论文标题
$ \ mathbb r $ $ divisors和shioda-tate公式的数值等效性用于算术品种
Numerical equivalence of $\mathbb R$-divisors and Shioda-Tate formula for arithmetic varieties
论文作者
论文摘要
让$ x $成为数字字段$ k $的整数环上的算术变化，并带有光滑的通用光纤$ x_k $。我们提供了一个公式，该公式将$ x $的第一个Arakelov-Chow矢量空间与Albanese品种的Mordell-Weil等级和$ x_k $的Mordell-Weil等级和Néron-Severi $ x_k $的Néron-Severi Group的等级相关联。这是用于椭圆表面的经典shioda-tate公式的较高维度和算术版本。我们证明，在$ x $上的数字算术$ \ mathbb {r} $ - $ x $上的分配正是主要主体的线性组合，这一事实得到了加强。该结果等同于算术相交配对的非分类性，部分证实了[GS94，猜想1]。
Let $X$ be an arithmetic variety over the ring of integers of a number field $K$, with smooth generic fiber $X_K$. We give a formula that relates the dimension of the first Arakelov-Chow vector space of $X$ with the Mordell-Weil rank of the Albanese variety of $X_K$ and the rank of the Néron-Severi group of $X_K$. This is a higher dimensional and arithmetic version of the classical Shioda-Tate formula for elliptic surfaces. Such analogy is strengthened by the fact that we show that the numerically trivial arithmetic $\mathbb{R}$-divisors on $X$ are exactly the linear combinations of principal ones. This result is equivalent to the non-degeneracy of the arithmetic intersection pairing in the argument of divisors, partially confirming [GS94, Conjecture 1].
