正特征函数域上的超越性及相关问题

In the study of transcendence over function fields with positive characteristic, the transcendence of typical functions related to the arithmetic and geometric structures of Drinfeld modules and the transcendence of special values of these functions are always among the central problems considered by many authors. In recent years, important progresses have been made on this subject, but there are still many problems remained open, the difficulty consists in that each of the four methods applied currently in the domain has its proper defects. Hence we need borrow outer tools from algebraic number theory, arithmetic algebraic geometry, algebraic groups etc, on the one hand to promote the fusion of current methods, and on the other hand to realize a breakthrough in methods, so that we can understand better the arithmetic and geometric structures of general function fields with positive characteristic, and finally may be led to the solution of some important problems, and this is precisely the ultimate goal of the present proposal.

在正特征函数域上的超越性研究中，与Drinfeld模的算术和几何结构有关的典型函数的函数超越性以及在特殊点处的值的超越性一直都是人们关心的中心问题之一。近几年来，人们在这一方面取得了重大进展，但遗留的问题还有许多，其困难在于该领域目前常用的四个方法各自均有缺陷。为此我们需借助源自代数数论、算术代数几何以及代数群等方面的外来工具，一方面促进现有方法的融合，另一方面则是要实现方法上的突破，以便能更好地理解一般正特征函数域的算术与几何结构，并最终导致若干重要问题的解决，而这正是本项目研究的终极目标。

在正特征函数域上的超越性研究中，与Drinfeld模的算术和几何结构有关的典型函数的函数超越性以及在特殊点处的值的超越性一直都是人们关心的中心问题之一。本项目围绕这一问题展开，在传统方法的融合和创新上取得了进展。我们得到一些新想法、发现一批新问题，在无理指数的计算、弱超越性判别准则、超二次形式幂级数连分式展式部分商的结构与算术性质等方面取得一些成果。在上述工作的基础上，项目组成员及其合作者现已在Advances in Mathematics，Proceedings of the London Mathematical Society，Journal für die Reine und Angewandte Mathematik等国际知名的数学专业杂志上发表论文14篇(均为SCI源刊收录)，另有几篇正在撰写中。借助该项目，还培养了4名博士研究生(其中3个已经毕业)，2个博士后(其中一个已经出站)。