辛空间中关于高度的有效结构定理
Effective structure theorems for symplectic spaces via height
摘要：
给定整体域$K$上的$2k$维含$N$变量的辛空间$(Z,F)$，$1 < 2k \leq N$，我们证明了有界高度的$(Z,F)$的辛基的存在性.这可以看作是辛空间的西格尔引理的一个版本。作为我们主要结果的推论，我们证明了$(Z,F)$到双曲平面的小高度分解的存在性，以及全迷向子空间的两个生成标志的存在性。这些类似物的已知结果二次空间。我们的论点的一个显著特点是，它同时适用于基本上任何领域的产品公式，代数封闭或不。事实上，我们证明了这些陈述的一个更一般的版本，其中规范高度被扭曲高度取代。所有的高度界限都是明确的。
英文摘要：
Given a $2k$-dimensional symplectic space $(Z,F)$ in $N$ variables, $1 < 2k \leq N$, over a global field $K$, we prove the existence of a symplectic basis for $(Z,F)$ of bounded height. This can be viewed as a version of Siegel's lemma for a symplectic space. As corollaries of our main result, we prove the existence of a small-height decomposition of $(Z,F)$ into hyperbolic planes, as well as the existence of two generating flags of totally isotropic subspaces. These present analogues of known results for quadratic spaces. A distinctive feature of our argument is that it works simultaneously for essentially any field with a product formula, algebraically closed or not. In fact, we prove an even more general version of these statements, where canonical height is replaced with twisted height. All bounds on height are explicit.
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