For nβ₯2, let
K=Qβ(Pn)=Qβ(T1β,β¦,Tnβ). Let E/K be the elliptic curve defined by a minimal Weiestrass equation
y2=x3+Ax+B, with A,BβQβ[T1β,β¦,Tnβ]. There's
a canonical height h^Eβ on E(K) induced by the divisor (O), where
O is the zero element of E(K). On the other hand, for each smooth
hypersurface Ξ in Pn such that the reduction mod Ξ of
E, EΞβ/Qβ(Ξ) is an elliptic curve with the
zero element OΞβ, there is also a canonical height
h^EΞββ on EΞβ(Qβ(Ξ)) that is
induced by (OΞβ). We prove that for any PβE(K), the equality
h^EΞββ(PΞβ)/degΞ=h^Eβ(P) holds for almost
all hypersurfaces in Pn. As a consequence, we show that for
infinitely many tβPn(Qβ), the specialization
map Οtβ:E(K)βEtβ(Qβ) is injective.Comment: updated versio