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?



Wild Ramification
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


