Higher Function Field Arithmetic

University Of Pennsylvania, Philadelphia PA

Investigators

Abstract

The study of symmetry is frequently useful in mathematics and its applications. Consideration of the symmetries of a system can yield insight into the behavior of related mathematical objects. This project seeks to answer the following question about certain highly symmetric objects: to what extent can they be distinguished from each other simply by examining them locally? The symmetries of such objects themselves form a geometric space, and the investigators aim to understand how the behavior of that space governs the answer to this overarching question. In particular, the project will explore whether symmetric spaces can be completely distinguished in the situation when any two symmetries can be connected by a line. Graduate students supported by the award will receive training to contribute towards the research, and the investigators will host a yearly event to encourage undergraduates from underrepresented groups to apply to mathematics Ph.D. programs. More precisely, the investigators study linear algebraic groups and their torsors, over function fields that are defined over a complete discretely valued field. They will study the relationships among three properties of a given linear algebraic group: whether the group satisfies the local-global principle for torsors, whether the group is R-trivial, and whether the group satisfies the weak approximation property. Implications among these properties have been shown in the case of groups over number fields, and this project aims to carry over those implications to groups over the function fields under consideration. Other goals include describing and bounding the obstruction to a local-global principle in terms of geometry and extending local-global principles to higher dimensions in order to obtain results about Brauer groups. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.