Critically separable rational maps in families

Abstract

Given a number field K, we consider families of critically separable rational maps of degree d over K possessing a certain fixed-point and multiplier structure. With suitable notions of isomorphism and good reduction between rational maps in these families, we prove a finiteness theorem which is analogous to Shafarevich's theorem for elliptic curves. We also define the minimal critical discriminant, a global object which can be viewed as a measure of arithmetic complexity of a rational map. We formulate a conjectural bound on the minimal critical discriminant, which is analogous to Szpiro's conjecture for elliptic curves, and we prove that a special case of our conjecture implies Szpiro's conjecture in the semistable case.

In this version, some notation and terminology has changed. In particular, this results in a slight change in the title of the paper. Many small expository changes have been made, a reference has been added, and a remark/example has been added to the end of section 3

References in corpus (2)

• Small rational points on elliptic curves over number fields
• Resultant and conductor of geometrically semi-stable self maps of the projective line over a number field or function field

Cited by in corpus (4)

• Good reduction and Shafarevich-type theorems for dynamical systems with portrait level structures
• Resultant and conductor of geometrically semi-stable self maps of the projective line over a number field or function field
• A dynamical Shafarevich theorem for rational maps over number fields and function fields
• A dynamical Shafarevich theorem for endomorphisms of ${\mathbb{P}}^N$