A theory of ordinary \(p\)-adic curves
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.
scientific article; zbMATH DE number 1034053
scientific article