
cites work: Q3931656
cites work: Q3766016
cites work: Special coordinate coverings of Riemann surfaces
cites work: On the Kodaira dimension of the moduli space of curves
cites work: Q4049198
cites work: Lifting curves over finite fields together with the characteristic correspondence Pi+Pi'
cites work: Logarithmic structures of Fontaine-Illusie. II
cites work: Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin
cites work: Arithmetic Moduli of Elliptic Curves. (AM-108)
cites work: Q3744437
cites work: Q3997989
cites work: Q4011193
cites work: The crystals associated to Barsotti-Tate groups: with applications to Abelian schemes
cites work: The geometry of the compactification of the Hurwitz scheme
cites work: Q4301584
cites work: Q4111838
cites work: Functors of Artin Rings
cites work: Q3665240
cites work: Q5625265
cites work: Q5539560
cites work: The Fenchel-Nielsen deformation
DOI: 10.2977/PRIMS/1195145686
reviewed by: Gerd Faltings
description: scientific article; zbMATH DE number 1034053