Arithmetic Properties of Function Fields

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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.