Order zeta functions of number rings and resolution of singularities

本课题旨在研究算术驱动的zeta函数的基本算术和解析不变量，如数环的阶zeta函数。后者是代数域中整数环的dirichlet型生成级数枚举顺序（带一的子带）。与Dedekind zeta函数的经典理论相反，这些函数的基本解析不变量——如收敛的横截面、极阶、特殊值等——在很大程度上是未知的。Bhargava的一个猜想对zeta函数的收敛横坐标以及数环中各阶多项式增长的程度有影响。Kaplan e.a.的最新论文对我们提到的不变量给出了一些估计。然而，阶ζ函数的显式公式只适用于小于5次的数域。本课题研究的zeta函数均具有自然的欧拉积分解，其因子为有理函数。已知的小次数域公式暗示了许多深刻的算术规律性、均匀性和对称性现象。他们详细的研究是提案的核心。研究相关欧拉因子及其乘积的既定方法将这些因子解释为合适的p进积分。因此，对这些积分的统一理解是理解全局ζ函数及其欧拉因子的算术性质的关键。相关超曲面的奇异性的解析是这项工作的核心工具。虽然Hironaka著名的定理保证了这种分辨率的存在，但现有的算法通常很快就会屈服于相关超曲面的复杂性和高维性。所提出的项目的一个关键思想是使用在数环的阶ζ函数的特定算术环境中出现的对称性和递归结构，以便在这种情况下设计和实现泰勒制作的奇点分辨率。该项目汇集了奇点分辨率领域经验丰富的研究人员和实践者以及群和环的zeta函数领域的专家。这两个pi各自专业知识的结合有望在当前具有高度国际相关性的渐近环理论领域取得重大进展。

The project aims at the study of fundamental arithmetic and analytic invariants of arithmetically motivated zeta functions such as order zeta functions of number rings. The latter are Dirichlet-type generating series enumerating order (subrings with one) of rings of integers in algebraic number fields.In contrast to the classical theory of the related Dedekind zeta function, the fundamental analytic invariants of these functions -- such as their abscissae of convergence, pole orders, special values etc -- are largely unknown. A conjecture attributed to Bhargava has implications on the abscissa of convergence of order zeta functions and hence on the degree of polynomial growth of orders in number rings. Newer papers by Kaplan e.a. yield some estimates for the invariants we mentioned. Explicit formulae for order zeta functions, however, are only known for number fields of degree less than five.The zeta functions studied in the project all have natural Euler product decompositions, whose factors are rational functions. The known formulae for number fields of small degree suggest a number of deep arithmetic regularity, uniformity, and symmetry phenomena. Their detailed study lies at the heart of the proposal.An established method to study the relevant Euler factors and their products interprets the factors as suitable p-adic integrals. A uniform understanding of these integrals therefore holds the key to the understanding both of the global zeta functions and the arithmetic properties of their Euler factors. Resolution of singularities of associated hypersurfaces are a central tool in this enterprise. Whilst Hironaka's celebrated theorem guarantees the existence of such resolutions, the existing algorithms usually soon yield to the complexity and high-dimensionality of the relevant hypersurfaces. A key idea of the proposed project is to use the symmetries and recursive structures occurring in the specific arithmetic context of order zeta functions of number rings in order to design and implement taylor-made resolutions of singularities in this context.The project brings together an experienced researcher and practitioner in the field of resolutions of singularities and an expert in the field of zeta functions of groups and rings. The combination of the two PIs' respective expertise promises significant progress in a field of asymptotic ring theory of high current international relevance.