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Number Theory and Geometry related to Algebraic Groups

与代数群相关的数论和几何

基本信息

批准号：
15340001
负责人：
YUKIE Akihiko
金额：
$ 6.08万
依托单位：
Tohoku University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

项目摘要

(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

(1)Yukie研究了与预齐次向量空间相关的Zeta函数，得到了判别式小于或等于X的五次域的个数的估计。(2)Hanamura研究了混合母题，得到了模簇的kunneth公式的一个结果。(3)石田建立了环簇的理想理论(环)与有理和实扇有关的理论。他还发现了代数簇爆破的一个真正的粉丝模拟。特别是对于有限数量的爆破，他证实了与代数簇的情况相似。在代数几何中，Zariski-Riemann拓扑可以定义在函数域的赋值环集上。他利用自由模和实向量空间上的所有加序的集合为有理和实数扇定义了这一概念。利用这种Zariski-Riemann拓扑，他证明了环簇的紧化的存在性。(4)Hara推广了…环R的理想I的紧闭的概念证明了广义行列式理想τ(I)的各种性质，从而为该理论奠定了基础。用这种方法证明了关于伴随丛整体生成的Fujita猜想的一个特例，以及关于正则局部环的理想的符号幂的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Galois共轭同源，则它称为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，数域上椭圆曲线的扭群是有限的。他研究了同源椭圆曲线之间的挠率是如何变化的。(6)Ogata用非常充分的线丛研究了Toric簇的射影正规性和定义理想的生成元的次数，得到了射影正规性的一些判据和定义理想的次数的一些估计。较少