Zeta functions of $\mathbb{F}_p$-Lie algebras and finite $p$-groups

TL;DR

This paper introduces a new method for computing zeta functions of Lie algebras over finite fields, providing explicit formulas and exploring their implications for finite p-groups and subgroup enumeration.

Contribution

Develops a general blueprint method for calculating zeta functions of -Lie algebras and applies it to new examples, linking to subgroup counting in finite p-groups.

Findings

Explicit formulas for zeta functions of various -Lie algebras

Connections established between zeta functions and subgroup enumeration in finite p-groups

Demonstration of the method's applicability to previously unstudied cases

Abstract

We study zeta functions enumerating subalgebras or ideals of Lie algebras over finite field of prime order $\mathbb{F}_p$. We first develop a general blueprint method for computing zeta functions of $\mathbb{F}_p$-Lie algebras, and demonstrate its practical applications in detail to obtain explicit formulas for various interesting new examples that are not covered in any known literature yet. For nilpotent cases this also provides zeta functions counting subgroups and normal subgroups of finite $p$-groups of exponent $p$ for almost all primes via the Lazard correspondence. We investigate their connections to the study of finite $p$-groups, and discuss what can be deduced from these finite Dirichlet polynomials.