Number Theory, Representation Theory, and Arithmetic Geometry

Princeton University, Princeton NJ

Investigators

Abstract

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. This project is part of an ongoing research program addressing fundamental questions in number theory and arithmetic geometry, with essential ingredients being employed from representation theory and the geometry of numbers. The proposed activity will heavily involve a number of graduate students, as well as undergraduate students and postdocs, who would be trained in the latest techniques in order to help advance the state of knowledge. Some of the goals would be to use recently developed as well as new techniques to understand the distribution of integer solutions to equations that have been studied for millennia, with some fundamental applications to distribution questions in algebraic number theory. A key ingredient will be a suitable development of sieve techniques based on the geometry of numbers. The proposed program is expected to lead to results on, e.g.: the distribution of integral solutions to classical equations such as the Thue equation and the Mordell equation; a positive proportion of cubic fields are not monogenic despite being locally so; at least 80% of all elliptic curves satisfy the BSD conjecture; new families of varieties over the rational numbers failing the Hasse principle and integral Hasse principle; and generalizations over other global fields. 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.