Let $\pi : E\to B$ be an elliptic surface defined over a number field $K$, where $B$ is a smooth projective curve, and let $P: B \to E$ be a section defined over $K$ with canonical height $\hat{h}_E(P)\not=0$. In this article, we show that the function $t \mapsto \hat{h}_{E_t}(P_t)$ on $B(\overline{K})$ is the height induced from an adelically metrized line bundle with non-negative curvature on $B$. Applying theorems of Thuillier and Yuan, we obtain the equidistribution of points $t \in B(\overline{K})$ where $P_t$ is torsion, and we give an explicit description of the limiting distribution on $B(\mathbb{C})$. Finally, combined with results of Masser and Zannier, we show there is a positive lower bound on the height $\hat{h}_{A_t}(P_t)$, after excluding finitely many points $t \in B$, for any "non-special" section $P$ of a family of abelian varieties $A \to B$ that split as a product of elliptic curves.

中文翻译：

---

标准高度和等分布的变化

令 $\pi : E\to B$ 是一个定义在数域 $K$ 上的椭圆曲面，其中 $B$ 是一条平滑的投影曲线，让 $P: B \to E$ 是一个定义在 $K 上的截面$ 具有规范高度 $\hat{h}_E(P)\not=0$。在本文中，我们表明 $B(\overline{K})$ 上的函数 $t \mapsto \hat{h}_{E_t}(P_t)$ 是从具有非$B$ 上的负曲率。应用Thuillier 和Yuan 的定理，我们得到点$t \in B(\overline{K})$ 的等分布，其中$P_t$ 是扭转，我们给出了$B(\mathbb{ C})$。最后，结合 Masser 和 Zannier 的结果，我们表明在排除有限多个点 $t \in B$ 后，高度 $\hat{h}_{A_t}(P_t)$ 存在正下界，对于任何“非特殊”