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

TL;DR

This paper investigates the relationship between heights of points on families of elliptic curves over projective space and their specializations, establishing a proportionality for almost all hypersurfaces and implications for injectivity of specialization maps.

Contribution

It proves a proportionality relation between canonical heights on the generic fiber and special fibers for almost all hypersurfaces, and shows the injectivity of specialization maps for infinitely many points.

Findings

Height equality holds for almost all hypersurfaces.

Specialization maps are injective for infinitely many points.

Results connect heights on generic and specialized elliptic curves.

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.