Number Theory and Geometry related to Algebraic Groups

(1)Yukie确定了双二次域的类数与调节子的乘积的密度。他还研究了其他密度定理。他还得到了具有有界判别式的五次域的个数的一个上界。(2)石田致力于将有理扇复理论推广到实域上的扇复理论。特别是，他用扇的形式解释了交换代数理论的理想理论，并致力于构造一个实扇的过程，它等价于代数簇的爆破。他还提出了粉丝的Zariski空间的概念，并证明了用与Nagata完成代数簇相同的方式来紧致实粉丝是可能的。(3)在CM椭圆曲线中，Nakamura对关于Galois群作用具有良好性质的Q-曲线进行了分类，并确定了Abel簇的结构。由这样的Q曲线得到。他还研究了奇异AB…的构造有理数域上更多的Elian曲面(4)Ogata研究了射影环簇的定义方程，得到了给出射影正规嵌入的充分线丛的张量个数的估计。(5)Hara在理论上引入了正特征环中奇点理论中的概念，利用Frobenius映射和紧闭包的概念，它应该对应于二次几何中的奇点和特征零点中的乘子理想。他还展示了它们之间的关系，并试图在正特征代数几何中找到应用。(6)佐藤研究了定义在数域上的椭圆曲线的二次扭曲的Mordell-well群的阶数的分布和相关问题。(7)长谷川从对称性的角度研究了离散可积系统的代数方面。特别是，他对离散Painleve方程的量子化研究较少

(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