LEAPS-MPS: Asymptotics for Number Fields

Arithmetic statistics is the area of number theory interested in studying the asymptotic behavior of arithmetic objects, that is, it studies the qualitative behavior of number-theoretic objects in aggregates rather than isolated instances. For such objects, arithmetic statistics asks questions such as how common they are or if it is possible to describe their distribution or density. The focus of this project is the study of problems in arithmetic statistics from algebraic and geometric perspectives. The PI's research and scholarship are closely intertwined with their service and outreach work in the mathematical community. This project focuses on two common threads in arithmetic statistics: obtaining asymptotics on the count of number fields and obtaining asymptotics on the count of rational points on algebraic varieties. The PI will obtain constants and error terms for specific instances of Malle's conjecture by counting arithmetic objects by parametrizing them and using sieving techniques. The PI will study the asymptotics of number fields coming from geometry by counting field extensions generated by algebraic points on plane curves and examining how the geometry of the curve affects these field extensions. The PI will explore problems in arithmetic statistics from the perspective of stacks motivated by the Batyrev—Manin conjecture. This project is jointly funded by the LEAPS program and the Established Program to Stimulate Competitive Research (EPSCoR). 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.