The first lemma in Lubin-Tate theory says the following:
Let $K$ be a local field, $A$ its ring
of integers, and $f\in A[[T]]$ be such
that $f(0) = 0$, $f'(0)$ is a
uniformizer, and $f$ induces Frobenius
over the residue field. Then there
exists a unique formal group law
$F_f(X,Y)\in A[[X,Y]]$ that makes $f$
into a formal $A$-endomorphism.
If you go over the details of the lemma, you can (I think) generalize it as follows:
If $R$ is any ring, $f\in R[[T]]$ such
that $f(0) = 0$ and $f'(0)\in R^\times$
(Edit: $u=f'(0)$ then $u^n – u\in R^\times$ for all $n$), then there exists a unique
formal group law $F_f(X,Y)\in R[[X,Y]]$ that makes $f$ into a formal
$R$-endomorphism.
The business about uniformizers and Frobenius in the Lubin-Tate lemma is just to ensure that everything converges on the maximal ideal of the ring of integers in the separable closure of $K$, so that you get an actual group.
So this is pretty cool—it says that you can take something purely analytic, $f$, and magically give it an algebraic structure. Specifically, the roots of the iterates $f^{(n)} = f\circ\cdots\circ f$ become a torsion $A$-module.
If the existence of $F_f$ generalizes like I think it does, a natural question is where does $F_f$ converge? I want to be able to answer the question for specific $f$, a simple example would be the following: if $R=\mathbb{C}$ and $f(z) = uz + z^2$, then what can you say about the convergence of $F_f$?
Edit: Okay, $\mathbb{C}$ was a bad choice, but suppose $R$ is a ring complete with respect to some $\mathfrak{a}$-adic topology. Would there be a reason not to study this case? Maybe the question I should be asking is, for what other $R$ and $f$ do people study these formal groups $F_f$?
Best Answer
I don't think you should be trying to interpret this stuff over the (real or) complex numbers. Discs do not have nice algebraic properties in the archimedean world as they do $p$-adically. Even $p$-adically, one never talks about an actual radius of convergence but just makes sure things converge on the maximal ideal and work there. For example, the Lubin--Tate series could even be a polynomial, with infinite radius of convergence, but still one just focuses on what it does inside the unit disc ($p$-adically).