Geometry of Numbers on Homogeneous Spaces and Generalized Hermite Constants

齐次空间上的数几何和广义厄米常数

基本信息

批准号：
12640023
负责人：
WATANABE Takao
金额：
$ 2.18万
依托单位：
Osaka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640023/
关键词：
Hermite constant Algebraic groud Reduction theory Flag variety Tamagawa number Adels geometry 節約理論旗多様体代数多様体の有理点

项目摘要

The purpose of this project is to study the distribution of rational points or integral points on an algebraic homogeneous space defined over a global field by using the method of geometry of numbers and adelic analysis. We obtained the following results. Let K be a global field, G a connected reductive K-algebraic group, Q a maximal K-parabolic subgroup of G and X = Q＼G a flag variety defined over K Denote by X(K) the set of K・rational points of X. If G(A)' and Q(A)' denote the unimodular parts of the adele groups of G and Q, respectively, then the quotient space Q(A)' ＼G(A)' is a locally compact space and contains X(K). By nsing the sirnple root corresponding to Q, one can define a beight function H on Y. for T >o, B(T) stands for the set of elements of Y whose heights are less than or equal tu T. Then the number N(T) = IB(T) ∩X(K) I is always finite. Main results are stated as follows1. If K is an algebraic number field, then the asymptotics N(T) 〜 ω (B(T)) τ (Q) / τ (G) (T→∞) holds. Here τ (G) and τ (Q) denotes the Tamaagwa number of G and Q, respectivaly, and ω (B(T)) stands for the volume of B(T) with respect to the Tamagawa measure ω ib Y2. We define the function γ on G(A)' by γ (g) = min { H(xg) I ×∈X(K) } for element g of G(A)' and denote by γ (G,Q,K) the maximum of γ.γ (G,Q,K) is called the fundamental Hermite constant. Satisfies some functorial properties, e.g., the invariance of scalar restrictions of K and some central extensions of G. Furthermore we generalized Rankin's inequality and the Minkowski-Hlawka bound to the fundamental Hermite constant

该项目的目的是研究通过使用数字和阿德里奇分析的几何方法在全球字段定义的代数均质空间上的合理点或积分点的分布。我们获得了以下结果。 Let K be a global field, G a connected reduced K-algebraic group, Q a maximum K-parabolic subgroup of G and X = Q＼G a flag variety defined over K Denote by X(K) the set of K・rational points of X. If G(A)' and Q(A)' denote the unimodular parts of the adele groups of G and Q, respectively, then the quotation space Q(A)' \G(A)' is局部紧凑的空间，包含x（k）。通过对与Q相对应的sirnple根，可以定义y上的beight函数h。对于t> o，b（t）代表y高度小于或相等的tu t的元素集。那么数字n（t）= ib（t）= ib（t）∩x（k）i总是有限的。主要结果表示如下1。如果k是代数数字段，则渐近学n（t）〜Ω（b（t））τ（q） /τ（g）（t→∞）保持。这里τ（g）和τ（q）表示G和Q，尊重和ω（b（t））的tamaagwa数量相对于tamagawa测量ωiby2而言，B（t）的体积代表了B（t）的体积。我们将γ（g）= min {h（xg）i×∈X（k）}在g（a）上定义g（a）'上的函数γ'的元素g（a）''，并用γ（g，q，k）表示γ.γ（g，q，k）的最大γ（g，q，q，q，k）的最大值称为基础hermite Hermite Hermite Hermite Hermite Hermite常数。满足某些功能属性，例如，K的标量限制和G的某些中心扩展。