Bushnell-Reiner zeta functions over two-dimensional semilocal rings

TL;DR

This work extends zeta-function theory from commutative two-dimensional regular local rings to noncommutative two-dimensional left arithmetical and semiperfect rings by filtration via invertible ideals. It introduces the proliferation formula and the lifted Hey formula, enabling explicit product representations of Bushnell-Reiner zeta functions in terms of Artin–Wedderburn data and arithmetical semigroup structures, with Morita invariance guiding the framework. The results yield concrete two-dimensional formulae, including effective calculations for Rump's regular semiperfect rings and reductions to one-dimensional cases, and recover Lustig’s classical commutative formula as a special case. The appendices provide essential one-dimensional calculations that underpin the two-dimensional theory, highlighting a robust local–global approach to zeta functions in noncommutative arithmetic geometry.

Abstract

Lustig gave an infinite product formula for the zeta function of a commutative two-dimensional regular local ring with finite residue field. We extend this to the noncommutative setting with a method based on filtration by an invertible ideal. One application gives an abstract two-dimensional analogue of Hey's formula. Another application provides effective formulae for zeta functions over Rump's two-dimensional regular semiperfect rings. In the appendices, we supplement these two-dimensional applications with requisite one-dimensional calculations.

Theorems & Definitions (29)

• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Lemma 3.3
• proof