Heights and the specialization map for families of elliptic curves over $\mathbb {P}^n$

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 Weierstrass 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.