Heights and the Specialization Map for Families of Elliptic Curves over P^n

Abstract

For $n\geq 2$, let $K=\overline{\mathbb{Q}}(\mathbb{P}^n)=\overline{\mathbb{Q}}(T_1, \ldots, T_n)$. Let $E/K$ be the elliptic curve defined by a minimal Weiestrass equation $y^2=x^3+Ax+B$, with $A,B \in \overline{\mathbb{Q}}[T_1, \ldots, T_n]$. There's a canonical height $\hat{h}_{E}$ 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 $\Gamma$ in $\mathbb{P}^n$ such that the reduction mod $\Gamma$ of $E$, $E_{\Gamma} / \overline{\mathbb{Q}}(\Gamma)$ is an elliptic curve with the zero element $O_\Gamma$, there is also a canonical height $\hat{h}_{E_{\Gamma}}$ on $E_{\Gamma}(\overline{\mathbb{Q}}(\Gamma))$ that is induced by $ (O_\Gamma)$. We prove that for any $P \in E(K)$, the equality $\hat{h}_{E_{\Gamma}}(P_\Gamma)/ \deg \Gamma =\hat{h}_{E}(P)$ holds for almost all hypersurfaces in $\mathbb{P}^n$. As a consequence, we show that for infinitely many $t \in \mathbb{P}^n(\overline{\mathbb{Q}})$, the specialization map $\sigma_t : E(K) \rightarrow E_t(\overline{\mathbb{Q}})$ is injective.