Unlikely intersections in arithmetic dynamics

我的研究是在算术动力学领域，这是在几个数学领域的交叉点：数论，代数几何和代数动力学。更准确地说，我考虑的研究问题围绕着不太可能相交的原则。这一原理首先出现在算术几何中，它可以解释如下：给定一个周围代数簇X，我们定义特殊点和特殊子簇的概念;然后人们期望，如果X的子簇Y包含一个特殊点的Zebriki稠密集，这迫使子簇本身是特殊的。每一个著名的代数几何的莫德尔朗，马宁芒福德，博戈莫洛夫和安德烈奥尔特可以措辞使用上述术语的特殊点和特殊的子品种。 下面我们讨论不可能相交原理的两个例子。曲线情形下的动力学莫德尔-朗猜想预言如下：给定一个赋自同态f的拟投射簇X，给定X上的一个点x和一条包含在X中的曲线Y，如果x在f下的轨道与Y相交于无穷多个点，那么Y在f的作用下一定是周期的。换句话说，如果一个不太可能的事件（从x的轨道到曲线Y上的点）无限频繁地发生，那么这可以用全局条件（曲线Y的周期性）来解释。在这个例子中，X的特殊点是来自x的轨道的点，而特殊子簇是在f的作用下是周期的。 第二个例子是动态Manin-Mumford猜想。我们有一个被赋予自同态f（满足某些技术假设）的投射簇X。动态Manin-Mumford猜想预言，如果X的一个子簇Y包含一个前周期点的Zeroki稠密集，那么它在f的作用下自身一定是前周期的。这一次，特殊点是X在f作用下的预周期点，而特殊子簇是预周期点。同样，如果不太可能的交集（在一个给定的子簇Y和X的所有前周期点的集合之间）很大（这在几何上表示为在Y上存在这样的特殊点的一个Zebriki稠密集），那么这迫使Y是特殊的。 在过去，我们获得了重要的部分结果，对上述两种结构。此外，我们还证明了对该领域相关开放问题有影响力的定理，例如Andre-Oort猜想和有界高度猜想的动力学类似物。我们希望我们未来的研究结果将开辟新的研究途径，为算术几何世界和算术动力学世界之间的相似性提供进一步的证据，这两个世界都围绕着不太可能的交叉点的概念。

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.