Variation of canonical height and equidistribution
Variation of canonical height and equidistribution
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DOI：
10.1353/ajm.2020.0012
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发表时间：
2017-01
影响因子：
1.7
通讯作者：
Laura Demarco;Niki Myrto Mavraki
中科院分区：
文献类型：
--
作者：
Laura Demarco;Niki Myrto Mavraki
设$\pi：E\to B$是数域$K$上定义的椭圆曲面，其中$B$是光滑的射影曲线，$P：B\to E$是定义在$K$上的截面，其标准高度为$\hat{h}_E（P）\not=0$.本文证明了$B（\overline{K}）$上的函数$t\mapsto\hat{h}_{E_t}（P_t）$是由$B$上具有连续半正度量的远度量化线丛导出的高度。证明建立在Silverman的工作和复杂的动力系统的结果。应用算术等分布定理（Chambert-Loir，Thuillier，and Yuan），我们得到了点$t\在B（\overline{K}）$中的等分布，其中$P_t$是挠率，并给出了$B（{\Bbb C}）$上极限分布的一个明确描述.最后，结合Masser和Zannier的结果，我们证明了：对于一族阿贝尔簇$A\to B$的任意非特殊截面$P$，在排除B$中的100个点$t\后，其高度$\hat{h}_{A_t}（P_t）$存在正的下界，从而解决了Zhang在1998年提出的一个猜想.
abstract: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 continuous semipositive metrics on $B$. The proof builds on workof Silverman and results from complex dynamical systems. Applying arithmetic equidistribution theorems (of Chambert-Loir, 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$B({\Bbb C})$. Finally, combined with results of Masser and Zannier, we show that---for any non-special section $P$ of a family of abelian varieties$A\to B$ that split as a product of elliptic curves---there is a positive lower bound on the height $\hat{h}_{A_t}(P_t)$, after excluding finitely many points $t\in B$, thus addressing a conjecture of Zhang from 1998.