Points on Curves Over Finite Fields and Motivic Stabilization

有限域上曲线上的点和动机稳定性

• 批准号: 1301690
• 负责人: Melanie Wood
• 金额: $ 33.8万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1301690&HistoricalAwards=false
• 关键词: Points; Curves; Over; Finite; Fields

The project is to study the distribution of points on curves over a fixed finite field and stabilization of sequences of moduli spaces, especially of curves, in the Grothendieck ring of varieties. The question of distribution of points on curves in a family over a fixed finite field is quite mysterious, even on the heuristic level, but many recent developments allow interesting access to special cases. These cases can provide the groundwork for development of the general theory. Related questions of counting points on varieties over finite fields have recently been determined to reflect a deeper structure in the Grothendieck ring of varieties, which will be studied and further developed in this project.A basic question about an equation is: how many solutions does it have? Many equations naturally fall into families of similar equations, for example, quadratic equations are ones in which all terms have degree at most two. This project studies how answers to the question of "how many solutions?" vary within a family and what consistent patterns develop as the family gets more complicated. The project also involves mentoring undergraduates, supporting infrastructure for networking and mentoring of women in mathematics, dissemination of mathematical ideas to middle school students, high school students, college non-math majors, and the general public, and mentoring and training graduate students

该项目旨在研究固定有限域上曲线上点的分布以及格罗腾迪克簇环中模空间序列（尤其是曲线）的稳定性。 即使在启发式水平上，固定有限域上的族曲线上的点分布问题也是相当神秘的，但许多最近的发展允许对特殊情况进行有趣的访问。 这些案例可以为一般理论的发展提供基础。 最近确定了有限域上簇的计数点的相关问题，以反映簇的格罗腾迪克环的更深层次结构，这将在本项目中进行研究和进一步发展。关于方程的一个基本问题是：它有多少个解有？ 许多方程自然地属于相似方程族，例如，二次方程是所有项的次数最多为二的方程。 该项目研究如何回答“有多少种解决方案？”的问题。家庭内部各不相同，随着家庭变得更加复杂，一致的模式也会发展。 该项目还包括指导本科生、支持女性数学网络和指导的基础设施、向中学生、高中生、大学非数学专业和公众传播数学思想，以及指导和培训研究生