Geometry of Numbers on Homogeneous Spaces and Generalized Hermite Constants

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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