Number Theory and Geometry related to Algebraic Groups

(1) Yukie研究了与预齐次向量空间相关的zeta函数，得到了判别式小于等于x的五次域个数的估计。(2)Hanamura研究了混合基，得到了模变体的Kunneth公式的一个结果。(3)石田建立了一种将环的理想理论与环的理性和真实扇形联系起来的理论。他还发现了代数变量膨胀的扇形模拟。特别是对于有限数量的膨胀，他证实了与代数变量情况的相似性。在代数几何中，Zariski-Riemann拓扑可以定义在函数域的赋值环集合上。他用自由模和实向量空间上所有加性阶的集合定义了这个概念。利用Zariski-Riemann拓扑，他证明了环型紧化的存在性。(4) Hara将环R的理想I的紧闭的概念推广到“I紧闭”，并证明了广义行列式理想τ(I)的各种性质，从而奠定了理论基础。利用该方法，他应用于关于伴随束整体生成的Fujita猜想的一个特例的证明，以及关于正则局部环理想的符号幂的Ein-Lazarsfeld-Smith比较定理的一个新证明。(5) Nakamura：如果一条CM椭圆曲线与它的所有伽罗瓦共轭都是等齐次的，它被称为q曲线，并且具有重要的性质。他对给定虚二次域的绝对类域上的所有q曲线进行了分类。众所周知，椭圆曲线在数域上的扭转群是有限的。他研究了等均椭圆曲线间的扭转变化。(6)绪方元研究了极充裕线束环变的定义理想的射影正态性和生成子的度，得到了一些定义理想的射影正态性准则和一些定义理想度的估计。少

(1) Yukie investigated zeta functions associated with prehomogeneous vector spaces and obtained an estimate of the number of quintic fields with discriminant less than or equal to X.(2) Hanamura investigated mixed motifs and obtained a result on the Kunneth formula for modular varieties.(3) Ishida established a theory relating ideal theory (of rings) to rational and real fans for toric varieties. Also he found a real fan analogue of blow-ups of algebraic varieties. Especially for a finite number of blow-ups, he confirmed similarities with the case of algebraic varieties. In algebraic geometry, Zariski-Riemann topology can be defined on the set of valuation rings of the function fields. He defined this notion for rational and real fans using the set of all additive orders on free modules and real vector spaces. Using this Zariski-Riemann topology, he proved the existence of the compactification of toric varieties.(4) Hara generalized the notion of tight closure of ideals I of rings R of … More positive characteristic to "I-tight closure" and proved various properties for the generalized determinantal ideal τ(I), thus made a foundation of the theory. Using this method, he applied to the proof of a special case of the Fujita conjecture on the global generation of adjoint bundles and to a new proof of the Ein-Lazarsfeld-Smith comparison theorem on the symbolic power of ideals of regular local rings.(5) Nakamura : If a CM elliptic curve is isogenous to all its Galois conjugate, it is called a Q-curve and has important properties. He classified all Q-curves over the absolute class field of a given imaginary quadratic field. It is well-known that the torsion group of an elliptic curve over a number field is finite. He investigated how the torsion changes among isogenous elliptic curves.(6) Ogata investigated projective normality and the degrees of the generators of the defining ideals of toric varieties by very ample line bundles, and opbtained some criteria of projective normality and some estimates of the degrees of the defining ideals. Less