Computing order zeta functions via resolution of singularities

For a number field K with ring of integers O, the order zeta function of K is a Dirichlet generating series enumerating orders, i.e. unital subrings of O of finite index. In comparison with the Dedekind zeta function of K, the order zeta function of K is poorly understood: for number fields of degree larger than 5, next to nothing general is known. Encoding this Dirichlet series as a p-adic integral, we develop computational tools to repeatedly resolve singularities until it is distilled to enumerating points on polyhedra and p-rational points of algebraic varieties. This is joint work with Anne Fruehbis-Krueger, Bernd Schober, and Christopher Voll.