Order zeta functions of number rings and resolution of singularities

该项目旨在研究算术激励Zeta函数的基本算术和解析不变量，例如数字环上的阶Zeta函数。后者是代数数域上整数环的Dirichlet型生成级数枚举阶(1个子环)，与经典的DedekindZeta函数理论不同，这些函数的基本分析不变量--如收敛截口、极阶、特殊值等--在很大程度上是未知的。Bhargava的一个猜想对数环中阶Zeta函数收敛的横坐标以及数环中阶的多项式增长度有一定的影响。卡普兰E.A.的较新论文。给出我们提到的不变量的一些估计。然而，阶Zeta函数的显式公式仅适用于次数小于5的数域，所研究的Zeta函数都具有自然的Euler积分解，其因子都是有理函数。已知的小次数数域的公式表明了许多深刻的算术正则性、一致性和对称性现象。他们的详细研究是提议的核心。一种已建立的研究相关欧拉因子及其乘积的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Zeta函数及其欧拉因子的算术性质的关键。求解相关超曲面的奇点是该企业的核心工具。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。该项目的一个关键思想是利用出现在数字环的阶Zeta函数的特定算术上下文中的对称性和递归结构来设计和实现Taylor提出的奇点分解。该项目汇集了群和环的Zeta函数领域的一位经验丰富的研究者和实践者。这两个私人投资机构各自的专长相结合，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.