Heights and Preperiodic Points of Polynomials over Function Fields

Let K be a function field in one variable over an arbitrary field F. Given a rational function φ ∈ K(z) of degree at least two, the associated canonical height on the projective line was defined by Call and Silverman. The preperiodic points of φ all have canonical height zero; conversely, if F is a finite field, then every point of canonical height zero is preperiodic. However, if F is an infinite field, then there may be non-preperiodic points of canonical height zero. In this paper, we show that for polynomial φ, such points exist only if φ is isotrivial. In fact, such K-rational points exist only if φ is defined over the constant field of K after a K-rational change of coordinates. Let K be a field with algebraic closure K̂, and let φ : P(K̂) → P(K̂) be a morphism defined over K. We may write φ as a rational function φ ∈ K(z). Denote the n iterate of φ under composition by φ. That is, φ is the identity function, and for n ≥ 1, φ = φ◦φ. A point x ∈ P(K̂) is said to be preperiodic under φ if there are integers n > m ≥ 0 such that φ(x) = φ(x). Note that x is preperiodic if and only if its forward orbit {φ(x) : n ≥ 0} is finite. If K is a number field or a function field in one variable, and if deg φ ≥ 2, there is a canonical height function ĥφ : P (K̂) → R associated to φ. (In this context, the degree deg φ of φ is the maximum of the the degrees of its numerator and denominator.) The canonical height gives a rough measure of the arithmetic complexity of a given point, and it also satisfies the functional equation ĥφ(φ(x)) = d · ĥφ(x), where d = deg φ. Canonical heights will be discussed further in Section 3; for more details, we refer the reader to the original paper [6] of Call and Silverman, or to the exposition in Part B of [9]. All preperiodic points of φ clearly have canonical height zero. Conversely, if K is a global field (i.e., a number field or a function field in one variable over a finite field), then ĥφ(x) = 0 if and only if x is preperiodic. This equivalence is very useful in the study of rational preperiodic points over such fields; see, for example, [4, 5]. However, if K is a function field over an infinite field, there may be points of canonical height zero which are not preperiodic. For example, if K = Q(T ) and φ(z) = z, then the preperiodic points in P(K̂) are 0, ∞, and the roots of unity in Q̂; however, all points in P(Q̂) have canonical height zero. Similarly, for the same field K, consider the function ψ(z) = Tz. In this case, the only K-rational points of canonical height zero are 0 and ∞, both of which are preperiodic. Nevertheless, in P(K̂), any point of the form aT with Date: October 20, 2005; revised December 12, 2005. 2000 Mathematics Subject Classification. Primary: 11G50. Secondary: 11D45, 37F10.