Voronoi Theory on Flag Varieties

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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).