Mathematical Sciences: Investigations in the Arithmetic of Curves and Surfaces

数学科学：曲线和曲面的算术研究

• 批准号: 9121727
• 负责人: Joseph Silverman
• 金额: $ 15.29万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-07-01 至 1996-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9121727&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Investigations; Arithmetic; Curves

This project supports research on a number of problems concerning the arithmetic geometry of curves and surfaces, as well as possible higher-dimensional generalizations. The research involves a detailed study of K3 surfaces with infinite automorphism group, centered around the canonical heights defined on such surfaces. The project will study the variation of the p-adic height on elliptic surfaces, extending recent results for the classical height, with possible applications to L-series. The research will also involve the study of various general position theorems in algebraic geometry, quantitative bounds for the number of rational points on curves of genus at least two, and the development of a new algorithm for computing canonical heights on elliptic curves which does not require a complete factorization of the discriminant. This project is in the general area of number theory and algebraic geometry. This is an area of research that focuses on the solutions sets, which are curves or surfaces in lower dimensions, of families of algebraic polynomials.

该项目支持关于曲线和曲面的算术几何的一些问题的研究，以及可能的高维推广。研究了具有无限自同构群的K3曲面，其中心是定义在这类曲面上的正则高度。该项目将研究椭圆曲面上p-进高的变化，推广了最近关于经典高度的结果，并可能应用于L级数。这项研究还将涉及代数几何中的各种一般位置定理的研究，亏格至少为2的曲线上有理点个数的定量界，以及不需要完全分解判别式的计算椭圆曲线上正则高度的新算法的发展。这个项目是在数论和代数几何的一般领域。这是一个专注于代数多项式族的解集的研究领域，解集是低维的曲线或曲面。