Zeta Functions: Algorithms and Applications

有限域上的矩函数给出了一种有趣的新方法来计算沿着一组品种的纤维的有理点的个数。就其本身而言，它包含了关于一组变种上有理点分布的关键信息。它也可以作为经典ζ函数（有理数）和Dwork的p进单位根ζ函数（p进超越）之间的几何桥梁。它是算术几何中的一个基本研究对象，将经典对象与p进对象联系起来，如p进模形式和p进l函数。本项目将结合p进方法和l进方法两种强大的方法，继续PI对zeta函数矩理论的发展，并探索其在算术镜像对称和编码理论中的实质性应用。PI正在扩大他与世界各地的研究生、博士后、初级和高级数学家的合作和互动，无论是在数学界，还是在计算机科学和工程学界。他的想法激励了大学和社区学院的研究人员和学生。在这个方向上的许多进一步的努力计划通过教学，写作和协作，以独立的方式。矩zeta函数是算术几何中一些非常深刻和复杂的问题的一个自包含的丢芬图重新表述（在计数有理点方面）。因此，它为跨领域合作和培养各级研究生提供了一个极好的会议场所。

Moment zeta function over finite fields gives interesting new ways to count the number of rational points along the fibres of a family of varieties. As such, it incorporates critical information about the distribution of rational points on a family of varieties.It also serves as a geometric bridge between classical zeta function (which is rational) and Dwork's p-adic unit root zeta function (which is p-adically transcendental). It is a fundamental object of study in arithmetic geometry, linking classical objects to p-adic objects, such as p-adic modular forms and p-adic L-functions.This project is to combine the two powerful approaches (p-adic methods and l-adic methods), to continue the PI's development of the theory of moment zeta function, and to explore its substantial applications in arithmetic mirror symmetry and coding theory.The PI is expanding his collaborations and interactions with graduate students, post-docs, junior and senior mathematicians worldwide, both within the mathematical community, and the computer science and engineering community.His ideas have motivated researchers and students, both at the university level and at the community college level. Many further efforts in this direction are planned via teaching, writing and collaborations in a self-contained manner.Moment zeta function is a self-contained diophantine reformulation (in terms of counting rational points) of some of the very deep and sophisticated problems in arithmetic geometry. As such, it provide an excellent meeting ground for collaborations crossing different fields and for training graduate students of all levels.