 

Dragos Ghioca and Niki Myrto Mavraki
Variation of the canonical height in a family of rational maps view print


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
f_{t}(z):=(z^{d}+t)/z
be a family of rational maps indexed by t. For each t=λ∈\barQ,
we let h_{fλ}(c(λ)) be the canonical height of c(λ) with respect to
the rational map f_{λ}; also we let h_{f}(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,
h_{fλ}(c(λ))h_{f}(c)⋅h(λ)≦ C.
In particular, we show that λ\mapsto h_{fλ}(c(λ)) is a Weil height on P^{1}.
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

