Geometry of Numbers on Homogeneous Spaces and Generalized Hermite Constants

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&gt；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。Hereτ(G)andτ(Q)denotes the Tamaagwa number of G and Q，respectivaly，andω(B(T))stands for the volume of B(T)with respect to the Tamagawa measureωib Y2。We define the functionγon G(A)‘byγ(G)=min{H(Xg)I×∈X(K)}for element g of G(A)’and denote byγ(G，Q，K)the maximum ofγ.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

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