BSD for a Large Class of Elliptic Curves

I’m giving up on the p-divisible group posts for awhile. I would have to be too technical and tedious to write anything interesting about enlarging the base. It is pretty fascinating stuff, but not blog material at the moment.

I’ve been playing around with counting fibration structures on K3 surfaces, and I just noticed something I probably should have been aware of for a long time. This is totally well-known, but I’ll give a slightly anachronistic presentation so that we can use results from 2013 to prove the Birch and Swinnerton-Dyer conjecture!! … Well, only in a case that has been known since 1973 when it was published by Artin and Swinnerton-Dyer.

Let’s recall the Tate conjecture for surfaces. Let {k} be a finite field and {X/k} a smooth, projective surface. We’ve written this down many times now, but the long exact sequence associate to the Kummer sequence

\displaystyle 0\rightarrow \mu_{\ell}\rightarrow \mathbb{G}_m\rightarrow \mathbb{G}_m\rightarrow 0

(for {\ell\neq \text{char}(k)}) gives us a cycle class map

\displaystyle c_1: Pic(X_{\overline{k}})\otimes \mathbb{Q}_{\ell}\rightarrow H^2_{et}(X_{\overline{k}}, \mathbb{Q}_\ell(1))

In fact, we could take Galois invariants to get our standard

\displaystyle 0\rightarrow Pic(X)\otimes \mathbb{Q}_{\ell}\rightarrow H^2_{et}(X_{\overline{k}}, \mathbb{Q}_\ell(1))^G\rightarrow Br(X)[\ell^\infty]\rightarrow 0

The Tate conjecture is in some sense the positive characteristic version of the Hodge conjecture. It conjectures that the first map is surjective. In other words, whenever an {\ell}-adic class “looks like” it could come from an honest geometric thing, then it does. But if the Tate conjecture is true, then this implies the {\ell}-primary part of {Br(X)} is finite. We could spend some time worrying about independence of {\ell}, but it works, and hence the Tate conjecture is actually equivalent to finiteness of {Br(X)}.

Suppose now that {X} is an elliptic K3 surface. This just means that there is a flat map {X\rightarrow \mathbb{P}^1} where the fibers are elliptic curves (there are some degenerate fibers, but after some heavy machinery we could always put this into some nice form, we’re sketching an argument here so we won’t worry about the technical details of what we want “fibration” to mean). The generic fiber {X_\eta} is a genus {1} curve that does not necessarily have a rational point and hence is not necessarily an elliptic curve.

But we can just use a relative version of the Jacobian construction to produce a new fibration {J\rightarrow \mathbb{P}^1} where {J} is a K3 surface fiberwise isomorphic to {X}, but now {J_\eta=Jac(X_\eta)} and hence is an elliptic curve. Suppose we want to classify elliptic fibrations that have {J} as the relative Jacobian. We have two natural ideas to do this.

The first is that etale locally such a fibration is trivial, so you could consider all glueing data to piece such a thing together. The obstruction will be some Cech class that actually lives in {H^2(X, \mathbb{G}_m)=Br(X)}. In fancy language, you make these things as {\mathbb{G}_m}-gerbes which are just twisted relative moduli of sheaves. The class in {Br(X)} is giving you the obstruction the existence of a universal sheaf.

A more number theoretic way to think about this is that rather than think about surfaces over {k}, we work with the generic fiber {X_\eta/k(t)}. It is well-known that the Weil-Chatelet group: {H^1(Gal(k(t)^{sep}/k(t), J_\eta)} gives you the possible genus {1} curves that could occur as generic fibers of such fibrations. This group is way too big though, because we only want ones that are locally trivial everywhere (otherwise it won’t be a fibration).

So it shouldn’t be surprising that the classification of such things is given by the Tate-Shafarevich group:

Ш \displaystyle (J_\eta /k(t))=ker ( H^1(G, J_\eta)\rightarrow \prod H^1(G_v, (J_\eta)_v))

Very roughly, I’ve now given a heuristic argument (namely that they both classify the same set of things) that {Br(X)\simeq} Ш {(J_\eta)}, and it turns out that Grothendieck proved the natural map that comes form the Leray spectral sequence {Br(X)\rightarrow} Ш{(J_\eta)} is an isomorphism (this rigorous argument might actually have been easier than the heuristic one because we’ve computed everything involved in previous posts, but it doesn’t give you any idea why one might think they are the same).

Theorem: If {E/\mathbb{F}_q(t)} is an elliptic curve of height {2} (occuring as the generic fiber of an elliptic K3 surface), then {E} satisfies the Birch and Swinnerton-Dyer conjecture.

Idea: Using the machinery alluded to before, we spread out {E} to an elliptic K3 surface {X\rightarrow \mathbb{P}^1} over a finite field. As of this year, it seems the Tate conjecture is true for K3 surfaces (the proofs are all there, I’m not sure if they have been double checked and published yet). Thus {Br(X)} is finite. Thus Ш{ (E)} is finite. But now it is well-known that if Ш{ (E)} being finite is equivalent to the Birch and Swinnerton-Dyer conjecture.

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