On one-dimensional formal group laws in characteristic zero

Abstract

Let ${\mathbb{K}}$ be a field of characteristic zero or, more generally, a ${\mathbb{Q}}$-algebra. A formal power series ${F(x,y)=x+y+ \sum_{i,j \geq 1} a_{i,j}x^iy^j \in \mathbb{K}{[\![} x, y{]\!]}}$ is called a one-dimensional formal group law if F(F(x, y), z) = F(x, F(y, z)). Using some elementary methods, we prove that for every one-dimensional formal group law F(x, y) there exists a formal power series ${f(x)=x+\sum_{n\geq 2}f_nx ^n \in \mathbb{K}{[\![} x, y{]\!]}}$ so that F(x, y) = f ⁻¹(f(x) + f(y)).