Unlikely Intersections in Arithmetic Dynamics

Combining ideas of Ihara-Serre-Tate, Lang [5] proved the following natural result. If a (complex, irreducible) plane curve C ⊂ A contains infinitely many points with both coordinates roots of unity, then C is the zero locus of an equation of the form xy = ζ, where a, b ∈ Z and ζ is a root of unity. In other words, if F ∈ C[x, y] is an irreducible polynomial for which there exist infinitely many pairs (μ, ν) of roots of unity such that F (μ, ν) = 0, then (modulo multiplying by a constant) F (x, y) is either a polynomial of the form x − ζy, or of the form xy − ζ, where a and b are non-negative integers and ζ is a root of unity. In particular, Lang’s result [5] provided the first instance when the Manin-Mumford Conjecture was proven: if a curve C ⊂ Gm contains a Zariski dense set of torsion points, then C is a torsion translate of a 1-dimensional torus. The proof of Ihara-Serre-Tate-Lang is a clever combination of various tools from mathematics (not only from number theory, but even basic complex analysis is used); for example, any graduate student in mathematics would benefit from reading their proof for the special case when the curve C is the graph of a polynomial. The result of [5] is the first instance of the principle of unlikely intersections in arithmetic geometry since Lang’s result may be interpreted as follows: if an unlikely event (such as the existence on the plane curve C of a point with both coordinates roots of unity) occurs infinitely often, then this must be explained by a global, geometric condition satisfied by C (in this case, C is a translate of a 1-dimensional algebraic subgroup of Gm by a torsion point). Other famous conjectures in number theory, such as the Mordell-Lang, the Bogomolov, or the André-Oort conjectures may be coined in the same terminology of unlikely intersections, which is also paraphrased as “special points and special subvarieties” since the special points in this case are the ones with both coordinates roots of unity and the only subvarieties containing a Zariski dense set of special points are the special subvarieties, which are torsion translates of algebraic subgroups. It is also natural to formulate a dynamical analogue of Lang’s result [5] by interpreting the roots of unity as preperiodic points under the action of the squaring map z 7→ z. We recall that given a self-map f on any set X, a point a is preperiodic if its orbit under the action of z 7→ f(z) is finite, i.e., f(a) = f(a) for some positive integers m < n (where in dynamics, unless otherwise noted, f ` always represents the `-th compositional iterate of f for each positive integer `). So, the dynamical reformulation of Lang’s result yields the following: if a plane curve C contains infinitely many preperiodic points for the map F : A −→ A given by F (x, y) = (x, y), then C must be preperiodic under the action of F . Furthermore, one can replace the squaring map by an arbitrary polynomial (or more generally a rational function) and ask whether the same result holds; this was conjectured in