In my class on modularity, I decided to explain what Wiles’ argument (in the minimal case) would look like for . There are two ways one can go with this. On the one hand, one can try to prove (say) Kronecker-Weber using Selmer groups, but avoiding any kind of circularity (by not assuming class field theory). On the other hand, one can allow oneself to be completely circular in an effort to concentrate on the technical details of Wiles’ arguments. This post concerns the latter, and we “prove” the following:
Theorem: Let be a number field which does not contain . Then the Galois group of the maximal abelian extension of unramified everywhere is isomorphic to the -part of the class group.
To prove this, we will (only) assume the following:
The first three assumptions for are equivalent to giving oneself the cyclotomic extensions and understanding their ramification properties. The last assumption, of course, contains every part of global class field theory (making the argument circular).
Let denote the Galois group of the maximal pro- abelian unramified extension of .
Let denote the corresponding group where ramification is allowed at some set of primes not containing , where one also insists that the order of inertia at primes in is at most . Formally, we have the universal deformation rings
and we also have the universal “modular” deformation rings
Here $M^F$ denotes the biggest finite quotient of $M$, is the obvious maximal open compact, and is the variant of such that is replaced by for . The half of global class field theory we are assuming gives us a compatible diagram of maps and . The Wiles-Greenberg formula gives us an equality:
where for this computation we use that . In order to annihilate the dual Selmer group, we need to annihilate classes in , which come from extensions . We can do this in the usual way, but we have to assume (again) that , since otherwise one cannot annihilate the class defined over using a prime . We see that we can annihilate the dual Selmer group with primes. What are the auxilary rings here? As rings, they are
the action on is via the inertia group at the auxiliary primes. To make this work, one needs local class field theory; this shows that inertia at is acting via . The action of on is given by the structure of as a module over . The compatibility of these actions is given by the compatibility of local and global class field theory. Moreover, if is the augmentation ideal of , then by definition, and by construction. Thus, in the usual way, one ends up with a map where:
edit: In order to see the “circularity” more clearly, one may compute the Selmer groups directly. The group is equal to , by definition. On the other hand, the group by the Kummer sequence is equal to , and thus the Greenberg-Wiles formula is equivalent to the equality:
or equivalently the claim that the maximal exponent -quotient of the class group captures all exponent -unramified extensions. (I guess this is very very slightly weaker than ).