Only two points

0:00Two points

Here is a curve: x + y = 1, where x and y are complex numbers. Look for points on it where x and y are both roots of unity, numbers whose powers eventually return to 1. There are infinitely many roots of unity, so you might expect many such points. There are exactly two. Here’s why. A root of unity has size one. So x lies on the unit circle around 0, and since y is 1 − x, x also lies on the unit circle around 1. Those circles cross at just two points, e^(±iπ/3), and both are sixth roots of unity. A coincidence like this is called an unlikely intersection. Result 016 in OpenAI’s math repository claims two major cases of the central conjecture about them: the Zilber–Pink conjecture.

0:53Counting dimensions

Count dimensions. Inside a space of dimension n, a piece of dimension a and a piece of dimension b should meet in dimension a + b − n. Our curve and a single point made of roots of unity, in the two-dimensional space of pairs x, y, should meet in dimension −1: not at all. When they meet anyway, the intersection is atypical. The pieces we test against are special: subgroups, shifted by points of finite order. The Zilber–Pink conjecture, from Boris Zilber and Richard Pink, building on Bombieri, Masser and Zannier, says that any subvariety has only finitely many maximal atypical pieces. Its simplest case, for points of finite order, is the Manin–Mumford conjecture. For pairs of nonzero complex numbers, Serge Lang proved it in 1965, crediting Ihara, Serre and Tate, and Michel Raynaud proved it for abelian varieties in 1983.

1:52Abelian varieties

Our opening example lives among pairs of nonzero numbers. Abelian varieties are the higher-dimensional cousins of elliptic curves: complex tori with a group law, defined by polynomial equations. For curves inside them, Philipp Habegger and Jonathan Pila proved Zilber–Pink in 2016. For higher-dimensional subvarieties, it was known only under extra conditions. The first paper claims the full abelian case over the algebraic numbers: for every abelian variety and every subvariety, only finitely many maximal atypical subvarieties, measured inside the smallest special subvariety that contains it.

2:33The Siegel threefold

The other three papers work in a moduli space. Each point of the Siegel threefold, A₂, is a principally polarized abelian surface. Special points are surfaces with complex multiplication. Special curves are families with extra symmetry of three kinds: a product with a CM elliptic curve, the square of an elliptic curve, or multiplication by a quaternion division algebra. A curve and a special curve in a three-dimensional space should not meet. Zilber–Pink predicts that a Hodge-generic curve, one not trapped in a smaller special surface, meets all special curves together, every one of infinitely many, in only finitely many points. Pila and Tsimerman handled special points in 2013. Christopher Daw and Martin Orr proved the curve case for curves that degenerate at the boundary in a particular way. The three papers claim the full curve case over the algebraic numbers, with no boundary assumption: one paper for each kind of special curve.

3:33How the proofs go

For abelian varieties, the paper first proves that the atypical points are not dense, then uses a reduction of Barroero and Dill to get finiteness. The engine is height theory, in the tradition of Vojta and Faltings, kept uniform across several scales at once. For the Siegel threefold, the strategy is Pila and Zannier’s. If a curve met special curves infinitely often, the meeting points would bring many Galois conjugates with them, and those would give too many rational points on a definable set, contradicting the Pila–Wilkie counting theorem. The missing piece was a height bound, to make the Galois orbits large. Two of the papers get it from period equations at many primes at once, centered at a point inside the curve rather than at the boundary. The E×CM paper uses a different period estimate. Either way, no boundary assumption is needed.

4:29The caveats

The caveats. These are four preprints, 170 pages, not peer reviewed, and with no Lean proof. The Siegel results are over the algebraic numbers, and none of the papers gives an explicit bound on how many atypical pieces there are. Zilber–Pink in general, for tori in higher dimensions and for other Shimura varieties, remains open. x + y = 1 has just two points made of roots of unity. Coincidences like that are rare, in every abelian variety, and on every Hodge-generic curve in A₂. If the proofs hold.