-因子的椭圆曲面与交点

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS
Laura Demarco, Niki Myrto Mavraki
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引用次数: 0

摘要

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。
Elliptic surfaces and intersections of adelic $\mathbb{R}$-divisors
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.
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来源期刊
ACS Applied Bio Materials
ACS Applied Bio Materials Chemistry-Chemistry (all)
CiteScore
9.40
自引率
2.10%
发文量
464
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