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

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