| Budget Amount *help |
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2005: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2004: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2003: ¥1,400,000 (Direct Cost: ¥1,400,000)
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| Research Abstract |
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).
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