Density measure of rational points on Abelian varieties
Density measure of rational points on Abelian varieties
复制标题
阿贝尔簇有理点的密度测度
DOI：
10.1017/s002776300000698x
复制
发表时间：
1999
影响因子：
0.8
通讯作者：
M. Waldschmidt
中科院分区：
文献类型：
--
作者：
M. Waldschmidt
摘要：设\(A\)是\(\mathbb{Q}\)上维度为\(g\)的一个单阿贝尔簇，设\(\ell\)是莫德尔 - 韦伊群\(A(\mathbb{Q})\)的秩。假定\(\ell\geq1\)。马祖尔的一个猜想断言，对于实拓扑，\(A(\mathbb{Q})\)在\(A(\mathbb{R})\)中的闭包包含原点的中性分支\(A(\mathbb{R})_0\)。只有在额外假设\(\ell\geq g^2 - g + 1\)的情况下这才是已知的。我们在此研究这个问题的一个定量细化：对于每个给定的正数\(h\)，\(A(\mathbb{Q})\)中内龙 - 泰特高度\(\leq h\)的点集是有限的，并且我们研究这些点如何分布在连通分支\(A(\mathbb{R})_0\)中。更一般地，我们考虑一个数域\(K\)（嵌入在\(\mathbb{R}\)中）上的阿贝尔簇\(A\)，以及\(A(K)\)中秩足够大的一个子群\(\Gamma\)。我们得到的密度有效结果依赖于一个丢番图逼近估计，即涉及阿贝尔对数的行列式的线性组合的一个下界。
Abstract Let be a simple Abelian variety of dimension g over ℚ, and let ℓ be the rank of the Mordell-Weil group (ℚ). Assume ℓ ≥ 1. A conjecture of Mazur asserts that the closure of (ℚ) into (ℝ) for the real topology contains the neutral component (ℝ)0 of the origin. This is known only under the extra hypothesis ℓ ≥ g2 - g + 1. We investigate here a quantitative refinement of this question: for each given positive h, the set of points in (ℚ) of Néron-Tate height ≤ h is finite, and we study how these points are distributed into the connected component (ℝ)0. More generally we consider an Abelian variety A over a number field K embedded in ℝ, and a subgroup Γ of (K) of sufficiently large rank. The effective result of density we obtain relies on an estimate of Diophantine approximation, namely a lower bound for linear combinations of determinants involving Abelian logarithms.