Arithmetic Properties of Global Fields

我的研究领域是数论。我感兴趣的是研究整数的算术性质和有限域上的多项式环。我计划在以下几个方面继续我的研究。第一个问题是关于广义瓦林问题的统一假设。对任意正整数S，k，n，设R，k(N)是正整数n表示为正整数的k次方和的次数。人们普遍认为，当k&gt；2和S&gt；k，Rs，k(N)有一个渐近公式时。设G(K)是S的最小值，使得Rs，k(N)有一个渐近公式。由于T.D.Wooley提出的嵌套有效同余方法和J.Bourain、C.Demeter和L.Guth提出的解耦方法的最新进展，可以大大提高G(K)的上界。这个问题可以推广到与系数项具有相同形式的多元齐次方程组，这样的系统称为广义Wering‘s系统。为了得到一般Wering系统G(K)的上界，我们需要假设一致假设，即某些局部解的存在性的假设。我的目标是证明这一假设。下一个领域涉及Drinfeld模块中的问题。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首次引入这一主题是为了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模组中表示出来。我计划研究Artin的原根猜想和Erdos-Pomance关于Drinfeld模的猜想。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-Pomance猜想是关于有限交换群的乘法群的素除数个数的分布。这两个猜想涉及整数的概率性质。它们在Drinfeld模块中的类似物是有趣的问题，还没有被仔细研究过。数论中有许多问题适合培养不同层次的学生。在入门级，学生可以从经典设置中的问题开始，然后对函数域设置中的类似问题进行数值实验。更老练、更有才华的学生可以研究数字数据来进行猜测，甚至可以证明这些猜想。

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.