16
$\begingroup$

Let $K$ be the imaginary quadratic field obtained by joining $\sqrt{-1}$ to the field of rational numbers $Q$. I would like to describe the extension $K^{ab}/Q^{ab}$, where for $F$ a number field, $F^{ab}$ denotes its maximal abelian extension (everything is taking place inside a big fixed field...).

More precisely I would like to know the Galois group and the ramification properties of such extension. Is this possible/easy? I suppose one should look at the kernel of the norm map between Idele class groups $N_{K/Q}:I_K\rightarrow I_Q$. But at the moment it is not clear to me how to get the answer. Any hint or comment would be appreciated. Thanks.

EDIT: Probably the idele class group of a number field $F$ should be denoted by $J_F$. Or by anything other than $I_F$...

$\endgroup$

3 Answers 3

16
$\begingroup$

Given that you want to know the structure of the Galois group and ramification, I think that you are best off working with the kernel of the norm map between connected components of idele class groups, as you yourself suggest.

These groups are very explicit: for $K := \mathbb Q(i)$, one obtains $\hat{\mathcal O}_K^{\times}/\{\pm 1,\pm i\}$, and for $\mathbb Q$ one obtains $\hat{\mathbb Z}.$ (Here $\hat{}$ denotes the profinite completion.) Apart from the diagonally embedded $\{\pm 1,\pm i\}$ quotient in the group for $K$, both groups factor as a product over primes, and the norm map is given component wise.

So the kernel of the norm map is equal to $$\bigl(\prod_p (\mathcal O_K\otimes_{\mathbb Z}\mathbb Z_p)^{\times, \text{Norm } = 1}\bigr)/ \{\pm 1,\pm i\}.$$

This should be explicit enough to answer any particular question you have.

$\endgroup$
2
  • 1
    $\begingroup$ That's really nice! I feel slightly embarrassed to admit it, but I had never thought about it this way: your local factors then give a very explicit description of the subgroup of image of Galois in $\text{GL}_2(\mathbb{Z}_p)$ on the Tate modules of the elliptic curve that fixes the cyclotomic extension. I somehow always assumed that this subgroup of $\text{GL}_2(\hat{\mathbb{Z}})$ would be much messier to write down. $\endgroup$
    – Alex B.
    Oct 20, 2010 at 2:40
  • $\begingroup$ @Alex: I do not understand your comment. What does the Tate module of some elliptic curve have to do with the group written by Emerton above? $\endgroup$
    – unknown
    Oct 21, 2010 at 1:10
12
$\begingroup$

In your particular case, $K^{ab}$ is completely understood, but your field is one of the very few for which such an explicit class field theory is known, so you got lucky.

I don't know your background, but to understand the answer you need to know something about the theory of complex multiplication. What I am going to say works for any imaginary quadratic field. The field $K^{ab}$ is generated by so-called Weber functions, usually just given by the $x$-coordinates of torsion points on any elliptic curve that has complex multiplication by the ring of integers of $K$. Actually, in your particular case, you are looking e.g. at the elliptic curve $y^2 = x^3+x$ and the maximal abelian extension is just generated by the torsion points (always true when $K$ has class number one).

You can read up on this in Silverman's "Advanced Topics in the Arithmetic of Elliptic Curves", Chapter II. Have a look particularly at example 5.8.1.

$\endgroup$
11
$\begingroup$

Use the theory of complex multiplication. $K^{ab}$ is the field generated by the torsion points of $y^2=x^3+x$.

$\endgroup$
1
  • $\begingroup$ Thanks! Can I then answer my question with this information? $\endgroup$
    – unknown
    Oct 20, 2010 at 0:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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