On torsion points of formal groups over a ring of Witt vectors

Abstract

Let k be a perfect field of characteristic p >0. Let W = W(k) be the ring of Witt vectors over k and K its quotient field. Let fit be the maximal ideal of the ring of integers of an algebraic closure of K. For a one-dimensional formal group F defined over W of finite height h, denote by Fpn the subgroup of F(ff0 of order dividing p". The operation of the Galois group G(K/K) on the Tate module T(F) = proj lim Fpn defines a p-adic representation G(K/K)---* GL h(Tlp) ( -Aut (T(F) ) ) , whose image we denote by H F. In [1], Fontaine raised a question concerning the characterization of H F as a closed subgroup of GLh(~,p). Let rn__> 1 and F be a formal group over W of height h, which corresponds to a Honda 's special element p +pm cl T +.. . +pm Ch_ ~ T h~ + c T h (cl, ..., ch_l~W, c: a unit of W) (cf. Sect. 2 and [-2]). In this paper we explicitly describe the Galois extension K(Fpm+~)/K by using some full formal groups (cf. Theorem 1). We also compute the ramification numbers of the extension (cf. Theorem 2). In particular our result gives an explicit description of K(Fp2)/K for all formal groups F over W of height h. By our result we can also determine/-/F for certain formal groups (cf. Sect. 4).