函数域Fq(T)上代数超曲面有理点的计数

代数超曲面的有理点是数学中重要的研究对象，对它们的计数是数论中非常有趣的课题，相关结果在许多数论问题中都起到了关键的作用。目前对于这一课题的研究主要集中在数域情形，而对于同等地位的函数域情形，该课题却未得到应有的重视。实际上，函数域中的各类计数问题具有重要的研究价值。最近项目组研究了函数域Fq(T)中多项式有理根个数的概率分布，并发现其与大数分解算法中的数域筛法有一定的联系。本项目将探讨函数域Fq(T)上代数超曲面有理点的计数问题，主要考虑三部分内容：首先，当超曲面方程的变元个数较多时，我们将研究相应的Hasse局部-整体原则。其次，当超曲面方程的变元个数较少时，我们将研究一族超曲面上有理点个数的一致上界估计。最后，对于具体超曲面，我们还将研究与之相应的Manin猜想。本项目属于分析、代数与几何的交汇领域，不仅自身具有重要的理论意义，同时也为函数域中许多其他计数问题的研究奠定了基础。

Rational points on algebraic hypersurfaces are important objects in mathematics. Counting the number of them is a very interesting area in number theory. Relevant results play crucial roles in many other arithmetic problems. So far, research in this area mainly focuses on the number field case. However, the function field case, which are of the same importance, is not getting enough attention it deserves. Actually，counting problems in function fields have a high research value. Recently, our project team has studied the probability distribution of rational roots of polynomials in function field Fq(T) and found it has some connection with the number field sieve in the problem of large number factoring. In this project, we will study the number of rational points on the algebraic hypersurfaces over function field Fq(T), and three topics will be considered. First, for hypersurfaces given by equations with a great many of variables, we will study the related Hasse local-global principle. Second, for hypersurfaces given by equations with not too many variables, we will study the uniform upper bound for the number of rational points lying on a family of algebraic hypersurfaces. Third, for some concrete algebraic hypersurfaces, related Manin's conjecture will be investigated. This project belongs to the crossing area of analysis，algebra and geometry. Besides its own theoretical significance，the project will also lay a foundation for studying other arithmetic problems in function fields.