The Dynamical Andre-Oort Conjecture for cubic polynomials

Abstract

In the moduli space of degree d polynomials, the special subvarieties are those cut out by critical orbit relations, and then the special points are the post-critically finite polynomials. It was conjectured that in the moduli space of degree d polynomials, the subvarieties containing a Zariski-dense set of special points are exactly these special subvarieties. In this article, we prove the first non-trivial case for this conjecture: the case of cubic polynomials.

At the time we were finishing writing this article, we learned that Charles Favre and Thomas Gauthier have independently obtained a proof of our main theorem using a different approach; we are grateful to both of them for sharing with us their preprint