New York Journal of Mathematics
Volume 19 (2013) 873-907

  

Dragos Ghioca and Niki Myrto Mavraki

Variation of the canonical height in a family of rational maps

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Published: November 22, 2013
Keywords: Heights, families of rational maps
Subject: Primary 11G50; Secondary 14G17, 11G10

Abstract
Let d ≧ 2 be an integer, let c ∈ \barQ(t) be a rational map, and let
ft(z):=(zd+t)/z
be a family of rational maps indexed by t. For each t=λ∈\barQ, 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∈\barQ,
|hfλ(c(λ))-hf(c)⋅h(λ)|≦ C.
In particular, we show that λ\mapsto hfλ(c(λ)) is a Weil height on P1. This improves a result of Call and Silverman, 1993, for this family of rational maps.

Acknowledgements

The research of the first author was partially supported by an NSERC grant. The second author was partially supported by Onassis Foundation.


Author information

Dragos Ghioca:
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
dghioca@math.ubc.ca

Niki Myrto Mavraki:
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
myrtomav@math.ubc.ca