OK, in our previous post I stated
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.
And then I said a bunch of stuff that sounded nothing like Theorem 1. What’s the connection?
Well, time for some caveats and remarks. First, a change of nomenclature. Instead of saying “Galois cover with Galois group H”, I’ll say “H-local system”. People who don’t already know what local systems are should take this as a definition. Second, notice that if I had said instead of Pic C, this would be easy. That’s because H-local systems are classified by , where the second formula is because H is abelian. But, by the Abel-Jacobi theorem, . A similar trick would work if I talked about for any r. On the other hand, I did say Pic(C) and I meant it. But I don’t actually mean that every H-local system on Pic(C) corresponds to one on C because, when building an H-local system on Pic(C) I have an infinite number of independent choices of what to do on each component. I only want to talk about the H-local systems where I make compatible choices on each component. That brings me to a definition. This definition isn’t quite the right one categorically, but it will be good enough for now.
Definition: A good H-local system on Pic C is an H-local system L with the folllowing property: let x be any point of C and let be the map of multiplication by C. Then .
This definition relates the behavior of L on and on . If I wanted to be really non-categorical, I could just say that I have isomorphisms by m(x), and I have taken the same element of on each component. I can now give a statement of Theorem 1 which contains the word bijection.
Theorem 1′ Consider the restriction map from Pic C to C along the standard inclusion . This is a natural bijection between good H-local systems on Pic C and H-local systems on C.
The word “natural” is doing a lot of work here. I won’t spell out the details, but if you take a look at our argument, you’ll see that I need to consider how nontrivial automorphisms of C affect this whole picture.
Ok, now on with the argument. Our first order of business is to give a better description of good H-local systems. I said that . We’ll need to understand what the isomorphism is. Choose some point x in C and some particular isomorphism . Now, let x move around a loop x(t) in C. We can naturally identify all of the different H-local systems with each other “by continuity”. But when we come back to x again, I will have a different identification of with — more precisely, it will have changed by the same element of H that we picked up in monodromy travelling through C.
We can say this in a more universal fashion, without choosing a basepoint in C to start at. Let be the multiplication map . For two H-local systems and on bases and , define to be the H-local system on whose fiber over is , where is the relation . Then the result of the previous paragraph is
In particular, let f be a rational function on C, with poles and zeroes . Then and are the same point of Pic(C); call this point x. Let D be an H-local system on C and let L be the corresponding good H-local system on Pic C. Then we get natural identifications
Now, we are finally close to seeing Theorem 2′. Remember Theorem 2′?
Theorem 2′ — Impose the additional condition that C is projective. Let f be an F-invariant meromorphic function on C. Then .
I’ve been sloppy about naturality throughout this discussion. But if I had been careful, you would have seen that the isomorphism between and was functorial enough that F acts naturally on both sides by the same element of H. But now, a final exercise for the reader: F acts on the tensor product of the by and F acts on the tensor product of the by . So as desired.
Let me make a few more remarks. How does this relate to the argument I tried to describe before? Before, I essentially embedded the H-local system D into a group G. This meant that, when F acted on D, I could talk about it multiplying any point of D by an element of G. In particular, I could talk about which element of G acted on each of the and . In the description I gave here, there is no action of F on the individual , only on their tensor product. But, if you are careful with the category theory, this is all you actually need.
Second, what if you want to prove the original Theorem 2 instead of Theorem 2′? Then, instead of Pic C, you need the generalized Jacobians of C. Let C be a smooth curve and it projective completion, with . Let , …, be nonnegative integers. Then the generalized Jacobian of is the quotient of the free abelian group generated by the points of C by the relations whenever and are the zeroes and poles of a meromorphic function f on C with f-1 vanishing to order at . Like the Picard group, the generalized Jacobian has the structure of an algebraic group. Presumably, one needs to state and prove an analogue of Theorem 1 relating H-local systems on punctured curves to H-local systems on Jacobians.
Finally, what would we need to make an argument like this work for number fields? I don’t really know, but here is a vague idea. It seems to me that part of the problem is that there isn’t really a number field analogue of C; only of C/F. Each “point” of a number field comes equipped with its own Frobenius and you can’t really think of them all as being the same automorphism or something. This reminds me, a little, of the existence of stacks which are not global quotients. But I have no idea whether that analogy is going anywhere…