Critically separable rational maps in families

TL;DR

This paper studies families of critically separable rational maps over number fields, establishing finiteness results and proposing a conjecture relating their arithmetic complexity to a global invariant, drawing parallels with elliptic curve theory.

Contribution

It introduces a finiteness theorem for critically separable rational maps with fixed structures and proposes a conjectural bound on their minimal critical discriminant, linking to Szpiro's conjecture.

Findings

Proved a finiteness theorem analogous to Shafarevich's theorem for these rational maps.

Defined the minimal critical discriminant as a measure of arithmetic complexity.

Showed that a special case of the conjecture implies Szpiro's conjecture in the semistable case.

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.