Voronoi Theory on Flag Varieties

本研究的目的是调查的Hermite常数附加到一个标志品种定义在一个全球性的领域，并开发一个类似的Voronoi理论在这个广义Hermite常数。1首先，我们研究了广义Hermite常数的Severi-Brauer品种。我们用K表示一个整体域，用V表示一个K-中心除代数D上的n维向量空间。然后从V定义一个Severi-Brauer簇X作为Grassman簇的扭曲形式。利用D上矩阵代数上的约化范数，我们可以定义V上的高度H和X的Hermite常数y（X）。然后我们得到了以下结果。(1)A与中村合作，利用D的zeta函数的某些特殊值给出了y（X）的下界. (2)In在D是四元数代数的情况下，利用数几何的一个引数给出了y（X）的一个上界。此外，在K是数域的情况下，我们引入了V上的四元数Humbert型的概念，并证明了关于y（X）的Voronoi型定理（与Coulangeon合作）。(3)We证明了Minkowski第二定理的一个类似物对V2上的高度H成立。其次，我们用线性子空间的最小扭曲高度刻画了广义Hermite-Rankin常数。设Gr（N，n ; K）是N维K向量空间中由n维子空间组成的Grassman簇.如果X是一个n维子空间，则Gr（X，m）表示X中m维子空间的Grassman簇。对于adele群GL_N（A）的每个元素g，我们有Gr（N，n ; K）上的扭曲高度H_g。定义GL_N（A）上的函数Γ为：Γ（g）=sup_Xinf_Y（H_g（Y）H_g（X）^（-m/n）），其中X在整个Gr（N，n ; K）上运行，Y在整个Gr（X，m）上运行.然后证明了sup_g（Γ（g）^2）=y（Gr（n，m ; K）），其中y（Gr（n，m ; K））表示Gr（n，m ; K）的广义Hermite常数.

The purpose of this research is to investigate the Hermite constant attached to a flag variety defined over a global field and develop an analog of Voronoi theory on this generalized Hermite constant.1 First, we studied the genreralized Hermite constant of a Severi-Brauer variety. We denote by K a global field and by V an n dimensional vector space over a K-central division algebra D. Then a Severi-Brauer variety X is defined from V as a twisted form of a Grassman variety. By making use the reduced norm on a matrix algebra over D, we can define the height H on V and the Hermite constant y(X) of X. Then we obtained the following results.(1)A lower bound of y(X) was given in terms of some special values of the zeta function of D (in collaboration with Nakamura).(2)In the case that D is a quaternion algebra, we gave an upper bound of y(X) by using an argument of the geometry of numbers. Furthermore, in the case of K a number field, we introduced the notion of quaternionic Humbert forms on V and proved the Voronoi type theorem with respect to y(X) (in collaboration with Coulangeon).(3)We showed that an analog of the Minkowski's second theorem holds for the height H on V.2 Second, we characterized the generalized Hermite-Rankin constant in terms of the minimal twisted height of linear subspaces. Let Gr(N, n ; K) be the Grassman variety consisting of n dimensional subspaces in an N dimensional K vector space. If X is an n dimensional subspace, then Gr(X, m) stands for the Grassman variety of m dimensional subspaces in X. For each element g of the adele group GL_N(A), we have the twisted height H_g on Gr(N, n ; K). Then the function Γ on GL_N(A) is defined to be Γ(g)=sup_X inf_Y (H_g(Y)H_g(X)^(-m/n)), where X runs over the whole Gr(N, n ; K) and Y runs over the whole Gr(X, m). Then we proved sup_g(Γ(g)^2) =y(Gr(n, m ; K)), where y(Gr(n, m ; K)) denotes the generalized Hermite constant of Gr(n, m ; K).