Research of Iwasawa Theory fof Formal Groups

岩泽理论最初是一种研究割圆域(塔)的方法，并很快被推广到任意数域。近年来，关于椭圆曲线的岩泽理论已经建立并得到了广泛的发展。数域的岩泽理论与乘法群有关，椭圆曲线的岩泽理论自然与椭圆曲线的群结构有关。我们在这个项目中的目的是将岩泽理论推广到形式群，其中包括作为特例的乘法和椭圆曲线群。为此，我们首先研究了(1)形式群的一般理论，作为该理论的重要例子，(2)数域(乘群)的岩泽理论，(3)椭圆曲线的岩泽理论(椭圆曲线群)。我们还尝试了用Maple对形式群进行数值计算。因为(2)我们主要关注Greenberg猜想及其与正规整基的关系。对于(3)，我们得到了椭圆曲线的Mu不变量的结果，推广了R.Greenberg的结果。

Iwasawa Theory started as a method for investigating (a tower of) cyclotomic fields and soon was generalized to arbitrary number fields. Recently, Iwasawa Theory for elliptic curves has been constructed and has developed extensively. Iwasawa Theory for number fields is related to multiplicative group and Iwasawa Theory for elliptic curves is naturally related to the group structure of elliptic curves. Our aim in this project was to generalize Iwasawa Theory for formal groups, which include multiplicative and elliptic curve groups as special cases. For that purpose, we first investigated(1) general theory of formal groupsand, as important examples of the theory,(2) Iwasawa Theory of number fields (multiplicative group),(3) Iwasawa Theory of elliptic curves (elliptic curve group).For (1) we mainly considered Honda's theory which classified formal groups over the integer ring. We also tried numerical computation for formal groups using Maple. For (2) our main concern was Greenberg's conjecture and its relation to normal integral bases. For (3) we obtained results for mu-invariants of elliptic curves, extending the former results of R.Greenberg.