2014/09/10 by Wei Pin Wong, Wong, Wei Pin
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.AG
paper · pdf · doi:10.48550/arxiv.1409.3255
updated version
openalex publication_date 2014/09/10 · arxiv created 2015/06/22 · arxiv updated 2015/06/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For n≥ 2, let K=ℚ(ℙn)=ℚ(T1, …, Tn). Let E/K be the elliptic curve defined by a minimal Weiestrass equation y2=x3+Ax+B, with A,B ∈ ℚ[T1, …, Tn]. There's a canonical height hE 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 Γ in ℙn such that the reduction mod Γ of E, EΓ / ℚ(Γ) is an elliptic curve with the zero element OΓ, there is also a canonical height hEΓ on EΓ(ℚ(Γ)) that is induced by (OΓ). We prove that for any P ∈ E(K), the equality hEΓ(PΓ)/ °Γ=hE(P) holds for almost all hypersurfaces in ℙn. As a consequence, we show that for infinitely many t ∈ ℙn(ℚ), the specialization map σt : E(K) → Et(ℚ) is injective.