What is the different in the cyclotomic tower over a finite ramified extension of Qp? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T01:44:30Z http://mathoverflow.net/feeds/question/54213 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/54213/what-is-the-different-in-the-cyclotomic-tower-over-a-finite-ramified-extension-of What is the different in the cyclotomic tower over a finite ramified extension of Qp? crocodile 2011-02-03T16:50:29Z 2011-02-04T16:41:06Z <p>If $K_n$ is the field <code>$\mathbb{Q}_p(\mu_{p^n})$</code>, then it's easy to see that the relative different <code>$\mathcal{D}(K_n / K_{n-1})$</code> is $(p)$ for all $n \ge 2$. </p> <p>What happens if I take an arbitrary, probably totally ramified, finite extension $L/\mathbb{Q}_p$ and look at the tower $L_n = LK_n$? It's clear that $\mathcal{D}(L_n / L_{n-1})$ divides $(p)$, and one can show (using a general result of Tate on $\mathbb{Z}_p$-extensions) that its valuation tends to 1 as $n \to \infty$; but is it true that it's equal to $(p)$ for all sufficiently large $n$?</p> http://mathoverflow.net/questions/54213/what-is-the-different-in-the-cyclotomic-tower-over-a-finite-ramified-extension-of/54220#54220 Answer by Laurent Berger for What is the different in the cyclotomic tower over a finite ramified extension of Qp? Laurent Berger 2011-02-03T18:22:54Z 2011-02-03T18:22:54Z <p>Take $K=Q_p$ and $L/K$ finite. It is known that the sequence ${ p^n v_p( \mathcal{D}(L_n/K_n) ) }_n$ is eventually constant (it's basically the valuation of the different of the extension $E_L/E_K$ which you get from the theory of the field of norms). This and the transitivity of the different should allow you to answer your question.</p> <p>As for the proof of my claim, see proposition 4.5 of <a href="http://www.math.jussieu.fr/~colmez/monodromie.pdf" rel="nofollow">http://www.math.jussieu.fr/~colmez/monodromie.pdf</a></p> http://mathoverflow.net/questions/54213/what-is-the-different-in-the-cyclotomic-tower-over-a-finite-ramified-extension-of/54225#54225 Answer by Mikhail Bondarko for What is the different in the cyclotomic tower over a finite ramified extension of Qp? Mikhail Bondarko 2011-02-03T19:06:47Z 2011-02-03T19:06:47Z <p>The answer is 'no'. The different is usualy distinct from $(p)$; if you have a tower for whose upper level the different equals $(p)$, then the same is true for all other levels (possibly, one needs $p>2$ here). See Proposition 1.3 in <a href="http://www.mathnet.ru/php/archive.phtml?wshow=paper&amp;jrnid=aa&amp;paperid=805&amp;option_lang=eng" rel="nofollow">http://www.mathnet.ru/php/archive.phtml?wshow=paper&amp;jrnid=aa&amp;paperid=805&amp;option_lang=eng</a></p> http://mathoverflow.net/questions/54213/what-is-the-different-in-the-cyclotomic-tower-over-a-finite-ramified-extension-of/54331#54331 Answer by Lubin for What is the different in the cyclotomic tower over a finite ramified extension of Qp? Lubin 2011-02-04T16:41:06Z 2011-02-04T16:41:06Z <p>Here's a relatively easy counterexample. Take $p=2$, $L=\mathbb{Q}_2(2^{1/3})$. I did some direct computation and saw that $v_2(\mathfrak{D}^{L_2}_L)=2/3$, $v_2(\mathfrak{D}^{L_3}_{L_2})=5/6$. Looks like there's a pattern. But there's a better way.</p> <p>We're in a situation where not only $L/\mathbb{Q}_2$ is tamely ramified of degree $3$, but also every $L_n/K_n$. Now we use the functoriality of the Hasse-Herbrand transition function: if $k\subset F\subset K$, then $\varphi^K_k=\varphi^F_k\circ \varphi^K_F$. Use the relation $\varphi^{L_n}_{\mathbb{Q}_2}=\varphi^{K_n}_{\mathbb{Q}_2}\circ \varphi^{L_n}_{K_n}= \varphi^L_{\mathbb{Q}_2}\circ \varphi^{L_n}_ L$ and the fact that a tamely ramified extension has all the transition-function action at the origin. That is, the function is $y=x$ for $x\le 0$, but $y=x/e$ for $x\ge 0$, where $e$ is the ramification index. So as real functions, $\varphi^L_{\mathbb{Q}_2}=\varphi^{L_n}_{K_n}$, namely this is just $y=x/3$. Consequently, the transition function of $L_n/L$ is gotten by conjugating that of $K_n/{\mathbb{Q}_2}$ with the tame transition function. The effect is to multiply all coordinates of vertex points by $3$.</p> <p>But we also know the transition function of $K_n$ over ${\mathbb{Q}_2}$: its vertices are at all $(2^{i-1}-1,i-1)$ for $2\le i\le n$.The new vertices are at $(3,3)$, $(9,6)$, $(21,9)$, etc. This means that the lower breaks of $L_n/L$ are at $3(2^{i-1}-1)$ for $2\le i\le n$, and in particular the unique break of $L_n/L_{n-1}$ is at $3(2^{n-1}-1)$.</p> <p>Now use the formula \begin{align}{ v_F(\mathfrak{D}^F_k)=\sum_{j\ge 0}\bigl(|G_j|-1\bigr) }\end{align} where the $G$'s are the lower ramification groups, and where in this case all the numbers being added up are $1$ or $0$, to see that $v_{L_n}\bigl(\mathfrak{D}^{L_n} _ {L_{n-1}}\bigr)=3(2^{n-1}-1)+1=3\cdot2^{n-1}-2$. Divide by the ramification number of $L_n$ over $\mathbb{Q}_2$ to get $1-1/(3\cdot 2^{n-2})$, agreeing with my computations for $n=2$ and $n=3$.</p> <p>It's not an issue of tame versus wild ramification in the extension $L/\mathbb{Q}_2$, either. I used $L=\mathbb{Q}_2(2^{1/4})$ to find that the numbers are $1-3/2^m$; the argument is similar but a bit more delicate, since you have no a priori idea of what the transition function of $L_n/K_n$ might be.</p>