One-dimensional algebraic formal groups
Robert F. Coleman · Pacific Journal of Mathematics · 1986
Let K be an algebraically closed field of characteristic zero.We shall call an element of K[[x l9 ... ,x n ]] algebraic if it is algebraic over K(x l9 ... 9 x n ).Thus a one-dimensional algebraic formal group is an element F e K[[x l9 x 2 ]] such that F is a formal group and Fis algebraic.As is well known, such formal groups arise from one-dimensional algebraic groups.Our intention is to show that this is the only way they arise.All formal groups mentioned in this note shall be one-parameter formal groups.DEFINITION.TWO algebraic formal groups F, F ' & K[[x v x 2 ]] are said to be algebraically isomorphic if there exists an algebraic element / e xK [[x]] such that/ Φ 0 and It is easy to see that there exists a unique element/* e xK" [[x]] such that/ o /* = JC.It then follows that