Mathematical Sciences: Arithmetic of Abelian Varieties

本研究集中于数论和算术代数几何中与阿贝尔变相关的问题。研究的主题包括一般阿贝尔变体的“modell - weil群”的确定，在数域上定义的阿贝尔变体上的扭转点的边界阶，以及当阿贝尔变体在其模域上具有模型理性时的准则的发展。在算术代数几何中，人们把数字方程解释为几何对象。如果方程足够好，这个几何物体有一种组合点得到新点的方法（即组合解得到新解）。当这种情况发生时，我们就有了所谓的阿贝尔变化。这类方程非常重要，包含大量的结构。私家侦探将在这次调查中花时间研究这种情况。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。这项研究无疑会产生更多的结果。

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.