Wild Ramification

The question is, loosely put, what is known about wild ramification?

Is there a semi-well-established theory of wild ramification that can be furthered in various specific situations? Or maybe there are some situations where we know a lot, but what we know doesn't apply to other situations? Or are all results about wild ramification ad hoc for whatever the author is interested in?

In short: what is there to know, and where is it written?

-
 Why was this made a community wiki ? Pete for example gives a very pertinent answer for which he should get some credit. – Chandan Singh Dalawat Feb 2 2010 at 4:42 James Borger is right that the question is quite vague -- maybe that's why it was made CW. (Better strategy in my opinion: make the question less vague!) I appreciate the support, but really, no big deal about the rep... – Pete L. Clark Feb 2 2010 at 8:57 "the question is quite vague" : some of the answers have been very precise... – Chandan Singh Dalawat Feb 2 2010 at 10:07 Perhaps users with sufficient reputation should be allowed to vote to unwikify a question? – Pete L. Clark Feb 2 2010 at 16:17 @Pete. A community wiki post is edited by many people. It is not possible to unwikify it so simply. There are question of authorship. So I suspect that the designers made it an irreversible decision. – Anweshi Feb 2 2010 at 17:56

The structure of wildly ramified abelian extensions of local fields is given by local class field theory (and conversely is where most of the content of LCFT resides): see Milne's notes or Serre's Corps Locaux.

Wildly ramified nonabelian extensions of local fields are "understood" in at least the following two senses [neither one of which connotes perfect understanding to me]:

1) The absolute Galois group of a local field is a topologically finitely presented profinite group, with known generators and relations.

2) Local Langlands for GLn (as proved by Harris-Taylor and Henniart) has something deep to say about the structure of wildly ramified extensions.

It would not be fruitful for me to elaborate, since there are other active MOers who are much more knowledgeable in these matters. This was really just a long comment.

Addendum: upon request, here is some bibliographical material for 1) above:

Jannsen, Uwe Über Galoisgruppen lokaler Körper. (German) [On Galois groups of local fields] Invent. Math. 70 (1982/83), no. 1, 53--69.

Jannsen, Uwe; Wingberg, Kay Die Struktur der absoluten Galoisgruppe p-adischer Zahlkörper. (German) [The structure of the absolute Galois group of p-adic number fields] Invent. Math. 70 (1982/83), no. 1, 71--98.

Wingberg, Kay Der Eindeutigkeitssatz für Demuskinformationen. (German) [The uniqueness theorem for Demushkin formations] Invent. Math. 70 (1982/83), no. 1, 99--113.

Let $k$ be a finite extension of ${\bf Q}_p$ with $p\neq 2$ and $\overline k$ be the algebraic closure of $k$. The study of the Galois group $G_k = G(\overline k/k)$ was initiated by K. Iwasawa [Trans. Amer. Math. Soc. 80 (1955), 448--469; MR0075239 (17,714g)] and continued mainly in several papers of A. V. Yakovlev and the reviewer. Yakovlev [Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 1283--1322; MR0236155 (38 #4453)] succeeded in the description of $G_k$ as a profinite group with generators and relations. But his result was unsatisfactory, since the structure of one relation which comes from Demushkin's relation for maximal $p$-extensions was rather complicated, especially in the case $n = [k: Q_p]\equiv 1\ (\text{mod}\,2)$. Therefore the reviewer considered the question once more [Dokl. Akad. Nauk SSSR 238 (1978), 19--22; MR0472776 (57 #12466)] and gave a cohomological characterisation of $G_k$ as a filtered group with the filtration by the inertia group $T_k$ and the $p$-Sylow group $V_k$ of the inertia group. The three papers under review can be considered as the final answer to the question of the structure of $G_k$. The progress with respect to the above-mentioned earlier papers consists in the following: (1) The authors give a description of $G_k$ with two relations, instead of three relations as in the earlier work on $G_k$, which simplifies the situation. (2) For the first time they give a satisfactory description of the case $n\equiv 1\ (\text{mod}\,2)$. (3) The study of $G_k$ is based on the notion of Demushkin formations, which was introduced by the reviewer [op. cit.]. The proof of the uniqueness of a Demushkin formation with given invariants which characterizes $G_k$ as a filtered group is given in full detail, based on structure theorems in module theory. Finally the authors give an explicit description of a Demushkin formation with given invariants by means of $n+3$ generators and two relations. The case $p=2$ remains open.

Reviewed by Helmut Koch

-
 Do you know a reference for your (1)? – Adam Topaz Feb 1 2010 at 23:51 I included one above. Koch's book Algebraic Number Theory also contains a treatment, with further references. – Pete L. Clark Feb 2 2010 at 1:02 "Local Langlands for $GL_n$ (as proved by Harris and Taylor)". Also by Guy Henniart, independently, albeit slightly later. His proof is apparently simpler. – Chandan Singh Dalawat Feb 2 2010 at 6:17

Let me give a low-brow answer to the question, and begin with my earlier answer (which got a couple of downvotes, so you have to take it with a pinch of salt). There I discussed quadratic extensions.

