Georgia Institute of Technology College of Sciences

Arithmetic Statistics is the study of arithmetic objects as they vary in a family. This is the framework used to analyze distributional properties of discriminants and class groups of number fields, ranks and Selmer groups of elliptic and hyperelliptic curves, low-lying zeroes and central values of L-functions, and many other key objects and invariants. An interrelated web of conjectures (Malle, Cohen--Lenstra, Poonen--Rains, Goldfeld, Katz--Sarnak, Loughran--Santens, and their many refinements) give quite precise conjectural descriptions (based on heuristical models) of the leading terms and asymptotic constants of the counts arising in these distributional questions. However, the secondary terms of the counts are much more mysterious, and very little is known (or even conjectured) about them.

In this talk, I will discuss some of the main distributional questions in arithmetic statistics, give some theoretical and practical explanations of why these secondary terms are so important, and end with explaining a series of results, joint with Manjul Bhargava, Takashi Taniguchi, and Jacob Tsimerman, in which some of these secondary terms are proved to exist, and computed.