基于Drinfeld模理论的函数域中数论问题研究

As well known that there is a close analogy between algebraic number fields and global function fields, and this analogy is even pronounced if we focus our attention on the genus theory and cyclotomic theory of them. This project is planed to explore the rich analogies that exist between them, and study the genus theory and the theory of Drinfeld module in global function fields. With the help of the genus theory and class field theory, we will present explicitly the genus fields for some special global function fields by the cyclotomic theory and the Carlitz module. Together Kummer theory with tips related to number theory, we will study the Dirichlet density of the set of primes, which meet certain conditions, in multiple radical extension of global function fields by simulating the ideal in the proof of Artin's primitive root conjecture in function field. With the help of the sgn-normalized Drinfeld theory developed by Hayes, we will attempt in this project to give a new proof of the Kronecker-Weber theorem in local fields with characteristic p without use of the Lubin-Tate formal groups. This project will give us better knowledge on the similarities between global function fields and algebraic number fields, and rich the arithmetic theory for global function.fields.

整体函数域与代数数域之间有诸多类似的性质，二者的研究相互启发与促进，本项目聚焦于二者相似性质的研究，重点关注函数域的亏格理论与Drinfeld模理论及其应用。利用整体域的亏格理论及类域论，结合函数域的分圆理论和Drinfeld模理论，明确构造几类整体函数域的亏格域，并研究涉及其中的若干函数域的类群结构及类数的整除性；利用Kummer理论及相关数论方法，模拟整体函数域中Artin原根猜想的证明思路，研究整体函数域上多重根式扩张中满足特定条件的素除子的Dirichlet密度；利用Hayes发展的sgn-normalized Drinfeld模理论，避开Lubin-Tate形式群理论，尝试给出特征为p的局部域上的Kronecker-Weber定理的全新证明。本项目的研究涉及代数数论、算术几何及分析等广泛领域，相关研究结果将会加深我们对代数数域与整体函数域相似性的认识，丰富整体函数域的算术理论。

整体域的类群结构与类数整除性是代数数论研究的核心课题与难点。本项目聚焦于一些特殊函数域的类群结构、类数整除性及Drinfeld模的算术理论等相关问题。利用整体域的Zeta函数与常值域扩张的性质，对于特定的素数l，研究了常值域扩张的除子类群的Sylow l-子群的结构，证明了Sylow l-子群为非平凡群的常值域扩张的存在性。利用函数域的Conner-Hurrelbrink正合六边形与类域论，深入研究了整体函数域三次循环扩域的类群结构与素数l次循环扩域的理想类群Sylow l-子群的结构。利用Kummer理论，研究了整体函数域双重根式扩张的算术性质，并给出了判断此种双重根式扩张类型的方法，结合Bilharz证明函数域上Artin猜想的方法，给出了满足特定条件的一类素除子集合Dirichlet密度的计算公式，由此证明该集合的Dirichlet密度大于0。结合Denis和Poonen关于Drinfeld模height的结果与Silverman和Ghioca等人研究椭圆曲线动力系统算术理论的方法，将经典Zsigmongdy定理在函数域上作了相应的模拟。