二次型空间的等价性与设计
Extremality and designs in spaces of quadratic forms
摘要：
一个著名的Voronoi定理刻画了极值二次型和欧氏格，即那些对于Hermite函数是局部极大值的格，是完美的和共序的。这种characterization已扩展到各种情况下，如家庭的格，部分格，亨伯特形式等，此外，有一个标准的极端格，发现文科夫，制定方面的球形设计已扩展的情况下，格拉斯曼和部分格。在这篇文章中，我们定义了一个一般的框架，其中有一个“Voronoi特征”，并通过适当的设计概念的“文科夫准则”。这一框架包含了许多有趣的情况，其中一个“Voronoi特征”已被证明。我们还讨论了相对于爱泼斯坦zeta函数的极值问题，我们扩展到我们的框架的表征最终zeta-极值制定的Delone和Ryshkov和一个标准的设计发现Coulangeon。
英文摘要：
A well known theorem of Voronoi caracterizes extreme quadratic forms and Euclidean lattices, that is those which are local maxima for the Hermite function, as perfect and eutactic. This caracterization has been extended in various cases, such that family of lattices, sections of lattices, Humbert forms, etc. Moreover, there is a criterion for extreme lattices, discovered by Venkov, formulated in terms of spherical designs which has been extended in the case of Grassmannians and sections of lattices. In this article, we define a general frame, in which there is a ``Voronoi characterization'', and a ``Venkov criterion'' through an appropriate notion of design. This frame encompasses many interesting situations in which a ``Voronoi characterization'' has been proved. We also discuss the question of extremality relatively to the Epstein zeta function, and we extend to our frame a characterization of final zeta-extremality formulated by Delone and Ryshkov and a criterion in terms of designs found by Coulangeon.
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