Arithmetic Properties of Global Fields

我的研究领域是数论。我对研究整数的算术性质和有限域上的多项式环很感兴趣。我计划在以下几个方面进行研究。第一个问题是关于广义韦林问题的统一假设。对于任意正整数s，k和n，设Rs，k(n)为正整数n的s次幂的和的表示形式的个数。人们普遍认为，当k bbbb2和s b>k, Rs,k(n)有一个渐近公式。设G(k)是最小的值s使得Rs，k(n)有一个渐近公式。由于T. D. Wooley引入的嵌套有效同余方法和J. Bourgain、C. Demeter和L. Guth开发的解耦方法的最新进展，人们可以在很大程度上锐化G(k)的上界。这个问题可以推广到多变量齐次方程组，其项的形式都是相同的，系数为1；这样的系统被称为一般沃林系统。为了得到一般Waring系统的G(k)的界，我们需要假设一致假设，即关于某些局部解存在的假设。我的目标是证明这个假设。下一个领域涉及Drinfeld模块的问题。椭圆曲线的研究在现代数论中占有重要地位。漏场模是椭圆曲线的函数场模拟。V. Drinfeld首先引入了这个问题来解决函数场中的朗兰兹猜想，并因此获得了菲尔兹奖。椭圆曲线中的许多问题都可以用德林菲尔德模的形式来表述。我计划研究Artin的原始根猜想和Erdos-Pomerance的猜想对于Drinfeld模。Artin的原始根猜想是关于素阶有限域的乘积群中的生成子的分布；Erdos-Pomerance猜想是关于有限阿贝尔群的乘积群的素数除数的分布。这两个猜想涉及整数的概率性质。它们在Drinfeld模块设置中的类似物是有趣的问题，但尚未被仔细研究。数论中有许多问题适合培养不同水平的学生。在入门阶段，学生可以从经典设置的问题开始，然后对函数场设置的类似问题进行数值实验。更复杂和有才华的学生可以研究数字数据来做出猜测，甚至证明它们。

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.