Zeta Functions and Algorithms

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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.