Counting Problems in Number Theory: Elliptic and Plane Quartic Curves over Finite Fields

University Of California-Irvine, Irvine CA

Investigators

Abstract

This research project will explore several questions concerning counting in number theory. One of the fundamental goals of number theory is to understand solution sets of polynomial equations. These equations define geometric objects, for example, the algebraic curves on which many modern cryptographic systems are based. Instead of focusing on solutions to a single equation, one can consider a family of equations and attempt to understand the average behavior, the extremal behavior, and in particularly nice settings, to compute the entire distribution of the number of solutions when varying through this family. Within families of curves there are certain properties that one might want to avoid, or select for, so it is an important problem to understand how often these properties arise. In the second half of the twentieth century, mathematicians began to develop the interplay between number theory and the theory of error-correcting codes. This project will explore this connection, using ideas from coding theory to understand families of algebraic curves. The questions examined in this research project are a part of the field of arithmetic statistics. The principal investigator will explore distributions of rational point counts for families of elliptic curves over finite fields, especially families that play a key role in cryptography. The project will build on earlier work studying rational point count distributions for elliptic curves over a fixed finite field, applying these ideas to answer statistical questions about Legendre curves and pairing-friendly elliptic curves. The project will also study the distribution of rational point counts for genus-3 curves over finite fields. Computational evidence suggests that there are more curves with many rational points than with few points. The principal investigator aims to use ideas from the theory of error-correcting codes to understand this asymmetry. A major theme of this project is finding boundaries between counting problems that have polynomial answers, ones where formulas are not polynomial but can be expressed in terms of quantities such as Fourier coefficients of modular forms, and problems where we can only hope for asymptotic answers. 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.