Rational Points and Heights

有理点和高

• 批准号: 0602333
• 负责人: Yuri Tschinkel
• 金额: $ 10.87万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0602333&HistoricalAwards=false
• 关键词: Rational; Points; Heights

The project addresses questions in Arithmetic Algebraic Geometry.It is proposed to study the distribution of rational points onalgebraic varieties over number fields, and to explore the interplaybetween global geometric invariants and arithmetic properties ofvarieties. The focus is on proving potential density for familiesof varieties, such as Fano varieties or Calabi-Yau varieties, andon establishing asymptotic formulas for the number of rational pointsof bounded height on equivariant compactifications of groups and homogeneous spaces.The proposed research ranges fromexplicit numerical experiments to advanced problems inmodern and rapidly developing areas of mathematics.This work will have a broader impact on the education of studentsand training of young mathematicians, as there is a rich supply ofconcrete questions and examples, consolidating our understanding of thesubject and leading to new structures.Increasingly, applications of Arithmetic Algebraic Geometryare spreading to vital areas of information storage, processing andtransmission; these applications rely on further advances onfundamental problems to be addressed in this proposal.

该项目解决了算术代数几何形状中的问题。它提议研究理性点在数量字段上的理性点的分布，并探索Varieties的全球几何不变性和算术性能之间的互动。重点是证明品种（例如Fano品种或Calabi-yau品种）的潜在密度，Andon为群体和同质空间的等效压缩的有限高度的合理点数量建立渐近点的渐近公式。高级问题的数学和快速发展的数学领域。这项工作将对学生和年轻数学家的培训的教育产生更广泛的影响，因为有很多混凝土的问题和示例，可以巩固我们对这些结构的理解，并导致新的结构。 ，算术代数几何形状的应用扩展到信息存储，加工和传输的重要领域；这些应用程序依赖于本提案中要解决的进一步进步问题。