Noether's Problem for Cremona Groups over algebraic number fields and its application to Number theory

研究了给定域上关于Cremona群的有限子群G的若干变量的有理函数域的定域的合理性的Noether问题。在G是六次传递置换群的情况下，我们肯定地解决了这个问题，并得到了固定域的显式描述。目前正在收集和准备一些论文的结果，尽管完成还需要一些时间。在研究期间，我们每年都会举办“伽罗瓦理论及相关课题”研讨会，讨论各种相关问题。分别在山形大学（2007年）、德岛大学（2008年）和金泽大学（2009年）举行。我们还每年在早稻田大学举办数论会议，并与许多这方面的专家交流，包括来自国外的专家。

We studied the Noether's Problem, which asks the rationality of the fixed field of the rational function field of several variables over a given field, with respect to a given finite subgroup G of the Cremona group. We solved this problem affirmatively in the case where G is one of the transitive permutation groups of degree six, and obtained the explicit description of the fixed field as expected. The results are now being collected and prepared in some papers, although it will take some time before the completion. During the period of the research, we had in each year a workshop entitled as "Galois theory and related topics", and discussed the various related problems.. They were held in the university of Yamagata (2007), Tokushima (2008), and Kanazawa (2009). We also had a conference on number theory each year at Waseda university and communicated with many experts of this subject, including those from foreign countries.