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Critically separable rational maps in families

Part of: Arithmetic algebraic geometry Arithmetic and non-Archimedean dynamical systems

Published online by Cambridge University Press:  12 October 2012

Clayton Petsche
Clayton Petsche*
Affiliation:
Department of Mathematics, Oregon State University, Corvallis OR 97331, USA (email: petschec@math.oregonstate.edu)
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Abstract

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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.

Keywords

arithmetic dynamicscritically separable rational mapscritical discriminantelliptic curvesSzpiro’s conjecture

MSC classification

Secondary: 37P15: Global ground fields 37P45: Families and moduli spaces 11G05: Elliptic curves over global fields
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 6 , November 2012 , pp. 1880 - 1896
Copyright
Copyright © Foundation Compositio Mathematica 2012

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