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

By Matthew Baker. 2006

97 citations recorded by the source corpus.

Abstract Let K be a function field, let φ ∈ K(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.

Subfields: Algebra and Number Theory, Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/222a3fb8db5bdebf4880666163db6880fa918fd0

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Citations in the corpus

Primitive divisors in arithmetic dynamics (2007, background): “Tom Tucker has pointed out to the authors that the results of this paper should be valid also for iteration of non-split functions over C(T ), and more generally over one-dimensional function fields of characteristic 0, since in this setting Benedetto [4] (for polynomial maps) and Baker [1] (for rat”
Potential Theory and Dynamics on the Berkovich Projective Line (2010, methodology): “These results, along with the energy minimization principle mentioned earlier, play a key role in applications of the theory to arithmetic dynamics over global fields (see [4] and [7]).”
Difference fields and descent in algebraic dynamics. I (2007, background): “…2 00 8 Difference fields and descent in algebraic dynamics, II Zoé Chatzidakis∗(CNRS - Université Paris 7) Ehud Hrushovski†(The Hebrew University) Introduction This second part of the paper strengthens the descent theory described in the first part [4] to rational maps and arbitrary base fields.”
Bifurcations, intersections, and heights (2014, methodology): “In this setting, one can say more: the set of points in P1(k) with height < b that lift to rational points in E(k) is Bifurcations, intersections, and heights 1033 finite for all b > 0; see [Baker 2009, Theorem B.9], where this is deduced from the conclusion of Theorem 1.2.”
Dynamical Degrees, Arithmetic Degrees, and Canonical Heights for Dominant Rational Self-Maps of Projective Space (2011, background): “The situation over function fields is subtler; see for example [7, 13].”
Heights and measures on analytic spaces. A survey of recent results, and some remarks (2010, background): “A theorem of Baker [4] shows that the converse is true: if φ is not isoconstant, then a point of height zero is then preperiodic; the proof relies on a detailed analysis of a Green function relative to the diagonal on P1 ×P1.”
Preperiodic points and unlikely intersections (2009, background): “The following result is also proved in [Bak09]:”
ABC implies primitive prime divisors in arithmetic dynamics (2012, background): “Baker’s result says that if K is a function field and ϕ ∈ K(x) is a non-isotrivial map with degϕ > 1, then α ∈ K is wandering if and only if hϕ(α) = 0.”