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

Seungjai Lee

TL;DR

This work develops a constructive blueprint for computing zeta functions counting subalgebras and ideals of finite-dimensional Lie algebras over $\mathbb{F}_p$, and situates these finite-field invariants as tractable proxies for the local $\mathbb{Z}_p$-zeta factors via the Lazard correspondence. It applies the method to a broad spectrum of algebras, deriving explicit formulas for nilpotent and semisimple cases (e.g., $\mathfrak{sl}_2(\mathbb{F}_p)$, $\mathfrak{gl}_2(\mathbb{F}_p)$, Heisenberg, maximal class algebras $M_c$, and free nilpotent rings $\mathfrak{f}_{c,d}$), as well as to Grenham rings and graded/solvable families. The article also connects these finite-field zeta functions to Higman’s PORC conjecture, presenting wild non-PORC behavior in examples tied to elliptic curves and cubic residues, and discusses the limits of $\,\mathbb{F}_p$-data as predictors of $\mathbb{Z}_p$-behavior. Overall, it provides a first systematic framework for $\mathbb{F}_p$-level zeta theory with concrete computations and clear links to $p$-group counting and arithmetic geometry, while outlining open questions about uniformity and depth of finite-quotient approximations. The results suggest that $\mathbb{F}_p$-zeta functions can serve as effective, arithmetic-prone proxies for understanding $p$-group growth and related uniformity phenomena.

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.

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

TL;DR

This work develops a constructive blueprint for computing zeta functions counting subalgebras and ideals of finite-dimensional Lie algebras over , and situates these finite-field invariants as tractable proxies for the local -zeta factors via the Lazard correspondence. It applies the method to a broad spectrum of algebras, deriving explicit formulas for nilpotent and semisimple cases (e.g., , , Heisenberg, maximal class algebras , and free nilpotent rings ), as well as to Grenham rings and graded/solvable families. The article also connects these finite-field zeta functions to Higman’s PORC conjecture, presenting wild non-PORC behavior in examples tied to elliptic curves and cubic residues, and discusses the limits of -data as predictors of -behavior. Overall, it provides a first systematic framework for -level zeta theory with concrete computations and clear links to -group counting and arithmetic geometry, while outlining open questions about uniformity and depth of finite-quotient approximations. The results suggest that -zeta functions can serve as effective, arithmetic-prone proxies for understanding -group growth and related uniformity phenomena.

Abstract

We study zeta functions enumerating subalgebras or ideals of Lie algebras over finite field of prime order . We first develop a general blueprint method for computing zeta functions of -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 -groups of exponent for almost all primes via the Lazard correspondence. We investigate their connections to the study of finite -groups, and discuss what can be deduced from these finite Dirichlet polynomials.

Paper Structure

This paper contains 29 sections, 22 theorems, 141 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background and motivations
  3. Main results and organization
  4. Applications and related works
  5. Lie algebras over $\mathbb{F}_p$ and finite $p$-groups of exponent $p$
  6. Applications on Higman's PORC conjecture
  7. Zeta functions of submodules and graded submodules under endomorphisms
  8. Assumptions and Notations
  9. Computing zeta functions of $\mathbb{F}_p$-Lie algebras
  10. Applications on semisimple cases: $\mathfrak{sl}_{2}$ and $\mathfrak{gl}_2$
  11. Computations in practice for various $\mathbb{F}_p$-Lie algebras
  12. Nilpotent $\mathbb{F}_p$-Lie algebras
  13. The Heisenberg Lie ring $H$, revisited
  14. The maximal class-$c$ Lie algebra $M_{c}$, counting ideals
  15. The maximal class-$c$ Lie algebra $M_{c}$, counting subalgebras
  16. ...and 14 more sections

Key Result

Theorem 1.2

Let $L$ be a nilpotent Lie ring. There exists an algebraic variety $Y^{*}$ defined over $\mathbb{Q},$ with irreducible subvarieties $E_{i}^{*},$$i\in T^{*}:=\{1,\,\ldots,\,t^{*}\},$ all of which are smooth and intersect normally, and rational functions $P_{I}^{*}(X,\,Y)\in\mathbb{Q}(X,\,Y),$$I\subse where $c_{p,\,I}^{*}=\left|\left\{ a\in\overline{Y^{*}}(\mathbb{F}_{p}):a\in\overline{E_{i}^{*}}\te

Theorems & Definitions (55)

  • Definition 1.1
  • Theorem 1.2: duSG/00, Theorem 1.3
  • Example 1.3
  • Definition 1.4
  • Remark 1.5
  • Theorem 2.1
  • Lemma 2.2
  • Example 2.3
  • Example 2.4
  • Lemma 2.5
  • ...and 45 more