We prove that hyperelliptic curves over Q of genus at least 2 have only finitely many rational points. Actually, we prove this for a more general class of high-genus curves over Q, including all solvable covers of P^1.
But wait, don’t we already know that, by Faltings? Of course we do. So the point of the paper is to show that you can get this finiteness in a different way, via the non-abelian Chabauty method pioneered by Kim. And I think it seems possible in principle to get Faltings for all curves over Q this way; though I don’t know how to do it.
Chabauty’s theorem tells you that if X/K is a curve over a global field, and the rank of J(X)(K) is strictly smaller than g(X), then X(K) is finite. That theorem statement is superseded by Faltings, but the method is not; and it’s been the subject of active interest recently, in part because Chabauty gives you more control over the number of points. (See e.g. this much-lauded recent paper of Katz, Rabinoff, and Zureick-Brown.)
But of course the Chabauty condition on the Mordell-Weil rank of J(X) isn’t always satisfied. So what do you do with curves outside the Chabauty range? A natural and popular idea is to exploit an etale cover Y -> X. All the rational points of X are covered by those of a finite list of twists of Y; so if you can show all the relevant twists of Y satisfy Chabauty, you get finiteness of X(K). And that seems pretty reasonable; there’s no reason two distinct points of X should lie on the same twist Y_e of Y, and once a twist Y_e has only one rational point, the stuff in J(Y) that’s not in J(X) doesn’t have any reason to have large Mordell-Weil rank. Even if J(X) has pretty big rank, it should get “diluted” by all that empty Prym-like space.
About 15 years ago (amazing how projects that old still feel like “things I was recently thinking about”) I spent a lot of time working on this. What I wanted to show was something like this: if X is a curve (let’s say with a given rational basepoint), there’s a natural etale cover X_n obtained by mapping X to its Jacobian and then pulling back the multiplication map [p^n]:J(X) -> J(X). In fact, the X_n fit together into a tower of curves over X. Is it the case that
for n large enough? And what about the twists of X_n? The idea was to work at the “top” of the tower, where you have actions of lots of p-adic groups; the action of T_p J(X) by deck transformations, the action of some symplectic quotient of Gal(K) on T_p J(X), making the Selmer group at the top a great big non-abelian Iwasawa module. I still think that makes sense! But I ended up not being able to prove anything good about it. (This preprint was as far as I got.)
That project was about abelian covers (the Jacobians) of abelian covers of X; so it was really about the metabelian quotient of the fundamental group of X. What Minhyong Kim did was quite different, working with the much smaller nilpotent quotient; and it turns out that here you can indeed show that, for certain X, some version of Chabauty applies with Jac(X) replaced by a “unipotent Albanese,” which is to the quotient of pi_1(X) by some term of the lower central series as Jac(X) is to the Jacobian. Very very awesome.
What are “certain X”? Well, Kim’s method applies to all curves whose Jacobians have CM, as proved in this paper by him and Coates.
Most hyperelliptic curves don’t have CM Jacobians. But now the etale cover trick comes to the rescue, because, by an offbeat result of Bogomolov and Tschinkel I have always admired, every hyperelliptic curve X has an etale cover Y which geometrically dominates the CM genus 2 curve with model y^2 = x^6 – 1! So the main point of our paper is to generalize the argument of Coates and Kim to apply to curves whose Jacobian has a nontrivial CM part. (This is familiar from Chabauty; you don’t actually need the Jacobian to have small rank, it’s enough for just one chunk of it to have small rank relative to its dimension.) Having done this, we get finiteness for Y and all its twists, whence for X.
There are further results by Poonen of Bogomolov-Tschinkel flavor; these allow us to go from hyperelliptic curves to a more general class of curves, including all curves which are solvable covers of P^1. But of course here’s the natural question:
Question: Does every algebraic curve over Q admit an etale cover which geometrically dominates a curve Y whose Jacobian has CM?
An affirmative answer would extend the reach of non-abelian Chabauty to all curves over Q, which would be cool!