John Baez sent me a very nice note urging me to try to put all of this class field theory for functions fields stuff into a more coherent order. At the same time, I got into a discussion on the n-Category Cafe with David Ben-Zvi about geometric class field theory, which I always thought the QR stuff was an example of. So, here is my attempt to set this in a better order. I won’t be relying on the earlier post, although I hope it will be informative to try to relate the low tech nature of that one to the higher tech view this time round.
In the end, the key fact we use will be the following statement from David Ben-Zvi’s lecture, which is almost what he calls the “first form of Geometric Class Field Theory”.
Theorem 1 — Let C be an algebraic curve and H a finite abelian group. Then there is a natural correspondence between covers of C with Galois group H and covers of Pic C with Galois group H.
There are a number of caveats, all of which will go in part two of this post. For now, let me explain what I always thought the main result of geometric class field theory was:
Let C be any space (eventually it will be a punctured Riemman surface) and let be a Galois cover of C with Galois group H. (We do NOT allow branching, that’s why I am going to permit punctures.) Let C and D both come equipped with autmorphisms, both of which I will name F, so that is equivariant. Let C/F denote the set of finite orbits of F. Let be a point of C/F. Choose a point .
Exercise — there is some element such that . Moreover, up to conjugation, h is determined by o.
We denote the conjugacy class of h by Frob(o).
I pause for a moment to make the connection with QR for function fields. H is Z/2. C and D are the points of two curves defined over . The curve C has genus zero and D is obtained from C by adjoining the square root of some polynomial f. F is the Frobenius. C/F is the set of primes of the curve C.
Exercise — Frob(o) is the QR symbol (f/o).
Now, suppose that H is abelian. Than Frob(o) is a well defined element of H. Let Div(C/F) denote the free abelian group generated by C/F; note that we can also think of this as the F equivariant elements of Div(C). Then, using the abelian nature of H a second time, we can extend Frob to a map of abelian groups from Div(C/F) to H. This map is called the Artin map. Here is the main result of geometric class field theory, as I thought I knew it.
Theorem 2 — Suppose that C is a smooth punctured curve, with its projective completion. Then, if f is a meromorphic function on which is F-equivariant and such that f-1 vanishes to sufficient order at the points of , then .
In particular, let’s say what this means when C is projective. There won’t be any interesting examples when C has genus zero, because a projective genus zero curve is simply connected and has no nontrivial covers. So we won’t be able to make nice elementary statements in terms of polynomials. Nontheless, the simplification we obtain by making empty will be worthwhile:
Theorem 2′ — Impose the additional condition that C is projective. Let f be an F-invariant meromorphic function on C. Then .
This is as far as we’ll get starting with QR and trying to work forwards. If you want to see more detail on how QR relates to Theorem 2, please see the previous post. In the second half of this story, we’ll start with Theorem 1 and try to work towards theorem 2′.