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

本研究计划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。这些方程定义了几何对象，例如，许多现代密码系统所基于的代数曲线。与其关注单个方程的解，不如考虑一组方程，并尝试理解它们的平均行为和极值行为，在特别好的情况下，计算在这个族中变化时解的整个分布。在曲线族中，有某些属性可能想要避免或选择，因此了解这些属性出现的频率是一个重要的问题。在20世纪下半叶，数学家开始发展数论和纠错码理论之间的相互作用。本项目将探索这种联系，使用编码理论的思想来理解代数曲线族。本研究项目所研究的问题是算术统计领域的一部分。主要研究者将探索有限域上椭圆曲线族的有理点计数分布，特别是在密码学中起关键作用的族。该项目将建立在早期研究椭圆曲线在固定有限域上的有理点计数分布的基础上，应用这些思想来回答关于勒让德曲线和配对友好型椭圆曲线的统计问题。该项目还将研究有限域上第三类曲线的有理点计数分布。计算证据表明，具有许多有理点的曲线比具有少量有理点的曲线更多。首席研究员的目标是利用纠错码理论来理解这种不对称性。这个项目的一个主要主题是找到具有多项式答案的计数问题之间的边界，那些公式不是多项式但可以用量来表示的问题，比如模形式的傅里叶系数，以及我们只能希望得到渐近答案的问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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.