Elliptic surfaces and intersections of adelic $\mathbb{R}$-divisors

IF 2.5 1区 数学 Q1 MATHEMATICS
Laura Demarco, Niki Myrto Mavraki
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引用次数: 0

Abstract

Suppose $\mathcal{E} \to B$ is a non-isotrivial elliptic surface defined over a number field, for smooth projective curve $B$. Let $k$ denote the function field $\overline{\mathbb{Q}}(B)$ and $E$ the associated elliptic curve over $k$. In this article, we construct adelically metrized $\mathbb{R}$-divisors $\overline{D}_X$ on the base curve $B$ over a number field, for each $X \in E(k)\otimes \mathbb{R}$. We prove non-degeneracy of the Arakelov-Zhang intersection numbers $\overline{D}_X\cdot \overline{D}_Y$, as a biquadratic form on $E(k)\otimes \mathbb{R}$. As a consequence, we have the following Bogomolov-type statement for the N\'eron-Tate height functions on the fibers $E_t(\overline{\mathbb{Q}})$ of $\mathcal{E}$ over $t \in B(\overline{\mathbb{Q}})$: given points $P_1, \ldots, P_m \in E(k)$ with $m\geq 2$, there exist an infinite sequence $t_n\in B(\overline{\mathbb{Q}})$ and small-height perturbations $P_{i,t_n}' \in E_{t_n}(\overline{\mathbb{Q}})$ of specializations $P_{i,t_n}$ so that the set $\{P_{1, t_n}', \ldots, P_{m,t_n}'\}$ satisfies at least two independent linear relations for all $n$, if and only if the points $P_1, \ldots, P_m$ are linearly dependent in $E(k)$. This gives a new proof of results of Masser and Zannier and of Barroero and Capuano and extends our earlier results. In the Appendix, we prove an equidistribution theorem for adelically metrized $\mathbb{R}$-divisors on projective varieties (over a number field) using results of Moriwaki, extending the equidistribution theorem of Yuan.
-因子的椭圆曲面与交点
假设$\mathcal{E} \to B$是定义在数域上的非等平凡椭圆曲面,对于光滑射影曲线$B$。设$k$表示函数场$\overline{\mathbb{Q}}(B)$, $E$表示相关的椭圆曲线$k$。在本文中,我们在一个数字字段上的基曲线$B$上为每个$X \in E(k)\otimes \mathbb{R}$构造幂度量的$\mathbb{R}$ -除数$\overline{D}_X$。在$E(k)\otimes \mathbb{R}$上证明了Arakelov-Zhang交数$\overline{D}_X\cdot \overline{D}_Y$的双二次型的非简并性。因此,对于$\mathcal{E}$ / $t \in B(\overline{\mathbb{Q}})$的纤维$E_t(\overline{\mathbb{Q}})$上的nsamron - tate高度函数,我们有以下bogomolov型声明:给定点$P_1, \ldots, P_m \in E(k)$和$m\geq 2$,存在一个无限序列$t_n\in B(\overline{\mathbb{Q}})$和专门化$P_{i,t_n}$的小高度扰动$P_{i,t_n}' \in E_{t_n}(\overline{\mathbb{Q}})$,使得集合$\{P_{1, t_n}', \ldots, P_{m,t_n}'\}$对所有$n$满足至少两个独立的线性关系,当且仅当点$P_1, \ldots, P_m$在$E(k)$中线性相关。这对Masser和Zannier以及Barroero和Capuano的结果给出了新的证明,并推广了我们之前的结果。在附录中,我们利用Moriwaki的结果,推广了Yuan的等分布定理,证明了(在数域上)射影变异上幂度量$\mathbb{R}$ -因子的一个等分布定理。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
4.50
自引率
0.00%
发文量
103
审稿时长
6-12 weeks
期刊介绍: The Journal of the European Mathematical Society (JEMS) is the official journal of the EMS. The Society, founded in 1990, works at promoting joint scientific efforts between the many different structures that characterize European mathematics. JEMS will publish research articles in all active areas of pure and applied mathematics. These will be selected by a distinguished, international board of editors for their outstanding quality and interest, according to the highest international standards. Occasionally, substantial survey papers on topics of exceptional interest will also be published. Starting in 1999, the Journal was published by Springer-Verlag until the end of 2003. Since 2004 it is published by the EMS Publishing House. The first Editor-in-Chief of the Journal was J. Jost, succeeded by H. Brezis in 2004. The Journal of the European Mathematical Society is covered in: Mathematical Reviews (MR), Current Mathematical Publications (CMP), MathSciNet, Zentralblatt für Mathematik, Zentralblatt MATH Database, Science Citation Index (SCI), Science Citation Index Expanded (SCIE), CompuMath Citation Index (CMCI), Current Contents/Physical, Chemical & Earth Sciences (CC/PC&ES), ISI Alerting Services, Journal Citation Reports/Science Edition, Web of Science.
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