I'm assuming that your base field $K$ is a finite extension of $\mathbb{Q}_p$ or of $\mathbb{F}_p((\pi))$, where $p$ is a prime number and $\pi$ is transcendental. (Very little will change if you allow $K$ to be a field complete for a discrete valuation with perfect residue field.) Let $k$ be the residue field of $K$.

Finite extensions $L|K$ can be unramified, (at worst) tamely ramified or wildly ramified.. The three cases correspond to $e=1$, $\operatorname{gcd}(e,p)=1$, $p|e$, where $e$ is the ramification index of $L|K$. There are uniquely determined subfields $K\subset L_0\subset L'\subset L$ such that $L_0|K$ is unramified, $L'|L_0$ is totally but tamely ramified, and $L|L'$ is totally ramified of degree $p^s$ for some $s\in\mathbb{N}$, so it is wildly ramified if $s>0$. (For me $0\in\mathbb{N}$; I want it to be an additive monoid.)

Unramified extensions can be completely understood in terms of extensions of the residue field $k$. When $k$ is finite as here, there is only one extension in each degree $n$, and it is obtained by adjoining a primitive $(q^n-1)$-th root of $1$, where $q=\operatorname{Card}(k)$. It follows that the maximal unramified extension of $K$ is obtained by adjoining primitive roots of $1$ of order prime to $p$.

Tamely ramified extensions are only slightly more complicated. It is not hard to show that if $L|K$ is totally but tamely ramified of degree $n$, then $L=K(\root n\of\varpi)$ for some uniformiser $\varpi$ of $K$, and not hard to determine when two uniformisers $\varpi$ and $\varpi'$ give the same extension. See for example Lecture 18 in my online notes arXiv:0903.2615. As shown there, the maximal tamely ramified extension $T|K$ is obtained by adjoining $\root n\of1$ and $\root n\of\varpi$ for all $n>0$ prime to $p$, where $\varpi$ is a fixed uniformiser of $K$. This allows you to write a simple presentation for the profinite group $\operatorname{Gal}(T|K)$ in which the generators have some arithmetic significance.

(If your base field had been $\mathbb{C}((t))$, all whose finite extensions are totally but tamely ramified, you would have been able to conclude at this point that an algebraic closure is obtained by adjoining an $n$-th root of $t$ for every $n>0$.)

What are all totally ramified extensions $L|K$ of degree $p^s$ ? Very little is known about the question. It is easy to see that there are only finitely many $L$ in the mixed characteristic case, and infinitely many $L$ in the equicharacteristic case, but there are exactly $p^s$ extensions when counted properly.

As Pete says, the abelian ones are given by local class field theory (in terms of index-$p^s$ subgroups of $K^\times$). (But even for $s=1$ there are extensions which are not galoisian, and I don't know them all.) The exponent-$p$ ones can be understood not only in terms of Class Field Theory, but also Kummer Theory or Artin-Schreier Theory.

The question is, loosely put, what is known about wild ramification?

The answer is, loosely put, not much. (Joking, eh ?)

Addendum (26/2/2010). Yesterday I came across a recent theorem of Abrashkin which says, roughly speaking, that if you know all the wildly ramified extensions (along with their filtration), then you know the local field.

More precisely, let $K$ be a local field of residual characteristic $p$, and $P|K$ the maximal pro-$p$-extension of $K$ --- the compositum of all $p$-extensions of $K$. The profinite group $G=\operatorname{Gal}(P|K)$ comes with the ramification filtration (in the upper numbering).

If $K',P',G'$ is another such triple, and if $\varphi:K\to K'$ is an isomorphism of local fields, then it induces an isomorphism of filtered pro-$p$-groups $G\to G'$.

Abrashkin's theorem says that, conversely, every isomorphism of filtered pro-$p$-groups $G\to G'$ comes from an isomorphism $K\to K'$ of local fields. In other words, the local field $K$ is completely determined by the filtered pro-$p$-group $G$. See Theorem A in his recent paper

This is a refinement of an earlier theorem of Mochizuki, who worked with $\operatorname{Gal}(\tilde K|K)$, where $\tilde K$ is a separable closure of $K$.

-
 Thanks for the update! In retrospect, I'm not sure it was good etiquette to edit right after you edited... But I do appreciate the interest (and the frequent revisions on this post)! – H. Hasson Feb 26 2010 at 5:48

It's hard to be helpful with such a vague question. (Compare: What is known about differentiability?) The first reference would have to be Serre's book Local Fields. For higher-dimensional varieties, Kazuya Kato has an expository paper Generalization of Class Field Theory, which has a section on higher-dimensional ramification theory.

-

There is the following work of Kerz and Saito:

http://arxiv.org/abs/1304.4400

In particular, one can view their Coro. II as a higher-dimensional analogue of the existence theorem, which encodes p-extensions (see Artin-Schreier-Witt theory), and hence wild ramification for varieties.

-