Mathematical Sciences: Arithmetic of Abelian Varieties

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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.