Arithmetic of formal groups and applications I: Universal norm subgroups

Abstract

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 (i.e., 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