论文标题
椭圆形表面和adelic $ \ mathbb {r} $ - 除数
Elliptic surfaces and intersections of adelic $\mathbb{R}$-divisors
论文作者
论文摘要
假设$ \ MATHCAL {E} \ to B $是一个在数字字段上定义的非异常椭圆表面，对于光滑的投影曲线$ b $。令$ k $表示函数字段$ \ overline {\ mathbb {q}}}}（b）$和$ e $ $ k $上的关联椭圆曲线。在本文中，我们在基本曲线$ b $上构造了$ \ mathbb {r} $ - 在e（k）\ otimes \ otimes \ mathbb {r} $中的每个$ x \ in number five上的$ \ overline {d} _x $。我们证明Arakelov-Zhang交叉数字$ \ OVERLINE {D} _X \ CDOT \ CDOT \ OVERLINE {D} _Y $是$ E（K）\ outimes \ otimes \ Mathbb {r} $上的biquadratic形式。结果，我们对纤维上的néron-tate高度发挥了以下bogomolov-type语句，$ e_t（\ overline {\ mathbb {q}} $ of $ \ mathcal {e} $ of $ \ mathcal {e} $ over $ t \ in b（in b） \ in E（k）$带有$ m \ geq 2 $，在b（\ overline {\ mathbb {q}}}）中存在一个无限序列$ t_n \ in e e_ {t_n} {$ oferline {q of eferline {i，t_n}'\ in eforce { $ p_ {i，t_n} $，以便设置$ \ {p_ {1，t_n}'，\ ldots，p_ {m，t_n}'\} $至少满足所有$ n $的至少两个独立的线性关系，并且仅当点$ p_1，\ ldots，\ ldots，p_m $ linear $ event $ event $ e（k）这给了Masser和Zannier以及Barroero和Capuano的新结果，并扩展了我们的早期结果。在附录中，我们使用Moriwaki的结果证明了用于射影品种（在一个数字字段）上的均值$ \ Mathbb {r} $的等均分布定理，扩展了Yuan的等均分布定理。
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 Néron-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.
