一些整体函数域的算术问题

The algebraic number fields and global function fields, which possess many similar properties, were called global fields. This project is planned to explore the rich analogies that exist between them. With the help of the genus theory, Conner-Hurrelbrink exact hexagon and Redei matrix for function fields, we will study the structure of ideal class group and divisor class group of cyclic extensions of prime degree l of global function fields. In order to give a relatively elementary proof of Brumer-Stark conjecture in function fields context, Hayes defined in 1985 normalizing field associated to a fixed sgn-normalized Drinfeld module, and its extension field which is analogue of classical cyclotomic function fields. We will also in this project study the arithmetic properties of theses fields and their subfields. Finally, we will study the relations between divisor class group and ideal class group for the subfields of some cyclotomic function fields, and try to present explicit analytic formulas for the class number of some special subfields of cyclotomic function fields. This project will give us better knowledge on the similarities between algebraic number fields and global function fields and the arithmetic properties of global function fields.

代数数域和整体函数域有很多类似的性质，它们统称为整体域。本项目着眼于二者相似性质的研究。利用函数域的亏格理论、Conner-Hurrelbrink正合六边形以及Redei矩阵，我们将研究整体函数域l次循环扩域的理想类群和除子类群的Sylow l-子群的结构；利用Drinfeld和Hayes发展的Drinfeld模理论及Rosen等人关于分圆函数域的研究成果，我们还将研究Hayes在证明函数域上的Brumer-Stark猜想过程中所定义的正规化域及其分圆扩域的算术性质；最后，我们将研究分圆函数域的子域的除子类群和理想类群类数的关系，尝试给出一些特殊子域明确的解析类数公式。本项目的研究将会加深我们对代数数域和整体函数域之间相似性的了解，有助于对整体函数域算术性质的认识。

研究了Kummer函数域与Artin-Schreier函数域的复合函数域中素除子的分歧状况, 由此给出了某些特殊情形下该复合函数域的亏格公式, 从而给出复合函数域亏格公式达到Castelnuovo不等式上界的一个例子, 进而说明Castelnuovo上界确实是最优上界. 在研究整体函数域Cohen-Lenstra预测的过程中, 发现利用反射定理可以研究一类数域子域的理想类群的Sylow p-子群, 建立了这类子域理想类群与单位群的p-rank之间的关系, 利用此结果研究了一类数域的整环的K2群p-rank. 利用整体域的Conner-Hurrelbrink正合六边形研究了一些整体函数域的理想类群的Galois模结构, 特别是研究了有理函数域l次循环扩域的Sylow l-子群的结构, 采用统一的方法证明Hasse关于数域情形下Sylow l-子群l-rank公式在整体函数域中的结果, 利用得到的相关结论, 研究了Galois群为Klein四元群的有理函数域的四次扩域与其二次子域理想类群类数之间的整除关系.