Unlikely intersections in arithmetic dynamics

我的研究领域是算术动力学，它是数论、代数几何和代数动力学几个数学领域的交叉点。更准确地说，我所考虑的研究问题围绕着不可能交集的原则。这个原理最早出现在算术几何中，它可以这样解释：给定一个环境代数变量X，我们定义了特殊点和特殊子变量的概念；那么我们期望，如果一个X的子变种Y包含特殊点的Zariski稠密集，这就迫使这个子变种本身是特殊的。莫德尔-朗、曼宁-芒福德、波格莫洛夫和安德烈-奥尔特在算术几何中的每一个著名猜想都可以用上述特殊点和特殊子变种的术语来表述。

My research is in the field of arithmetic dynamics, which is at the intersection of several mathematical areas: number theory, algebraic geometry and algebraic dynamics. More precisely, the research questions I consider revolve around the principle of unlikely intersections. This principle first appeared in arithmetic geometry and it can be explained as follows: given an ambient algebraic variety X, we define the notion of special points and of special subvarieties; then one expects that if a subvariety Y of X contains a Zariski dense set of special points, this forces the subvariety to be itself special. Each of the famous conjectures in arithmetic geometry of Mordell-Lang, Manin-Mumford, Bogomolov and of Andre-Oort can be phrased using the above terminology of special points and special subvarieties. We discuss below two of the instances of this principle of unlikely intersections. The Dynamical Mordell-Lang Conjecture in the case of curves predicts the following: given a quasiprojective variety X endowed with an endomorphism f, given a point x on X and a curve Y contained in X, if the orbit of x under f intersects Y in infinitely many points, then Y must be periodic under the action of f. In other words, if an unlikely event (which is the landing on the curve Y of a point from the orbit of x) occurs infinitely often, then this is explained by a global condition (which is the periodicity of the curve Y). In this example, the special points of X are the points from the orbit of x, while the special subvarieties are the ones which are periodic under the action of f. Our second example is the Dynamical Manin-Mumford Conjecture. We have a projective variety X endowed with an endomorphism f (which satisfies certain technical hypotheses). The Dynamical Manin-Mumford Conjecture predicts that if a subvariety Y of X contains a Zariski dense set of preperiodic points, then it must be itself preperiodic under the action of f. This time, the special points are the preperiodic points of X under the action of f, while the special subvarieties are the preperiodic ones. Again, if the unlikely intersection (between a given subvariety Y and the set of all preperiodic points of X) is large (which geometrically is expressed by the existence of a Zariski dense set of such special points on Y), then this forces Y to be special. In the past we obtained important partial results towards both conjectures described above. Also, we proved impactful theorems towards related open questions in the field, such as the dynamical analogues of the Andre-Oort Conjecture and of the Bounded Height Conjecture. We hope our future results will open new avenues of research, providing further evidence of the similarities between the world of arithmetic geometry and the world of arithmetic dynamics, both worlds revolving around the concept of unlikely intersections.