Mathematical Sciences: Arithmetic of Abelian Varieties

这项研究集中在数论和算术代数几何中与阿贝尔簇相关的问题。研究课题包括确定一般阿贝尔变种的“Mordell-Weil群”，定义在数域上的阿贝尔变种上扭点的有界阶，以及当阿贝尔变种在其模数域上有一个有理模型时，发展准则。在算术代数几何中，人们将数中的方程解释为几何对象。如果方程足够好，这个几何对象有一种方法来组合点以获得新点(即组合解以获得新解)。当这种情况发生时，人们就会有所谓的阿贝尔变种。这些类型的方程非常重要，并且包含大量的结构。在这项研究中，私家侦探将花费时间研究这种情况。尽管她是一个相对较新的博士，但她已经取得了一些出色的成绩。这项研究无疑会带来更多的结果。

This research concentrates on questions in number theory and arithmetical algebraic geometry related to abelian varieties. Topics for research include the determination of "Mordell-Weil groups" of generic abelian varieties, bounding orders of torsion points on abelian varieties defined over number fields, and the development of criteria for when an abelian variety has a model rational over its field of moduli. In arithmetical algebraic geometry one interprets the equations in number as a geometric object. If the equations are nice enough this geometric object has a way of combining points to get new points (i.e. combining solutions to get new solutions). When this happens one has what is called an abelian variety. These types of equations are very important and contain huge amounts of structure. The P.I. will spend her time during this research studying this situation. Even though she is a relatively new PhD she already has some excellent results to her credit. This research will undoubtedly result in many more.