Number Theory and Geometry related to Algebraic Groups

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(1) Yukie determined the density of the product of the class number and the regulator of biquadratic fields. Also he investigated other density theorems. He also obtained an upper bound for the number of quintic fields with bounded discriminant(2) Ishida worked on a generalization of the theory of complexes for rational fans to fans over real fields. In particular, he interpreted the ideal theory of commutative algebra theory in terms of fans, and worked on a construction of a process for real fans which is equivalent to blowups for algebraic varieties. He also formulated the notion of Zariski space for fans and established that it is possible to compactify real fans in the same manner as Nagata's completion of algebraic varieties(3) Among CM elliptic curves, Nakamura classified Q-curves which have nice properties with respect to the Galois group action and determined the structure of the Abelian varieties. obtained by such Q-curves. He also investigated the construction of singular Ab … More elian surfaces over the field of rational numbers(4) Ogata investigated defining equations of projective toric varieties and obtained an estimate of the number of tensors of an ample line bundle which give projectively normal imbeddings. He also obtained an estimate of the number of degrees of generators of the defining ideal and determined varieties which give rise to generators of the defining ideals of the highest degree(5) Hara introduced the notions in singularity theory in positive characteristic ring theoretically which should correspond to singularities in birational geometry and multiplier ideals in characteristic zero using the notion of Frobenius map and the tight closure. He also showed the relation between them and tried to find applications to algebraic geometry in positive characteristic(6) Sato investigated the distributions of the ranks of the Mordell-Well groups of quadratic twists of elliptic curves defined over number fields and related problems(7) Hasegawa investigated algebraic aspects of discrete integrable systems from the viewpoint of symmetry. In particular, he investigated quantization of discrete Painleve equations Less