Critically separable rational maps in families

Details Critically separable rational maps in families Critically separable rational maps in families

2011-09-14

arxiv.org/abs/1109.3226v2

Mathematics

Number Theory

In this version, some notation and terminology has changed. In particular, this results in a slight change in the title of the

Scientific paper

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.

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