Kummer Congruences and Formal Groups Original Research Article

Let R be an integral domain and let f(X) = (f1(X), ..., fn(X)) be an n-tuple of power series in n variables X = (x1, ..., xn) such that dfj set membership, variant circled plus R[[X]]dxi, f(X) ≡ 0 (mod deg 1), and J(f) = ((∂fi/∂xJ(0)) is invertible over R. We can form the formal group Ff(X, Y) = f−1(f(X) + f(Y)). A priori, the coefficients of Ff are in K, the quotient ring of R. T. Honda (J. Math. Soc. Japan22, 1970, 213-246) and M. Hazewinkel ("Formal Groups and Applications," Academic Press, Orlando, FL, 1978) give some sufficient conditions for Ff(X, Y) to be defined over R in the form of functional equations for the coefficients of the fi. 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.