Knowlpedia

Theorem

Formal group logarithm

Over a Q-algebra, a commutative formal group law has a unique strict isomorphism to the additive law.

Statement

Let RR be a Q\mathbb Q-algebra and let FF be a commutative nn-dimensional over RR. There is a unique tuple

log⁡F(X)=X+terms of total degree at least 2\log_F(X)=X+\text{terms of total degree at least \(2\)}

such that

log⁡F(F(X,Y))=log⁡F(X)+log⁡F(Y).\log_F(F(X,Y))=\log_F(X)+\log_F(Y).

The formal group logarithm is therefore a strict from FF to the nn-dimensional .

Construction from invariant differentials

In one dimension, set

ωF(X)=(∂F∂Y(X,0))−1dX.\omega_F(X) = \left(\frac{\partial F}{\partial Y}(X,0)\right)^{-1}dX.

This is the invariant differential of FF, and its coefficientwise formal integral normalized by log⁡F(0)=0\log_F(0)=0 is the logarithm:

log⁡F(X)=∫0XωF.\log_F(X)=\int_0^X\omega_F.

Division by positive integers explains the Q\mathbb Q-algebra hypothesis. In several commuting dimensions, the matrix of invariant one-forms is formally closed and integrates to the logarithm coordinatewise.

Inverse exponential

The linear term of log⁡F\log_F is the identity, so the gives an inverse

exp⁡F=log⁡F−1.\exp_F=\log_F^{-1}.

It satisfies exp⁡F(U+V)=F(exp⁡F(U),exp⁡F(V))\exp_F(U+V)=F(\exp_F(U),\exp_F(V)).

Scope

The theorem requires commutativity. A noncommutative in characteristic zero is instead put into , where its tangent remains visible. Over rings not containing Q\mathbb Q, logarithm coefficients may have forbidden denominators, and height and other integral phenomena survive.

References
  1. Michiel Hazewinkel, Formal Groups and Applications, AMS Chelsea Publishing, 2012. AMS book record. Relevant: Chapters 1–2, logarithms in characteristic zero.
  2. Serge Lang, Algebra, revised third edition, Springer, 2002. Relevant: formal power series and commutative formal groups over characteristic-zero rings.