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Voronoi Theory on Flag Varieties

Voronoi Theory on Flag Varieties
旗帜品种的 Voronoi 理论
批准号:
15540026
负责人:
WATANABE Takao
金额:
$2.3万
依托单位:
依托单位国家:
日本
项目类别:
Grant-in-Aid for Scientific Research (C)
财政年份:
2003
资助国家:
日本
项目状态:
已结题
起止时间:
2003 至 2005

项目摘要

项目成果

WATANABE Takao的其他基金

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中文摘要
翻译
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.
英文摘要
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).
期刊论文(40)
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会议论文
On the best bound of the minimal twisted height of linear subspaces
关于线性子空间最小扭曲高度的最佳界
DOI: --
发表时间:
期刊: Archiv der Mathematik (in press)
影响因子: --
作者: [Jonathan Beck, Hiraku Nakajima, T.Watanabe]
通讯作者: T.Watanabe
The normalization constants of a certain invariant measure On GL_n(D_A)
GL_n(D_A) 上某个不变测度的归一化常数
DOI: --
发表时间: 2004
期刊: Manuscripta Math. 115
影响因子: --
作者: [Hiraku Nakajima, Kota Yoshioka, Y.Nakamura]
通讯作者: Y.Nakamura
A survey and a complement of fundamental Hermite constants
基本埃尔米特常数的调查和补充
DOI: --
发表时间: 2004
期刊: Contemporary Math.Amer.Math.Soc. 344
影响因子: --
作者: [T.Watanabe, T.Watanabe]
通讯作者: T.Watanabe
Minkowski's second theorem over a simple algebra
关于简单代数的闵可夫斯基第二定理
DOI: --
发表时间:
期刊: Monatshefte fur Mathematik (in press)
影响因子: --
作者: [Hiraku Nakajima, Kota Yoshioka, T.Watanabe]
通讯作者: T.Watanabe
11
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    • 财政年份:
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    • 批准号:
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    • 项目类别:
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    • 资助金额:
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