People

Takao WATANABE

Email twatanabe(@math.sci.osaka-u.ac.jp)
Research
Algebraic number theory
Keywords Adele, Hermite constant, algebraic group
Office Science Building B-546 (Toyonaka Campus)
URL http://www.math.sci.osaka-u.ac.jp/~twatanabe/

My current interest is the Geometry of Numbers. The Geometry of Numbers was founded by Hermann Minkowski in the beginning of the 20th century. Minkowski proved a famous theorem known as "Minkowski's convex body theorem", which asserts that "there exists a non-zero integer point in V if V is an o-symmetric convex body in the n-dimensional Euclidean space whose volume is greater than 2^n". When V is an ellipsoid, this theorem is refined as follows. Let A be a non-singular 3 by 3 real matrix and K(c) the ellipsoid consisting of points x such that the inner product (Ax, Ax) is less than or equal to c > 0. For i = 1,2,3, we define the constant c_i as the minimum of c > 0 such that K(c) contains i linearly independent integer points. Then c_1, c_2, c_3 satisfies the inequality c_1c_2c_3 <= 2|det A|^2. This is called "Minkowski's second theorem". A similar inequality holds for any n-dimensional ellipsoid. Namely, if A is a non-singular n by n real matrix and K(c) is the n-dimensional ellipsoid defined by (Ax, Ax) <= c, we can define c_i for i = 1,2, ..., n as the minimum of c > 0 such that K(c) contains i linearly independent integer points. Then the inequality c_1c_2...c_n <= h(n)|det A|^2 holds for any A. The optimal upper bound h(n) does not depend on A, and is called Hermite's constant. We know h(2) = 4/3, h(3) = 2, h(4) = 4, ..., h(8) = 256, but h(n) for a general n is not known. A recent major topic of this research area is the determination of h(24). In 2003, Henry Cohn and Abhinav Kumar proved that h(24) = 4^24. (Incidentally, h(3) was essentially determined by Gauss in 1831, and h(8) was determined by Blichfeldt in 1953. If you would determine h(9), then your name would be recorded in treatises on the Geometry of Numbers.) Now I study (an analogue of) the Geometry of Numbers on algebraic homogeneous spaces. One of my results is a generalization of Minkowski's second theorem to a Severi-Berauer variety. In addition, I am interested in the reduction theory of arithmetic subgroups, automorphic forms, the algebraic theory of quadratic forms and Diophantine approximation.