The intersection of a curve with a union of translated codimension-two subgroups in a power of an elliptic curve
The intersection of a curve with a union of translated codimension-two subgroups in a power of an elliptic curve
复制标题
曲线与平移余维数（椭圆曲线幂中的两个子群）的并集的交点
DOI：
--
复制
发表时间：
2008
期刊：
影响因子：
--
通讯作者：
Evelina Viada
中科院分区：
文献类型：
--
作者：
Evelina Viada
设E是椭圆曲线。嵌入E的幂A中的不可约代数曲线C如果不包含在A的任何真代数子群中，则称为弱横截曲线;如果不包含在这样的子群的任何平移中，则称为横截曲线。假设E和C定义在代数数上。首先，我们证明了一个横截曲线C的代数点，接近于所有的余维数2的代数子群的Eg的并通过在一个子群G的有限秩A中的点平移是一个有界的高度集。接近度的概念使用高度函数定义。如果G是平凡的，则假设C是弱横截的就足够了。文章的核心是介绍一种确定这些集合的有限性的方法。从一条横曲线C的归一化高度的一个几何下界，我们推出上述集合是有限的。当g ≤ 3时，存在这样的下界。关于代数子群的余维数，我们的结果是最好的。
Let E be an elliptic curve. An irreducible algebraic curve C embedded in a power A of E is called weak-transverse if it is not contained in any proper algebraic subgroup of A, and transverse if it is not contained in any translate of such a subgroup. Suppose E and C are defined over the algebraic numbers. First we prove that the algebraic points of a transverse curve C that are close to the union of all algebraic subgroups of E g of codimension 2 translated by points in a subgroup G of A of finite rank are a set of bounded height. The notion of closeness is defined using a height function. If G is trivial, it is sufficient to suppose that C is weak-transverse. The core of the article is the introduction of a method to determine the finite- ness of these sets. From a conjectural lower bound for the normalized height of a transverse curve C , we deduce that the sets above are finite. Such a lower bound exists for g ≤ 3. Concerning the codimension of the algebraic subgroups, our results are best possible.