Zeta Functions and Algorithms

摘要奖DMS-0400647 Wan本建议的中心目的是给出一个系统的研究的p-adic变化的zeta函数的环面超曲面在有限域上。这自然有三个方面：理论、应用和算法。我们期望在所有三个方面都取得重大进展。在理论方面，PI先前证明了Dwork的单位根zeta函数是p-adic亚纯函数，但除了在更简单的阿贝尔秩为1的情况下，没有关于其零点的信息。我们提出了在一般非交换高阶情形下，得到关于单位根zeta函数零点的有效信息，解决了这一基本问题。这应该与几何非交换p-adic L-函数和几何Iwasawa理论有关。因此，我们将给出一些重要的例子，如一族Kloosterman和和族Calabi-Yau超曲面的详细深入的研究，解决该领域的一些公开问题。在应用方面，我们建议调查算术镜像对称性：zeta函数的Calabi-Yau品种和zeta函数的镜像品种之间的关系。Candelas等人已经沿着这个方向进行了沿着的实验/计算研究，他们通过研究Picard-Fuch方程的周期发现了关于这种zeta函数的一些有趣的结构。我们建议给出一个严格的数学证明。最近，沿着这条路线，我们提出了几个显然更难的算术镜像猜想，包括一般的斜率镜像猜想和小斜率镜像猜想。沿着这条路，我们将建立一般牛顿多边形的分解定理与GKZ主判别式之间的联系，从而获得关于GKZ判别式的算术信息。在算法方面，我们计划利用一些更好的p-adic理论和计算交换代数来获得更快更好的p-adic算法，包括更难的奇异情况。拟议的研究将涉及几个colloborator和研究生。数论中的一个基本问题是了解有限域上多项式方程的解的个数。除了其内在的理论兴趣，这个问题与数学的几个主要分支有联系。它在编码理论、密码学和组合学中也有重要的应用。我们建议对这一基本问题进行系统的研究，并获得更好的理解。我们期望在理论、应用和算法方面都取得重大进展。因此，我们将解决一些开放的问题，在该地区，建立新的联系与其他领域，如p-adic L-函数，计算交换代数，环面几何和镜像对称。由于该提案的多样性和范围，拟议的研究将涉及几个colloborator和研究生。希望拟议的研究也将导致几本专著或书籍，这将使这个美丽的主题连贯和可访问。

Abstract for award DMS-0400647 of WanThe central aim of this proposal is to give a systematic study of the p-adic variation of zeta functions of toric hypersurfaces over finite fields. This naturally has three aspects: theory, applications and algorithms. We expect to make significant progresses on all three fronts. On the theoretical side, the PI previously proved that Dwork's unit root zeta function is p-adic meromorphic but no information about its zeros is known except in the simpler abelian rank one case. We propose to obtain effective information about zeros of the unit root zeta function in the general non-abelian higher rank case, settling this basic problem. This should be related to the conjectural non-abelian p-adic L-functions and geometric Iwasawa theory. As a consequence, we shall give a detailed in-depth study of some important examples such as a family of Kloosterman sums and a family of Calabi-Yau hypersurfaces, settling some open problems in the area. On the application side, we propose to investigate the arithmetic mirror symmetry: the relation between the zeta function of a Calabi-Yau variety and the zeta function of its mirror variety. An experimental/computational study along this direction has already been made by Candelas etc who disovered some interesting structures about such zeta functions via the study of the periods of the Picard-Fuch equations. We propose to give a rigorous mathematical proof of their conjectures. Recently, along this line, we proposed several apparently harder arithmetic mirror conjectures, including the generic slope mirror conjecture and the small slope mirror conjecture. We plan to make these conjectures precise in a general form and provide substantial evidence to them, opening new areas of research.Along the way, the link between our decomposition theorems for generic Newton polygons and the GKZ principal discriminant would be established, obtaining arithmetic information about the GKZ discriminant. For the algorithmic side, we plan to exploit some of the finer p-adic theory and computational commutative algebra to obtain faster and better p-adic algorithms for the zeta functions, including the harder singular case. The proposed research would involve several colloborators and graduate students. A fundamental problem in number theory is to understand the number of solutions of a polynomial equation over a finite field. In addition to its intrinsic theoretic interest, this problem has connections to several major branches of mathematics. It also has important applications in coding theory, crpyptography, and combinatorics. We propose to give a systematic study of this fundamental problem and obtain an improved understanding. We expect to make significant progresses on both the theory, application and algorithmic aspects of the problem. As a consequence, we would settle several open problems in the area, establishing new links with other areas such as p-adic L-function, computational commutative algebra, toric geometry and mirror symmetry. Due to the diversity and the scope of this proposal, the proposed research would involve several colloborators and graduate students. It is hoped that the proposed research would also lead to several monographs or books which would make this beautiful subject coherent and accessible.