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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?

I've tried looking around for Maybe the answer question I should be asking is, for what other $R$ and found a lot of interesting stuff on $f$ do people study these formal schemes etc., but I don't want to know about the general theory. None of the things I found seem to talk about the radius of convergence of this groups $F_f$ that you get, so I was just wondering if anyone has looked at it.F_f$? 5 update question The first lemma in Lubin-Tate theory says the following: Let$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$(but not a root of unity, though that might not matter), Edit:$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}$-adic topology. Would there be a reason not to study this case? I've tried looking around for the answer and found a lot of interesting stuff on formal schemes etc., but I don't want to know about the general theory. None of the things I found seem to talk about the radius of convergence of this$F_f$that you get, so I was just wondering if anyone has looked at it. 4 deleted 4 characters in body The first lemma in Lubin-Tate theory says the following: Let$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$into a formal group endomorphism.$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$(but not a root of unity, though that might not matter), then there exists a unique formal group law$F_f(X,Y)\in R[[X,Y]]$that makes$f$into a formal group endomorphism.$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$? I've tried looking around for the answer and found a lot of interesting stuff on formal schemes etc., but I don't want to know about the general theory. None of the things I found seem to talk about the radius of convergence of this$F_f\$ that you get, so I was just wondering if anyone has looked at it.

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