Toroidal methods for computing zeta functions of groups and rings

计算群和环的 zeta 函数的环形方法

• 批准号: 238710579
• 负责人: Professor Dr. Christopher Voll
• 金额: --
• 依托单位: Arbeitsgruppe Algebra, insbesondere Gruppentheorie
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2013
• 资助国家: 德国
• 起止时间: 2012-12-31 至 2016-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/238710579?language=en
• 关键词: Toroidal; methods; computing; zeta; functions

Originally introduced in the 1980s in the area of subgroup growth, the study of zeta functions of groups and rings has since evolved into a deep theory that combines methods from algebra, combinatorics, algebraic geometry, and other areas of mathematics. The explicit computation of such zeta functions, however, remains extremely challenging. The main objective of this project is to develop toroidal methods for computing and analyzing zeta functions of groups and rings. More precisely, the zeta functions that we consider admit Euler product factorisations into local zeta functions indexed by primes and we seek to compute these local factors. Our first main goal is to develop and implement an algorithm for computing such local zeta functions under non-degeneracy assumptions with respect to certain associated Newton polyhedra. Our algorithm will combine algebro-geometric computations and methods from convex geometry. Our second main goal is to develop methods for studying analytic properties of local zeta functions of groups and rings, again under non-degeneracy assumptions. In particular, we will develop methods for studying the local pole spectra of such zeta functions. The third main goal of our project is to apply our methods to study zeta functions of interesting and challenging classes of groups and rings. These computations will both stimulate further developments of our algorithms and they will provide a testing ground for conjectures in the area. All software and databases developed as part of this project will be made freely available.

最初是在20世纪80年代在子群增长领域引入的，对群和环的zeta函数的研究已经发展成为一个结合了代数、组合学、代数几何和其他数学领域方法的深入理论。然而，这种ζ函数的显式计算仍然极具挑战性。本项目的主要目标是发展计算和分析群和环的zeta函数的环面方法。更准确地说，我们考虑的zeta函数承认欧拉乘积分解成由素数索引的局部zeta函数，我们寻求计算这些局部因子。我们的第一个主要目标是开发和实现一种算法，用于在有关某些相关牛顿多面体的非退化假设下计算这种局部zeta函数。我们的算法将结合代数几何计算和凸几何的方法。我们的第二个主要目标是发展研究群和环的局部ζ函数解析性质的方法，同样是在非简并假设下。特别是，我们将开发研究这种zeta函数的局部极谱的方法。我们项目的第三个主要目标是应用我们的方法来研究有趣且具有挑战性的群和环类的zeta函数。这些计算将刺激我们算法的进一步发展，并将为该领域的猜想提供一个试验场。作为该项目的一部分开发的所有软件和数据库将免费提供。