Number Theory and its Development to Discrete Mathematics

关于本研究主题的核心课题；数论，特别是与阿贝尔域相关的Hasse问题[A]，算术，代数几何[B]及其在离散数学中的应用[C]，我们于2008~2010年每年8月和1月在Saga举办了数论研讨会。研究组织者在南南大学呆了两年半的时间，与两校的博士学者共同进行研究。在Hasse的问题上，组织者通过与NUCES的博士学者的合作，获得了某些纯六场和纯八场族的单性性表征。在对代数几何码的研究中，Cooperative Uehara提出了构造一类不同于已知的单点型代数几何码的新码的思想。同时，Uehara应用了一元型代数几何码的求值码的概念，提出了一种从代数数域上的整数环构造码的方法，并给出了一些这样的码的显式例子[C]。Cooperative Miyazaki研究了Buchsbaum品种的最小自由分辨率，并根据Castelnuovo-Mumford正则性对Buchsbaum品种进行了分类[B]。Cooperative Terai研究了多项式环中的无平方单项式理想stanley - reisner理想[B]。片山合作社。利用三次域和二次域的单位群结构，确定了三次和四次有限辛群，并在Saga 2011年的研讨会上发表了这些结果。Newman， Shanks和Williams在20世纪80年代确定了平方阶的有限辛群。Katayama还研究了单位圆内的k形全等多边形的数量，其中顶点从圆的n个分划点中选择[C]。Cooperative Taguchi研究了截断离散估值环（=:tdvr’s）的分支理论和mod p伽罗瓦表示的（不）存在性。在前者上，Taguchi（与T. Hiranouchi联合）证明了tdvr a的有限扩展的范畴等价于提升a的完全离散估值域的有限扩展的受限分支的范畴。在后者上，Taguchi（与H. Moon联合）证明了某些二次域的2维模2伽罗瓦表示的不存在性[B]。少

On the core subjects of this research theme ; Number Theory, specifically Hasse's problem related to Abelian fields[A], Arithmetic, Algebraic geometry[B] and its application to Discrete Mathematics[C], we held the Workshop on Number Theory in Saga in each August and January of 2008~2010. The research organizer stayed at NUCES during two and half years to work the joint research with PhD scholars at both campuses. On Hasse's problem, the organizer obtained the characterization on the monogeneity for certain family of pure sextic and pure octic fields by the joint work with PhD scholars at NUCES. In our research on algebraic geometry codes, Cooperative Uehara gave an idea of constructing a new class of algebraic geometry codes different from the known codes of one-point type. Also, applying the concept of evaluation codes, which generalize one-point-type algebraic geometry codes, Uehara invented a method of constructing codes from integer rings on algebraic number fields, and presented s … More ome explicit examples of such codes[C]. Cooperative Miyazaki studied the minimal free resolution of Buchsbaum varieties and obtained a classification of the Buchsbaum variety in terms of the Castelnuovo-Mumford regularity[B]. Cooperative Terai studied Stanly-Reisner ideals, which are squarefree monomial ideals in polynomial rings[B]. Cooperative Katayama . has determined finite symplectic groups of cube and 4th order, using the structure of the unit groups of cubic and quadratic fields, and announced these results at the workshop in Saga 2011. Newman, Shanks and Williams determined finitesymplectic groups of square order in1980's. Katayama also investigated the number of congruent k-polygons inscribed in a unit circle, where the vertices chosen from n division points of the circle[C]. Cooperative Taguchi studied the ramification theory of truncated discrete valuation rings (= : tdvr's) and the (non)existence of mod p Galois representations. On the former, Taguchi proved (jointly with T. Hiranouchi) that the category of finite extensions of a tdvr A is equivalent to a category of finite extensions, with restricted ramification, of a complete discrete valuation field which lifts A. On the latter, Taguchi proved (jointly with H. Moon) the non-existence of 2-dimensional mod 2 Galois representations for some quadratic fields[B]. Less