Unlikely intersections in arithmetic dynamics

我的研究领域是算术动力学，这是几个数学领域的交集：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。这一原理最早出现在算术几何中，它可以解释为：给定一个环境代数簇X，我们定义了特殊点和特殊子簇的概念；然后人们期望，如果X的一个子簇Y包含一个Zariski稠密的特殊点集，这就迫使这个子簇本身是特殊的。Mordell-lang、Manin-Mumford、Bogomolov和Andre-Oort的算术几何中的每一个著名猜想都可以用上面的特殊点和特殊子簇来表述。 下面我们讨论这一不太可能相交原理的两个例子。在曲线情形下的动力学Mordell-Lang猜想预言：给定一个具有自同态f的拟投射簇X，给定X上的一个点X和X中包含的一条曲线Y，如果x在f下的轨道与Y在无限多个点上相交，则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(即某一点从x的轨道着陆在曲线Y上)无限频繁地发生，那么这可以用一个整体条件(即曲线Y的周期)来解释。在这个例子中，X的特殊点是X的轨道上的点，而特殊的子簇是在f的作用下是周期性的。 我们的第二个例子是动力学Manin-Mumford猜想。我们有一个射影簇X，它有一个自同态f(它满足某些技术假设)。动力Manin-Mumford猜想预言，如果X的一个子簇Y包含一个预周期点的Zariski稠密集，那么它一定是在f的作用下它自身是准周期的。这一次，特定点是X在f作用下的预周期点，而特殊的子簇是准周期的。同样，如果不太可能的交集(给定的子簇Y和X的所有准周期点集之间)很大(几何上由Y上这样的特殊点的Zariski稠密集的存在来表示)，那么这迫使Y是特殊的。 在过去，我们对上述两个猜想都得到了重要的部分结果。此外，我们还证明了该领域相关公开问题的有效定理，如Andre-Oort猜想和有界高度猜想的动力学类比。我们希望我们未来的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可能相交的概念。

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.