Extending methods from Lubin-Tate theory - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T12:38:42Z http://mathoverflow.net/feeds/question/21786 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/21786/extending-methods-from-lubin-tate-theory Extending methods from Lubin-Tate theory Sean Kelly 2010-04-18T23:16:31Z 2010-04-23T10:37:03Z <p>The first lemma in Lubin-Tate theory says the following:</p> <blockquote> <p>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.</p> </blockquote> <p>If you go over the details of the lemma, you can (I think) generalize it as follows:</p> <blockquote> <p>If $R$ is any ring, $f\in R[[T]]$ such that $f(0) = 0$ and $f'(0)\in R^\times$ (<strong>Edit</strong>: $u=f'(0)$ 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.</p> </blockquote> <p>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.</p> <p>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.</p> <p>If the <em>existence</em> of $F_f$ generalizes like I think it does, a natural question is where does $F_f$ <em>converge</em>? 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 $F_f$?</p> <p><strong>Edit:</strong> 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 $F_f$?</p> http://mathoverflow.net/questions/21786/extending-methods-from-lubin-tate-theory/21802#21802 Answer by KConrad for Extending methods from Lubin-Tate theory KConrad 2010-04-19T01:24:01Z 2010-04-19T01:24:01Z <p>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). </p> http://mathoverflow.net/questions/21786/extending-methods-from-lubin-tate-theory/22328#22328 Answer by olli_jvn for Extending methods from Lubin-Tate theory olli_jvn 2010-04-23T10:37:03Z 2010-04-23T10:37:03Z <p>It does probably not work in the way you wish.</p> <p>The crucial bit where convergence is used later in Lubin-Tate is to realize the Galois action via these series [a]. The first-level roots will lie in m minus m^2, so you really want convergence in this radius.</p> <p>For finding roots itself I think convergence really is not the problem, but having some roots will not help you much if you cannot make the machine realizing the Galois action through power series work.</p> <p>If you look at the proof (e.g. Yoshida's notes or Milne's notes), at some point one uses that</p> <p>f(X^q) - f(X)^q</p> <p>is zero modulo p, which is quite crucial for the convergence - as you say above. But this is not just a sufficient condition, I think if you try a series over some other ring where this fails, you'll really inavoidably get a series which doesn't converge in m. Still your Lubin-Tate polynomial (as proposed by KConrad as a good example above) has roots and all that, good, but there is no way to make the formal O_K-module series [a] act on them. I don't quite remember so well, but I am not even so sure whether it is certain that the extension made from the roots of a Lubin-Tate polynomial/powerseries is Galois anymore in general.</p>