HOPF GALOIS KUMMER THEORY OF FORMAL GROUPS

Abstract. Let f: F! G be an isogeny between finite n-dimensional formal groups defined over R, the valuation ring of some field extension K of Qp. Let H be the R-Hopf algebra which arises from this isogeny. For such H, we classify Gal (H), the group of H-Galois objects. Let M be the maximal ideal of R, and let P(F, K) denote the n-tuples of M under the group operation induced by F. Our main result is the construction of an isomorphism from the cokernel of P ( f) to Gal (H), where P ( f) is the induced map from P(F, K) to P(G, K). In geometric language Gal (H) describes the group of isomorphism classes of principal homogeneous spaces for Spec (H) over Spec (R). Geometric methods have been used by Mazur to establish the above isomorphism, but the proof is nonconstructive. A geometric approach has also provided a formula for the cardinality of Gal (H). We give an alternative derivation of this result using formal group techniques. Introduction. Given a field K and a finite abelian group G a question in classical field theory is to determine Gal (K, G), the isomorphism classes of Galois field extensions of K with group G. When K has characteristic zero and contains a primitive nth root of unity, classical Kummer theory asserts that Gal (K, Cn) = U(K)=U(K)n where U(K) denotes the units of K and Cn is the cyclic grou