Random maximal isotropic subspaces and Selmer groups

调查人员建议研究分布的塞尔默群体在家庭的阿贝尔品种，建设的观察（与埃里克雨）的p-塞尔默群的椭圆曲线是一个交叉的最大各向同性子空间在一个局部紧凑的二次空间。 目标是构建一个随机模型，该模型将产生对由素数幂塞尔默群、Shafarevich-Tate群和Mordell-Weil群组成的包的行为的结构解释，然后通过证明尽可能多的预测来证明模型的合理性。该项目是古希腊人开始的广泛研究计划的一部分，理解当变量必须是有理数（整数或负数的比值）时方程的解。 它将提供洞察的性质，解决方案的程度3方程的2个变量：虽然这是最简单的情况下尚未了解，了解它已经逃避了研究人员超过世纪。 调查员还将继续编写两本关于方程有理解问题的研究生教科书。

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.