A Manin-Mumford Theorem for the Maximal Compact Subgroup of a Universal Vectorial Extension of a Product of Elliptic Curves
A Manin-Mumford Theorem for the Maximal Compact Subgroup of a Universal Vectorial Extension of a Product of Elliptic Curves
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椭圆曲线乘积通用向量延展的最大紧子群的Manin-Mumford定理
DOI：
10.1093/imrn/rnz207
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发表时间：
2021
影响因子：
1
通讯作者：
Jones G
中科院分区：
文献类型：
--
作者：
Jones G
研究了代数簇与椭圆曲线乘积的泛向量扩张的极大紧子群的交。对于这个交集，我们给出一个Manin-Mumford类型的语句。这回答了一些问题所提出的Corvaja，马瑟，赞尼尔，这产生了与他们的调查交叉的代数曲线的最大紧凑子群的各种代数群。特别是他们证明，这些交叉点是有限的普遍矢量扩展的椭圆曲线。使用Khovanskii的零估计结合分层结果的Gabrielov-Vorobjov和最近的工作的作者，我们得到有效的界限，这个交叉只依赖于程度的代数簇和组的维数。作为推论，我们得到了某些交换簇的可加扩张的Manin-Mumford型的新的一致结果。
We study the intersection of an algebraic variety with the maximal compact subgroup of a universal vectorial extension of a product of elliptic curves. For this intersection we show a Manin–Mumford-type statement. This answers some questions posed by Corvaja–Masser–Zannier, which arose in connection with their investigation of the intersection of an algebraic curve with the maximal compact subgroup of various algebraic groups. In particular they proved that these intersections are finite for universal vectorial extensions of elliptic curves. Using Khovanskii’s zero-estimates combined with a stratification result of Gabrielov–Vorobjov and recent work of the authors, we obtain effective bounds for this intersection that only depend on the degree of the algebraic variety and the dimension of the group. As a corollary, we obtain new uniform results of Manin–Mumford type for additive extensions of certain abelian varieties.