Let d ≥ 2 be an integer, let c ∈ ℚ(t) be a rational map, and let be a family of rational maps indexed by t. For each t = λ ∈ ℚ, we let ĥfλ (c(λ)) be the canonical height of c(λ) with respect to the rational map f λ; 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 ∈ ℚ |ĥfλ (c(λ)) - ĥf(c) · h(λ)| ≤ C. In particular, we show that λ→ ĥfλ (c(λ)) is a Weil height on ℙ1. This improves a result of Call and Silverman, 1993, for this family of rational maps.
|Original language||English (US)|
|Number of pages||35|
|Journal||New York Journal of Mathematics|
|State||Published - Nov 28 2013|
- Families of rational maps
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