Zeta Functions: Algorithms and Applications

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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.