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edited Apr 19 2010 at 2:31 |
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$?
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edited Apr 19 2010 at 2:14 |
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.
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edited Apr 18 2010 at 23:54 |
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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3
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edited Apr 18 2010 at 23:33 |
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.
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)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.
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 $f(z) = z uz + z^2$, 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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edited Apr 18 2010 at 23:27 |
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 $f$ incudes 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.
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), then there exists a unique
formal group law $F_f(X,Y)\in
R[[X,Y]]$ that makes $f$ into a formal
group 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 $f(z) = z + z^2$, 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 thing 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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asked Apr 18 2010 at 23:16 |
Extending methods from Lubin-Tate theory
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 $f$ incudes 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.
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), then there exists a unique
formal group law $F_f(X,Y)\in
R[[X,Y]]$ that makes $f$ into a formal
group 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 $f(z) = z + z^2$, 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 thing 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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