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Dragos Ghioca and Niki Myrto Mavraki
Variation of the canonical height in a family of rational maps view print
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| Published: |
November 22, 2013
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| Keywords: |
Heights, families of rational maps |
| Subject: |
Primary 11G50; Secondary 14G17, 11G10 |
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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.
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| Acknowledgements
The research of the first author was partially supported by an NSERC grant. The second author was partially supported by Onassis Foundation.
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| 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
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