QR and Geometric CFT II

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 Pic^0 C instead of Pic C, this would be easy. That’s because H-local systems are classified by Hom(\pi_1(C), H)=Hom(H_1(C, Z), H), where the second formula is because H is abelian. But, by the Abel-Jacobi theorem, H_1(Pic^0 C, Z) \cong H_1(C, Z). A similar trick would work if I talked about Pic^r C 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 m(x): Pic C \to Pic C be the map of multiplication by C. Then m(x)_* L \cong L.

This definition relates the behavior of L on Pic^{r-1} C and on Pic^r C. If I wanted to be really non-categorical, I could just say that I have isomorphisms H_1(Pic^{r-1} C) \cong H_1(Pic^{r} C) by m(x), and I have taken the same element of Hom(H_1(Pic^r C), H) 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 C \to Pic^1 C. 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 m(x)_* L \cong L. We’ll need to understand what the isomorphism is. Choose some point x in C and some particular isomorphism m(x)_* L \cong L. Now, let x move around a loop x(t) in C. We can naturally identify all of the different H-local systems m(x(t))_* L with each other “by continuity”. But when we come back to x again, I will have a different identification of m(x)_* L with L — 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 \mu be the multiplication map C \times Pic C \to Pic C. For two H-local systems L_1 and L_2 on bases B_1 and B_2, define L_1 \otimes_H L_2 to be the H-local system on B_1 \times B_2 whose fiber over (x_1, x_2) is ((B_1)_{x_1} \times (B_2)_{x_2})/\sim, where \sim is the relation (h y_1, y_2)=(y_1, h y_2). Then the result of the previous paragraph is
\mu^* L=L|_C \otimes_H L

In particular, let f be a rational function on C, with poles f^{-1}(\infty)=\{p_1, \dots, p_r \} and zeroes f^{-1}(0)=\{z_1, \dots z_r\}. Then p_1+ \ldots + p_r and z_1 + \ldots + z_r 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
L_x \cong L_{e} \otimes_H D_{z_1} \otimes_H \cdots \otimes_H D_{z_r} \cong L_{e} \otimes_H D_{p_1} \otimes_H \cdots \otimes_H D_{p_r}

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 Artin(f^{-1}(0))=Artin(f^{-1}(\infty)).

I’ve been sloppy about naturality throughout this discussion. But if I had been careful, you would have seen that the isomorphism between L_{e} \otimes_H D_{z_1} \otimes_H \cdots \otimes_H D_{z_r} and L_{e} \otimes_H D_{p_1} \otimes_H \cdots \otimes_H D_{p_r} 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 D_{z_i} by Artin(f^{-1}(0)) and F acts on the tensor product of the D_{p_i} by Artin(f^{-1}(\infty)). So Artin(f^{-1}(0))=Artin(f^{-1}(\infty)) 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 D_{z_i} and D_{p_i}. In the description I gave here, there is no action of F on the individual D_{z_i}, 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 \bar{C} it projective completion, with \bar{C} \setminus C= \{ x_1, \dots, x_k \}. Let m_1, …, m_k be nonnegative integers. Then the generalized Jacobian of (C, m) is the quotient of the free abelian group generated by the points of C by the relations z_1 + \dots + z_k=p_1 + \dots + p_k whenever z_i and p_i are the zeroes and poles of a meromorphic function f on C with f-1 vanishing to order m_i at x_i. 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…