Rational points on algebraic varieties

代数簇的有理点

• 批准号: RGPIN-2017-03970
• 负责人: Mckinnon, David
• 金额: $ 2.19万
• 依托单位: University of Waterloo
• 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=683600
• 关键词: Rational; points; algebraic; varieties

One of the fundamental problems of number theory is to describe the****set of rational or integer solutions to Diophantine equations, which****are polynomial equations in several variables with integer****coefficients. My research program investigates the distribution of****rational solutions to systems of Diophantine equations, in several****directions.*******Paul Vojta has made some wide-ranging conjectures on what kinds of****solutions Diophantine equations should have, based on the geometric****properties of the solution sets of these equations. In my future****research, I propose to study these conjectures, to improve on my****previous proofs of various special cases of them, and to use existing****results to gain further insight into the solutions of Diophantine****equations.*******In particular, I am interested in the distribution of rational points****on K3 surfaces. I have already proven many results in this area,****including (with Logan and van Luijk) a proof that the rational points****on many diagonal quartic surfaces are dense in the real and Zariski****topology, and a proof of the celebrated Batyrev-Manin Conjecture that****is conditional on Vojta's Main Conjecture.*******I have, in joint work with Michael Roth, investigated how close two points with rational coordinates can get to one another, in terms of the geometry of the object the points lie on. Even more, we have obtained some results in which one of the points doesn't have rational coordinates, but instead has coordinates that are the roots of polynomials with rational coefficients. This has proven to be deep and interesting work, and I am continuing to work on proving more interesting results in this area. *****

数论的基本问题之一是描述丢番图方程的有理解或整数解的****组，这些方程****是具有整数****系数的多个变量的多项式方程。 我的研究项目在几个****方向上研究了丢番图方程组的****有理解的分布。********Paul Vojta 根据这些方程解集的几何****性质，对丢番图方程应该具有什么样的****解做出了一些广泛的猜想。 在我未来的****研究中，我建议研究这些猜想，改进我****之前对它们的各种特殊情况的证明，并利用现有的****结果进一步了解丢番图****方程的解。********特别是，我对K3曲面上有理点****的分布感兴趣。 我已经证明了这一领域的许多结果，****包括（与 Logan 和 van Luijk 一起）证明了许多对角四次曲面上的有理点****在实数和 Zariski****拓扑中是稠密的，以及著名的 Batyrev-Manin 猜想的证明****，该猜想****以 Vojta 的主要猜想为条件。********我与 Michael Roth 合作，研究了如何 根据点所在物体的几何形状，具有有理坐标的靠近的两个点可以相互接近。 更重要的是，我们获得了一些结果，其中一个点没有有理坐标，而是具有有理系数多项式的根坐标。 事实证明，这是一项深入而有趣的工作，我将继续努力证明该领域更有趣的结果。 *****