Toroidal methods for computing zeta functions of groups and rings

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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.