$p$-Adic Asymptotic Subalgebra Enumeration

TL;DR

This paper introduces $p$-adic asymptotics for counting subalgebras and subgroups, linking these to local zeta functions, and proves key properties of their poles in specific algebraic contexts.

Contribution

It establishes the smallest real pole of local zeta functions for residually nilpotent algebras and analyzes its properties in graded cases, addressing conjectures by Rossmann.

Findings

Identified the smallest real pole for local zeta functions in residually nilpotent algebras.

Proved the simplicity and residue of this pole in graded algebras.

Provided a detailed description of $p$-asymptotic behavior in these algebraic structures.

Abstract

We introduce the notion of $p$-adic asymptotics, or $p$-asymptotics, to the context of finite-index subgroup and subalgebra enumeration. For finitely generated groups and finite-dimensional algebras, we connect these asymptotics with the poles of their associated local zeta functions. Our two main results establish the smallest real pole for local zeta functions associated with residually nilpotent algebras, as well as its simplicity and residue whenever this algebra is graded. We thereby provide proof to parts of two conjectures raised by Rossmann and give a precise description of the $p$-asymptotic behaviour inside these algebras.