A one-dimensional commutative <formal group law> over a ring $R$ is a series $F(X,Y)\in R[[X,Y]]$ satisfying
$$
F(X,0)=X,
\qquad F(X,Y)=F(Y,X),
\qquad F(F(X,Y),Z)=F(X,F(Y,Z)).
$$
An isomorphism from $F$ to $G$ is a series $h(T)=uT+O(T^2)$ with $u\in R^*$ and
$$
h(F(X,Y))=G(h(X),h(Y)).
$$

Over a characteristic-zero field $K$, every such formal group is isomorphic to the additive formal group. Differentiate the associativity identity and define the invariant differential
$$
\omega_F(T)=\left(\frac{\partial F}{\partial Y}(T,0)\right)^{-1}dT.
$$
Termwise integration is possible in characteristic zero; the <formal logarithm>
$$
\log_F(T)=\int_0^T\omega_F
$$
has leading term $T$. Invariance of $\omega_F$ gives
$$
d\log_F(F(X,Y))=d\log_F(X)+d\log_F(Y),
$$
and evaluation at $(0,0)$ removes the integration constant. Hence $\log_F(F(X,Y))=\log_F(X)+\log_F(Y)$. Its unit linear coefficient gives a compositional inverse, so it is an isomorphism to $X+Y$. Therefore any two one-dimensional commutative formal groups over $K$ are isomorphic.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 125 / 2 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-125/2/a/solution