Topology and Geometry of the Berkovich Ramification Locus for Rational Functions

TL;DR

This paper studies the complex structure of ramification loci for rational functions over non-Archimedean fields using Berkovich analytic curves, revealing new behaviors absent in classical complex analysis.

Contribution

It introduces a detailed analysis of the ramification locus for self-maps of the projective line over non-Archimedean fields, highlighting new non-algebraic ramification phenomena.

Findings

Ramification locus can be a closed analytic subspace, not just a divisor.

New ramification behaviors emerge in non-Archimedean settings.

Approach combines concrete and combinatorial techniques.

Abstract

Given a nonconstant holomorphic map f: X -> Y between compact Riemann surfaces, one of the first objects we learn to construct is its ramification divisor R_f, which describes the locus at which f fails to be locally injective. The divisor R_f is a finite formal linear combination of points of X that is combinatorially constrained by the Hurwitz formula. Now let k be an algebraically closed field that is complete with respect to a nontrivial non-Archimedean absolute value. For example, k = C_p. Here the role of a Riemann surface is played by a projective Berkovich analytic curve. As these curves have many points that are not algebraic over k, some new (non-algebraic) ramification behavior appears for maps between them. For example, the ramification locus is no longer a divisor, but rather a closed analytic subspace. This article initiates a detailed study of the ramification locus for self-maps f: P^1 -> P^1. This simplest first case has the benefit of being approachable by concrete (and often combinatorial) techniques.