Arithmetic Statistics, Fourier Analysis, and Equidistribution

Number theory is the study of whole numbers, and ratios of whole numbers (called rational numbers). In particular, number theorists since antiquity have been interested in finding whole number and rational number solutions to equations, such as y^2 = x^3 + 22 - that is, can a square be exactly 22 more than a cube? This project largely concerns the study of the probability that random equations of various types have whole number or rational number solutions. For example, a recent result of this kind proven during the period of the previous grant is that most equations of the form y^2 = a x^4 + b x^3 + c x^2 + d x + e, where a,b,c,d,e are whole numbers, do not possess any rational number solutions for x and y. The goal of the current project is to develop techniques for proving similar results for other types of classical equations of interest in number theory. The proposed activity would heavily involve a number of graduate students, undergraduates, and postdocs, who would be trained in the latest techniques in order to help address basic questions that advance the field. This project is part of an ongoing research program addressing fundamental questions in number theory and arithmetic geometry, with essential ingredients from representation theory and analysis. During the period of this award, this program is expected to lead to further results on the distribution of Galois groups of various types of algebraic objects, thus yielding new effective, quantitative forms of Hilbert irreducibility; new results on the distribution of Selmer groups of general (not necessary odd degree) hyperelliptic curves, and new results on the behavior of rational and integral points on curves and higher-dimensional varieties corresponding to representations having a ring of invariants that is not necessarily free. This will involve combining techniques from group theory, representation theory, geometry of numbers, Fourier analysis, and more, yielding an exciting interplay of techniques that we expect will have applications beyond just the problems mentioned. 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.