Hopf Galois Kummer theory of formal groups

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Let ƒ : F → G be an isogeny between finite n -dimensional formal groups defined over R , the valuation ring of some field extension K of Q p . 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 ( ƒ ) to Gal( H ), where P ( ƒ ) 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.