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（称为函数域）上的多项式环。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。******一个例子是多维的韦林问题。对于任意正整数s，k和n，设R_s，k(n)为正整数n作为正整数s次幂的和的表示形式的个数。人们普遍认为，当k>2和s>k, R_s,k(n)有一个渐近公式。设u(k)是使R_s，k(n)有渐近公式的最小值s。G. H. Hardy和J. E. Littlewood首先得到了u(k)的界2k^2+2k-3。此后，u(k)的上界被许多人改进。这一问题是由S. T. Parsell在整数的多维环境下公式化的。我想锐化函数场情况下u(k)的上界，通过采用T. D. Wooley发明的函数场中的有效同余法。******第二个例子是西顿序列。给定一个正整数序列a，如果对所有a， a‘ in a, a+a’都是不同的，我们说a是一个西顿集合。如果任意正整数n可以写成A的h个元素的和，那么西顿集A就称为h阶的西顿基。西顿在他的傅里叶分析中引入了这个概念，以研究某些有界系数的幂级数。P. Erdos推测有阶数3的西顿基。这个猜想至今仍然存在。我们可以对函数域提出同样的问题。我的目标是使用概率方法来证明Erdos猜想和其他相关问题对于函数域的情况是成立的。******第三类例子是德林菲尔德模块中的问题。椭圆曲线的研究在现代数论中占有重要地位。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。我计划研究关于Drinfeld模的Koblitz猜想和Erdos-Pomerance猜想。这两个猜想考虑的是椭圆曲线类似物尚未得到证明的德林菲尔德模的概率性质。我对它们的研究将阐明关于椭圆曲线的原始猜想。******数论中有很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模拟问题进行数值实验。更成熟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。在我的训练下，许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.