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

本研究项目将探讨数论中有关计数的几个问题。数论的基本目标之一是理解多项式方程的解集。这些方程定义几何对象，例如，许多现代密码系统所基于的代数曲线。与其专注于单个方程的解，人们可以考虑一族方程，并尝试理解平均行为、极端行为，并在特别好的环境中，计算在这个族中变化时解的数量的整个分布。在曲线族中，人们可能希望避免或选择某些属性，因此了解这些属性出现的频率是一个重要的问题。在二十世纪下半叶，数学家们开始发展数论和纠错码理论之间的相互作用。本课题将探索这种联系，利用编码理论的思想来理解代数曲线族。本研究课题所研究的问题是算术统计领域的一部分。主要研究人员将探索有限域上椭圆曲线族的有理点计数的分布，特别是在密码学中起关键作用的曲线族。该项目将建立在研究固定有限域上椭圆曲线的有理点数分布的早期工作的基础上，应用这些思想来回答有关勒让德曲线和配对友好的椭圆曲线的统计问题。该项目还将研究有限域上亏格3曲线的有理点数的分布。计算证据表明，有理点较多的曲线多于较少有理点的曲线。主要研究人员旨在利用纠错码理论中的思想来理解这种不对称性。这个项目的一个主要主题是在具有多项式答案的计数问题和我们只能希望得到渐近答案的问题之间找到界限，其中公式不是多项式的，但可以用模形式的傅立叶系数等数量来表示。这个奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.