Arithmetic Properties of Global Fields

全局字段的算术属性

• 批准号: RGPIN-2020-03915
• 负责人: Kuo, Wentang
• 金额: $ 1.31万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=716567
• 关键词: Arithmetic; Properties; Global; Fields

My research area is number theory. I am interested in studying arithmetic properties of integers and the ring of polynomials over the finite fields. I plan to pursue my research on the following areas. The first question concerns the uniform hypothesis for the generalized Waring's problem. For any positive integers s, k and n, let Rs,k(n) be the number of representations of the positive integer n as the sum of s kth powers of positive integers. It is widely expected that when k>2 and s>k, Rs,k(n) has an asymptotic formula. Let G(k) be the smallest value s such that Rs,k(n) has an asymptotic formula. Due to recent progress on the nested efficient congruencing method introduced by T. D. Wooley and the decoupling method developed by J. Bourgain, C. Demeter and L. Guth, one can largely sharpen the upper bounds for G(k). This problem can be generalized to systems of homogenous equations in multi-variables, whose terms are of the same form with the coefficient one; such a system is called the general Waring's system. In order to obtain a bound for G(k) for the general Waring's system, we need to assume the uniform hypothesis, an assumption on the existence of certain local solutions. My goal is to prove this hypothesis. The next area concerns problems in Drinfeld modules. The study of elliptic curves is important in modern number theory. Drinfeld modules are the function field analogue of elliptic curves. V. Drinfeld first introduced this subject to solve the Langlands conjecture in function fields, and was awarded the Fields medal for his work. Many questions in elliptic curves can be formulated in the Drinfeld module setting. I plan to study Artin's primitive root conjecture and Erdos-Pomerance's conjecture for Drinfeld modules. Artin's primitive root conjecture is about the distribution of the generators in the multiplicative groups of finite fields of prime order; Erdos-Pomerance's conjecture is about the distribution of number of prime divisiors of the multiplicative groups of finite abelian groups. These two conjectures concern probabilistic properties of the integers. Their analogues in the setting of Drinfeld modules are interesting problems and have not been carefully studied yet. There are many questions in number theory that are suitable for training different levels of students. At the entry level, students can start with problems in the classical setting and then conduct numerical experiments for the analogous problems in the function field setting. More sophisticated and talented students can study numerical data to make conjectures, or even prove them.

我的研究领域是数论。我对研究有限域上整数和多项式环的算术性质很感兴趣。我计划在以下几个方面进行研究。 第一个问题是关于广义华林问题的一致假设。对于任何正整数s，k和n，设Rs，k（n）是正整数n表示为正整数s的k次幂和的个数。当k>2，s>k时，Rs，k（n）有一个渐近公式。设G（k）是使Rs，k（n）有渐近公式的最小值s。由于T. D. Wooley和J. Bourgain，C. Demeter和L. Guth，人们可以大大地锐化G（k）的上界。这个问题可以推广到多变量齐次方程组，其项与系数1具有相同的形式;这样的系统被称为广义华林系统。为了得到一般Waring系统的G（k）的界，我们需要假设一致性假设，即存在某些局部解的假设。我的目标是证明这个假设。 下一个领域涉及Drinfeld模中的问题。椭圆曲线的研究在现代数论中占有重要地位。Drinfeld模是椭圆曲线的函数域模拟。V. Drinfeld首先介绍了这一主题，以解决函数域中的朗兰兹猜想，并因其工作而被授予菲尔兹奖。椭圆曲线中的许多问题都可以在Drinfeld模的设置下公式化。我计划研究Drinfeld模的Artin原根猜想和Erdos-Pomerance猜想。Artin的原根猜想是关于素数阶有限域的乘法群的生成元的分布; Erdos-Pomerance的猜想是关于有限交换群的乘法群的素因子数的分布。这两个定理涉及整数的概率性质。它们在Drinfeld模中的类似物是一个有趣的问题，尚未被仔细研究。 数论中有许多问题，适合培养不同层次的学生。在入门阶段，学生可以从经典设置中的问题开始，然后在函数域设置中对类似问题进行数值实验。更复杂和更有才华的学生可以研究数字数据来制作图表，甚至证明它们。