Applications of Analytic Uniformity in Arithmetic Statistics

This project will investigate questions in arithmetic statistics, an area of mathematics concerned with determining the "typical" properties of objects in number theory. In practice, many questions in arithmetic statistics can be fruitfully understood by studying how the underlying objects decompose into simpler objects. Much of the research in this project will be aimed at studying these kinds of decompositions and, particularly, the process of assembling properties of these simpler objects to provide answers to complex questions. The PI will also continue his established record of undergraduate and graduate mentorship. He will also organize special sessions and workshops aimed at bringing together early career researchers in both analytic and algebraic number theory. More technically, this project will focus on the development of uniform bounds (both upper and lower) on number fields and class groups. It will consider the applications of these bounds to central problems in arithmetic statistics, like Malle's conjecture on the number of fields with a given Galois group and the ell-torsion conjecture on the size of the class group. Another focus will be the development of zero density estimates for Artin L-functions. These density estimates will play a role in the proofs of the uniform bounds and will have other applications, for example, to irreducibility in certain families of random polynomials, which the PI will explore. 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.