Convergence polygons for connections on nonarchimedean curves

TL;DR

This survey explores the convergence polygons of differential equations on nonarchimedean curves, connecting their properties to formal classifications, cohomology indices, ramification, and automorphism lifting problems.

Contribution

It provides a comprehensive overview of convergence polygons in nonarchimedean geometry, including new technical results and connections to various classification and ramification topics.

Findings

Convergence polygons relate to formal classification of differential equations.

Index formulas link de Rham cohomology to convergence properties.

Ramification and automorphism lifting are connected to convergence polygon analysis.

Abstract

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.