由CM数域产生的格中最短向量范数的一个界
A Bound on the Norm of Shortest Vectors in Lattices Arising from CM Number Fields
摘要：
本文部分地讨论了由任何CM数域产生的欧氏格族中向量的长度的特征问题。我们在这些格上定义了一个修正的二次型，即加权范数，它包含标准场迹作为特例。使用这种修改后的二次型，我们得到了一个绑定的域范数的任何向量，有一个最小的长度在任何这些格，在一个基础的单位的环的整数的领域。对任意CM数域F，我们证明了存在F的有限元素集，使得可以在F的整数环的每个主理想中找到极小向量的集合.我们解释我们的结果的Hilbert模形式的渐近行为，并考虑我们的定理的计算影响。此外，我们展示了如何我们的结果可以应用到特定的克雷格的差格问题，这要求我们找到的极小向量格所产生的分圆数域。
英文摘要：
This paper partially addresses the problem of characterizing the lengths of vectors in a family of Euclidean lattices that arise from any CM number field. We define a modified quadratic form on these lattices, the weighted norm, that contains the standard field trace as a special case. Using this modified quadratic form, we obtain a bound on the field norm of any vector that has a minimal length in any of these lattices, in terms of a basis for the group of units of the ring of integers of the field. For any CM number field F, we prove that there exists a finite set of elements of F which allows one to find the set of minimal vectors in every principal ideal of the ring of integers of F. We interpret our result in terms of the asymptotic behavior of a Hilbert modular form, and consider some of the computational implications of our theorem. Additionally, we show how our result can be applied to the specific Craig's Difference Lattice problem, which asks us to find the minimal vectors in lattices arising from cyclotomic number fields.
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