Let d ≧ 2 be an integer, let c ∈ arQ(t) be a rational map, and let ft(z):=(zd+t)/zbe a family of rational maps indexed by t. For each t=λ∈arQ, we let hfλ(c(λ)) be the canonical height of c(λ) with respect to the rational map fλ; also we let hf(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∈arQ, |hfλ(c(λ))-hf(c)⋅h(λ)|≦ C. In particular, we show that λmapsto hfλ(c(λ)) is a Weil height on P1. This improves a result of Call and Silverman, 1993, for this family of rational maps.
展开▼
