Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Fix a finite set $S$ of places of $\mathbb Q$. Let $G_{\mathbb Q,S}$ be the Galois group of the maximal extension of $\mathbb Q$ unramified outside S$. I believe that it is an open question whether this group is topologically finitely generated, that is, contains a dense, finitely generated subgroup.

Is there a standard conjecture about whether $G_{\mathbb Q,S}$ should be finitely generated or not?
Has anyone published an opinion?

JSE mentions a stronger conjecture: $|S|$ generators suffice; further, the existence of a special set of generators labeled by $S$ and inert at the labeling place. Is this refinement standard? (Note that a finite extension would suffice to disprove this hypothesis.)

Weakening the hypothesis, we could restrict the ramification at $S$ to be tame.

share|improve this question
2  
This seems to be an old conjecture of Shafarevich. This much is mentioned in Barry Mazur's article in "Modular Forms and Fermat's Last Theorem." –  Vesselin Dimitrov Sep 24 '12 at 19:11

1 Answer 1

up vote 4 down vote accepted

I think you can look at page 532 of (the first version of) J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, Springer, 1999. They explicitly write that we do not even know precisely what to conjecture for arbitrary number field. They also add "many mathematicians (including the authors) tend to think of $G_S$ as being not finitely generated". In their notation, $G_S$ is the Galois group of the maximal unramified outside of $S$ extension of a number field $K$ which has been fixed once and for all (and $S$ is finite). I do not know if the special case $K=\mathbb{Q}$ is somehow different, but I doubt it (after all, their $G_S$ is a subgroup of your $G_{\mathbb{Q},S'}$ for $S'=S\cap\mathrm{Ram}(K/\mathbb{Q})$...)

share|improve this answer
    
For the second edition, it's on page 623. –  Kevin Ventullo Nov 6 '12 at 5:53
    
WOW! Ninety pages more. I hope they are not all taken by the preface to the second edition... –  Filippo Alberto Edoardo Nov 6 '12 at 9:53
    
Surprising. I thought that it was widely believed that those groups were finitely generated. –  Joël Nov 6 '12 at 15:11
    
One (secondary) question remains: was the finite generation really a conjecture of Shafarevich (and then, where), or is Mazur's saying so in "modular forms and Fermat's last theorem" a mistake ? –  Joël Nov 6 '12 at 15:13
    
I honestly do not know. Do you have a precise reference for Mazur saying so, is it in the "Deformations" paper? –  Filippo Alberto Edoardo Nov 6 '12 at 15:31

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.