Arithmetic Properties of Function Fields

我的研究领域是数论。数论中的一个重要课题是有限域上的多项式环，称为函数域。它与整数环非常相似。关于整数的许多结果可以从这些类似物中得到。 多维的瓦林问题就是一个例子。对任意正整数S，k，n，设R_S，k(N)为正整数n表示为正整数的k次方和的个数。人们普遍期望当k&gt；2和S&gt；k，R_S时，k(N)有一个渐近公式。设u(K)是S的最小值，使得R_S，k(N)有一个渐近公式。G.H.Hardy和J.E.Littlewood最先得到了u(K)的界2k^22k-3。从那时起，u(K)的上界得到了许多人的改进。这个问题是由S.T.Parsell在整数上的多维环境下提出的。本文采用T.D.Wooley发明的函数域上的有效同余方法，锐化函数域中u(K)的上界。 第二个例子是Sidon序列。给定一个正整数序列A，如果对所有的a，a‘in A，a’都是不同的，则称A是Sidon集。Sidon集A称为h阶Sidon基，如果任何正整数n都可以表示为A的h个元素的和。Sidon在他的《傅立叶分析》一书中引入了这一概念，以研究某些系数有界的幂级数。P.鄂尔多斯猜想存在一个3阶的Sidon基。这一猜想至今仍然有效。我们可以对功能字段提出同样的问题。我的目的是用概率论的方法证明鄂尔多斯猜想和其他相关问题适用于函数域情形。 第三类例子是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。我打算研究Drinfeld模上的Koblitz猜想和Erdos-Pomance猜想。这两个猜想涉及到Drinfeld模的概率性质，其椭圆曲线的类似还没有被证明。我对它们的研究将有助于阐明椭圆曲线上最初的猜想。 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的问题开始，然后对函数域设置的模拟问题进行数值实验。更老练、更有才华的学生可以研究数字数据来做出猜测，甚至证明它们。在我的训练下，很多学生都取得了成功。

My research area is number theory. An important subject in number theory is the ring of polynomials over a finite field, called a function field. It bears a close resemblance to the ring of integers. Many results about integers can be asked of these analogues. One example is the multi-dimensional Waring’s problem. For any positive integers s, k and n, let R_s,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, R_s,k(n) has an asymptotic formula. Let u(k) be the smallest value s such that R_s,k(n) has an asymptotic formula. G. H. Hardy and J. E. Littlewood were the first to obtain the bound 2k^2+2k-3 for u(k). Since then, the upper bound of u(k) has been improved by many people. This problem has been formulated in the multi-dimensional setting over the integers by S. T. Parsell. I would like to sharpen the upper bound of u(k) in the function field case, by adopting the efficient congruencing method in function fields, invented by T. D. Wooley. A second example is a Sidon sequence. Given a sequence of positive integers A, we say that A is a Sidon set if for all a, a’ in A, a+a’ are distinct. A Sidon set A is called a Sidon basis of order h if any positive integer n can be written as a sum of h elements of A. S. Sidon introduced this concept in his work in Fourier analysis, to investigate certain power series with bounded coefficients. P. Erdos conjectured that there is a Sidon basis of order 3. This conjecture is still open today. We can ask the same question for function fields. My goal is to use probabilistic methods to prove that Erdos’ conjecture and other related questions hold for the function field case. A third class of examples are 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 Koblitz's conjecture and Erdos-Pomerance’s conjecture on Drinfeld modules. These two conjectures regard probabilistic properties of Drinfeld modules whose elliptic curve analogues have not been proved yet. My work on them will shed light on the original conjectures on elliptic curves. 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 analogue problems with the function field setting. More sophisticated and talented students can study the numerical data to make conjectures, or even prove them. Many students have been successful under my training.