Unlikely intersections in arithmetic dynamics

算术动力学中不太可能的交叉点

• 批准号: RGPIN-2018-03690
• 负责人: Ghioca, Dragos
• 金额: $ 2.04万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=682950
• 关键词: Unlikely; intersections; arithmetic; dynamics

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.

我的研究领域是算术动力学领域，它是数论、代数几何和代数动力学等几个数学领域的交叉点。更准确地说，我考虑的研究问题围绕着不可能交叉的原则。这个原理首先出现在算术几何中，可以解释如下：给定一个环境代数簇 X，我们定义特殊点和特殊子簇的概念；那么人们期望如果 X 的子品种 Y 包含特殊点的 Zariski 稠密集，则这会迫使该子品种本身是特殊的。 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使用上述特殊点和特殊子簇的术语来表达。 ******我们在下面讨论这个不可能交叉原则的两个实例。曲线情况下的动力学莫代尔-朗猜想预测如下：给定一个赋予自同态 f 的拟射影簇 X，给定 X 上的点 x 和 X 中包含的曲线 Y，如果 f 下 x 的轨道与 Y 相交于无穷多个点，则 Y 在 f 的作用下必定是周期性的。换句话说，如果一个不太可能发生的事件（即 x 轨道上的点落在 Y 曲线上）无限频繁地发生，那么这可以通过全局条件（即曲线 Y 的周期性）来解释。在这个例子中，X的特殊点是来自x轨道的点，而特殊子簇是在f作用下具有周期性的点。 *****我们的第二个例子是动力学马宁-芒福德猜想。我们有一个具有自同态 f 的射影簇 X（满足某些技术假设）。动态马宁-芒福德猜想预测，如果 X 的子品种 Y 包含 Zariski 前周期点的稠密集，那么在 f 的作用下它本身一定是前周期的。这次，特殊点是X在f作用下的前周期点，而特殊子品种是前周期点。同样，如果不太可能的交集（给定的子品种 Y 和 X 的所有前周期点的集合之间）很大（在几何上通过 Y 上此类特殊点的 Zariski 稠密集的存在来表示），那么这会迫使 Y 变得特殊。******在过去，我们对上述两个猜想都获得了重要的部分结果。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，例如安德烈-奥尔特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交叉的概念。