Arithmetic Properties of Global Fields

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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.