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.
