Kummer Congruences and Formal Groups

Abstract

Let R be an integral domain and let f( X) = ( f 1( X), ..., f n ( X)) be an n-tuple of power series in n variables X = ( x 1, ..., x n ) such that df j ∈ ⊕ R[[ X]] dx i , f( X) ≡ 0 (mod deg 1), and J( f) = ((∂ f i /∂ x J (0)) is invertible over R. We can form the formal group F f ( X, Y) = f −1( f( X) + f( Y)). A priori, the coefficients of F f are in K, the quotient ring of R. T. Honda ( J. Math. Soc. Japan 22, 1970, 213-246) and M. Hazewinkel ("Formal Groups and Applications," Academic Press, Orlando, FL, 1978) give some sufficient conditions for F f ( X, Y) to be defined over R in the form of functional equations for the coefficients of the f i . This paper considers the conserve question: Given a commutative formal group F( X, Y) defined over a ring R, what necessary conditions must be satisfied by the coefficients of the logarithm of F( X, Y)? These results generalize the results of C. Snyder ( Rocky Mountain J. Math. 15, No. 1. 1985, 1-11) in the one dimensional case.