Variation of the canonical height in a family of rational maps

Dragos Ghioca, Niki Myrto Mavraki

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

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 ĥ (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 |ĥ (c(λ)) - ĥf(c) · h(λ)| ≤ C. In particular, we show that λ→ ĥ (c(λ)) is a Weil height on ℙ1. This improves a result of Call and Silverman, 1993, for this family of rational maps.

Original languageEnglish (US)
Pages (from-to)873-907
Number of pages35
JournalNew York Journal of Mathematics
Volume19
StatePublished - Nov 28 2013

Keywords

  • Families of rational maps
  • Heights

ASJC Scopus subject areas

  • Mathematics(all)

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