Concrete Arithmetic Applications of Analytic Number Theory

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Number theory is the branch of mathematics concerned with the study of the integers. Two of the fundamental objects considered in this research project are prime numbers and polynomial equations with integer solutions, both of which play a central role in cryptography. A theme of modern mathematics is to consider a family of related questions and to understand how the solution changes as the question varies within the family, with an eye toward revealing the structure underlying any individual question. In number theory, the families are typically discrete -- there is no way to continuously turn one integer into another without leaving the integers -- but one may take a broad view and ask about the proportion of questions or objects satisfying a certain property; this is the viewpoint of the subfield of number theory known as arithmetic statistics. This project will explore various questions within arithmetic statistics and number theory in general, often from different angles, with the goal being to see the extent to which these different approaches complement and inform each other.More specifically, the key objects of consideration are primes, L-functions, number fields, and elliptic curves. This project will study the growth of the average size of Selmer groups of elliptic curves with level structure using tools from the geometry of numbers as developed by Bhargava. It will consider quadratic twists of elliptic curves, in particular the extent to which analytic and algebraic non-vanishing results can be intersected, and it will consider similar problems for twists of elliptic curves by nonabelian number fields. It will also study the distribution of values of Artin L-functions, obtaining applications to class numbers, extreme value problems, and geometry.