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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Cuntz, Joachim / Huybrechts, Daniel / Hwang, Jun-Muk

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A finiteness theorem for canonical heights attached to rational maps over function fields

Matthew Baker
  • School of Mathematics, Georgia Institute of Technology, Atlanta GA 30332-0160, USA. e-mail:
Published Online: 2009-01-08 | DOI: https://doi.org/10.1515/CRELLE.2009.008


Let K be a function field, let φK(T) be a rational map of degree d ≧ 2 defined over K, and suppose that φ is not isotrivial. In this paper, we show that a point P ∈ ℙ1 () has φ-canonical height zero if and only if P is preperiodic for φ. This answers affirmatively a question of Szpiro and Tucker, and generalizes a recent result of Benedetto from polynomials to rational functions. We actually prove the following stronger result, which is a variant of the Northcott finiteness principle: there exists ε > 0 such that the set of points P ∈ ℙ1 (K) with φ-canonical height at most ε is finite. Our proof is essentially analytic, making use of potential theory on Berkovich spaces to study the dynamical Green's functions g φ, υ (x, y) attached to φ at each place υ of K. For example, we show that every conjugate of φ has bad reduction at υ if and only if g φ, υ (x, x) > 0 for all , where denotes the Berkovich projective line over the completion of υ. In an appendix, we use a similar method to give a new proof of the Mordell-Weil theorem for elliptic curves over K.

Received: 2006-05-31

Revised: 2007-10-22

Published Online: 2009-01-08

Published in Print: 2009-01-01

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 2009, Issue 626, Pages 205–233, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/CRELLE.2009.008, January 2009

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