Arithmetic of formal groups and applications I: Universal norm subgroups
Arithmetic of formal groups and applications I: Universal norm subgroups
复制标题
形式群的算术及其应用 I：通用范数子群
DOI：
10.1007/bf01389244
复制
发表时间：
1987
影响因子：
3.1
通讯作者：
P. Schneider
中科院分区：
文献类型：
--
作者：
P. Schneider
形式乘法群的范数映射理论被称为局部类域理论。在1972年Mazur [14]在他的推广岩泽理论的阿贝尔品种在数域相当清楚，这是绝对至关重要的这样一个全球性的理论，首先要了解规范地图的正式集团重视阿贝尔品种。他实现了这一认识的情况下，正式团体的乘法型利用理论的proalgebraic集团。后来，Lubin和罗森[12]对Mazur的结果给出了一个完全初等的方法.在此期间Hazewinkel [8]已经解决了一维交换形式李群的情况下，他的方法包括在一个明确的和非常复杂的研究性质的幂级数系数的对数的一个正式的群体法律的手段更高的分歧理论。本着同样的精神，Vvedenskij [18]和Konovalov [10]谴责并稍微扩展了Hazewinkel的结果。设K/Q是具有剩余类域·的整数环R的有限扩张，Koo/K是具有整数环R和F '的分歧Zv扩张. Gal（Koo/K）。设i/R表示有限维d的光滑连通交换形式R-群（即交换形式李群）。设K_1/K是K_0/K中p ~“度的中间层，R是它的整数环，G_i是：-Gal（K./K. K）它的伽罗瓦群，那么我们知道-从ff作为形式李群的描述，i（R，）G--（R），和
The theory of the norm map for the formal multiplicative group is well-known as local class field theory. In 1972 Mazur [14] in his generalization of Iwasawa theory to abelian varieties over number fields made quite clear that it is absolutely crucial for such a global theory first to understand the norm map for the formal groups attached to abelian varieties. He achieved this understanding in case of formal groups of multiplicative type by using the theory of proalgebraic groups. Later on, Lubin and Rosen [12] gave a completely elementary approach to Mazur's results. In the meantime Hazewinkel [8] had settled the case of one-dimensional commutative formal Lie groups; his method consists in an explicit and very complicated study of the properties of the power series coefficients of the logarithm of a formal group law by means of higher ramification theory. In the same spirit Vvedenskij [18] and Konovalov [10] reproved and slightly extended Hazewinkel's results. Let K/Q, be a finite extension with ring of integers R and residue class field• and Koo/K be a ramified Zv-extension with ring of integers R~ and F'.= Gal (Koo/K). Let i/R denote a smooth connected commutative formal R-group of finite dimension d (ie, a commutative formal Lie group). If K,/K is the intermediate layer of degree p" in Koo/K, R, its ring of integers, and G,:---Gal (K./K) its Galois group then we know-from the description of ff as a formal Lie group that i (R,) G------(R), and