On commutative nonarchimedean Banach fields

We study the problem of whether a commutative nonarchimedean Banach ring which is algebraically a field can be topologized by a multiplicative norm. This can fail in general, but it holds for uniform Banach rings under some mild extra conditions. Notably, any perfectoid ring whose underlying ring is a field is a perfectoid field.Comment: 14 pages; v7: includes corrections to published versio

Convergence polygons for connections on nonarchimedean curves

This is a survey article on ordinary differential equations over nonarchimedean fields based on the author's lecture at the 2015 Simons Symposium on nonarchimedean and tropical geometry. Topics include: the convergence polygon associated to a differential equation (or a connection on a curve); links to the formal classification of differential equations (Turrittin-Levelt); index formulas for de Rham cohomology of connections; ramification of finite morphisms; relations with the Oort lifting problem on automorphisms of curves. The appendices include some new technical results and an extensive thematic bibliography.Comment: v2: final refereed version; one appendix withdrawn, other minor correction

The Hochschild-Serre property for some p-adic analytic group actions

Let $H \subseteq G$ be an inclusion of $p$-adic Lie groups. When $H$ is normal or even subnormal in $G$, the Hochschild-Serre spectral sequence implies that any continuous $G$-module whose $H$-cohomology vanishes in all degrees also has vanishing $G$-cohomology. With an eye towards applications in $p$-adic Hodge theory, we extend this to some cases where $H$ is not subnormal, assuming that the $G$-action is analytic in the sense of Lazard.Comment: 9 pages; v2: simpler counterexample added, other minor correction

Automorphisms of perfect power series rings

Let R be a perfect ring of characteristic p. We show that the group of continuous R-linear automorphisms of the perfect power series ring over R is generated by the automorphisms of the ordinary power series ring together with Frobenius; this answers a question of Jared Weinstein.Comment: 4 pages; v3: final submitted versio

Computing zeta functions via p-adic cohomology

We survey some recent applications of p-adic cohomology to machine computation of zeta functions of algebraic varieties over finite fields of small characteristic, and suggest some new avenues for further exploration.Comment: 18 pages; to appear in the Proceedings of ANTS-VI (Algorithmic Number Theory Symposium, 2004