Resultant and conductor of geometrically semi-stable self maps of the projective line over a number field or function field

TL;DR

This paper investigates the relationship between the minimal resultant divisor and conductor of self-maps of the projective line over number and function fields, exploring semi-stability, minimality, and dynamical analogs to elliptic surface theorems.

Contribution

It introduces a dynamical analog to elliptic surface discriminant bounds, analyzes semi-stability and minimality conditions, and constructs counterexamples in degree two maps.

Findings

Semi-stable bad reduction coincides with singular reduction.

The associated Lattes map of a semi-stable elliptic curve has unstable bad reduction.

Counterexample to the dynamical analog of the discriminant-conductor bound in degree two maps.

Abstract

We study the minimal resultant divisor of self-maps of the projective line over a number field or a function field and its relation to the conductor. The guiding focus is the exploration of a dynamical analog to Theorem 1.1, which bounds the degree of the minimal discriminant of an elliptic surface in terms of the conductor. We study minimality and semi-stability, considering what conditions imply minimality (Theorem 4.4) and whether semi-stable models and presentations are minimal, proving results in the degree two case (Theorems 4.6, 4.7). We prove the singular reduction of a semi-stable presentation coincides with the bad reduction (Theorem 3.1). Given an elliptic curve over a function field with semi-stable bad reduction, we show the associated Lattes map has unstable bad reduction (Theorem 3.6). Degree 2 maps in normal form with semi-stable bad reduction are used to construct a counterexample (Theorem 2.1) to a natural dynamical analog to Theorem 1.1. Finally, we consider the notion of "critical bad reduction," and show that a dynamical analog to Theorem 1.1 may still be possible using the locus of critical bad reduction to define the conductor (Theorem 4.10, Theorem 4.13).