Variation of the Canonical Height in a Family of Rational Maps

Let d ≥ 2 be an integer, let c ∈ Q̄(t) be a rational map, and let ft(z) := z+t z be a family of rational maps indexed by t. For each t = λ ∈ Q̄, we let ĥfλ (c(λ)) be the canonical height of c(λ) with respect to the rational map fλ; also we let ĥf (c) be the canonical height of c on the generic fiber of the above family of rational maps. We prove that there exists a constant C depending only on c such that for each λ ∈ Q̄, ∣∣∣ĥfλ (c(λ))− ĥf (c) · h(λ)∣∣∣ ≤ C. In particular, we show that λ 7→ ĥfλ (c(λ)) is a Weil height on P 1. This improves a result of Call and Silverman [4] for this family of rational maps.