Random maximal isotropic subspaces and Selmer groups

随机最大各向同性子空间和 Selmer 群

• 批准号: 1069236
• 负责人: Bjorn Poonen
• 金额: $ 37.28万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1069236&HistoricalAwards=false
• 关键词: Random; maximal; isotropic; subspaces; Selmer

The investigator proposes to study the distribution of Selmer groups in a family of abelian varieties, by building on the observation (made with Eric Rains) that the p-Selmer group of an elliptic curve is an intersection of maximal isotropic subspaces in a locally compact quadratic space. The goal is to construct a random model that will yield structural explanations for the behavior of the package consisting of prime power Selmer groups, Shafarevich-Tate groups, and Mordell-Weil groups, and then to justify the model by proving as many of its predictions as possible.The project is part of an extensive research program begun by the ancient Greeks, to understand the solutions to equations when the variables are required to be rational numbers (ratios of whole numbers or their negatives). It will provide insight into the nature of solutions to degree 3 equations in 2 variables: although this is the simplest case not yet understood, an understanding of it has eluded researchers for over a century. The investigator will also continue work on two graduate textbooks on the subject of rational solutions to equations.

研究人员建议通过观察（与 Eric Rains 合作）来研究 Selmer 群在阿贝尔簇族中的分布，即椭圆曲线的 p-Selmer 群是局部紧二次方程中最大各向同性子空间的交集空间。 目标是构建一个随机模型，该模型将为由素数幂 Selmer 群、Shafarevich-Tate 群和 Mordell-Weil 群组成的包的行为提供结构解释，然后通过证明尽可能多的预测来证明该模型的合理性该项目是古希腊人开始的一项广泛研究计划的一部分，旨在了解当变量需要为有理数（整数或其负数的比率）时方程的解。 它将深入了解 2 个变量的 3 次方程解的本质：虽然这是尚未理解的最简单的情况，但一个多世纪以来研究人员一直未能理解它。 研究人员还将继续编写两本关于方程有理解主题的研究生教科书。