I’ll start with thanking Google/Chrome. I sent in a request to fix the CSS rendering issue that happened with the blog and the next day it was fixed (maybe a coincidence?).
We start with our standing assumptions. Let be an elliptic curve with CM by for some quadratic imaginary (with normalized iso ). We will need the following fact. Suppose is defined over a number field . Let be a prime of and the reduction of mod . An element is in the image of the natural map if and only if commutes with every element in the image of the map.
The proof is a straightforward follow your nose type argument by cases. Caution: You must consider the case that is an order in a quaternion algebra because is defined over a field of positive characteristic. Recall that last time we said that is the Hilbert class field of . Thus without loss of generality we may assume from here on that is defined over (because ).
Here is something that seemed bizarre to me the first time I saw it. Let be a prime of degree of and a prime of lying over . Suppose has the property that the natural map is an isogeny of degree ( lies over ) and the reduction is purely inseparable. This happens for all but finitely many . Then there is an isogeny lifting the -th power Frobenius. This should feel funny because we’re lifting the Frobenius map to characteristic , i.e. the following diagram commutes:
The proof is to use that we already know there is some whose reduction is purely inseparable of degree and hence factors as where the second map of degree and hence an automorphism. We’re done if we can check that this automorphism lifts, but we can do this by checking that it commutes with everything in the image of by the above lemma. There’s quite a bit of work checking things element-wise to finish this off.
A special case is the following. For all but finitely many primes such that (i.e. is principal) the is unique such that the endomorphism descends to the -the power Frobenius.
We want to now think about generating abelian extensions of using the torsion points of . Fix a finite map defined over (called a Weber function for ). These are easy to come by. For example, if , then if our model is , then works. Here is another beautiful relation between CM theory and class field theory. Recall are the -torsion points with respect to the normalized map , i.e. the set of such that for all .
For any integral ideal of the field is the ray class field of of conductor . An immediate corollary is that . Hence we get the amazing fact that the maximal abelian extension of can be made by taking and adjoining to the numbers and the -coordinates of the torsion points of the Weierstrass model (as long as ). The proof of this is pretty long and involves all the stuff from earlier in the post, so we’ll skip it.
Last time we said that if we don’t just take -coordinates and instead adjoin all torsion coordinates we get which is an abelian extension of . It is interesting that doing this does not in general produce an abelian extension of . Another interesting corollary to this is that we get a situation in which we can adjoin all torsion points. Given any with class number we get a chain and hence .