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Zeta functions of solvable Lie algebras over finite fields -- with calculations in detail

Seungjai Lee

TL;DR

The article computes and analyzes the subalgebra and ideal zeta functions $\zeta_{L}^{*}(s)$ for all solvable $\mathbb{F}_{q}$-Lie algebras of dimension $\le 4$, providing explicit symbolic formulas that depend on finite-field point counts of auxiliary varieties. A unified RRDF-based method is developed to enumerate finite-index subalgebras and ideals by decomposing into diagonal cells and solving polynomial constraints, yielding concrete results for each isomorphism type. The work demonstrates that, in dimensions $\le 3$, zeta functions are PORC (and in many nilpotent cases uniformly so), while in dimension $4$ non-PORC behavior arises, illustrating richer arithmetic phenomena tied to splitting behavior over primes. It also investigates uniformity across local factors, introducing notions like $\mathcal{O}_{\mathfrak{p}}$- and $\mathbb{F}_{q}$-periods, and discusses the extent to which $L$-structures influence the period and the count of points on associated varieties. Overall, the paper provides a comprehensive explicit data set linking solvable Lie algebra structure to zeta-function uniformity, guiding future explorations of zeta functions over finite fields and their uniformity properties.

Abstract

Let $L$ be a solvable Lie algebra of dimension less than or equal to 4 over finite fields. We compute and record, in explicit symbolic form, the zeta functions enumerating subalgebras or ideals of $L$, and study their properties. We also discuss the implications of our data, in particular in relation to the general theory of Lie algebras over finite fields and zeta functions of Lie algebras over commutative rings.

Zeta functions of solvable Lie algebras over finite fields -- with calculations in detail

TL;DR

The article computes and analyzes the subalgebra and ideal zeta functions for all solvable -Lie algebras of dimension , providing explicit symbolic formulas that depend on finite-field point counts of auxiliary varieties. A unified RRDF-based method is developed to enumerate finite-index subalgebras and ideals by decomposing into diagonal cells and solving polynomial constraints, yielding concrete results for each isomorphism type. The work demonstrates that, in dimensions , zeta functions are PORC (and in many nilpotent cases uniformly so), while in dimension non-PORC behavior arises, illustrating richer arithmetic phenomena tied to splitting behavior over primes. It also investigates uniformity across local factors, introducing notions like - and -periods, and discusses the extent to which -structures influence the period and the count of points on associated varieties. Overall, the paper provides a comprehensive explicit data set linking solvable Lie algebra structure to zeta-function uniformity, guiding future explorations of zeta functions over finite fields and their uniformity properties.

Abstract

Let be a solvable Lie algebra of dimension less than or equal to 4 over finite fields. We compute and record, in explicit symbolic form, the zeta functions enumerating subalgebras or ideals of , and study their properties. We also discuss the implications of our data, in particular in relation to the general theory of Lie algebras over finite fields and zeta functions of Lie algebras over commutative rings.
Paper Structure (16 sections, 41 theorems, 668 equations)

This paper contains 16 sections, 41 theorems, 668 equations.

Table of Contents

  1. Introduction
  2. Background and motivations
  3. Main results and applications
  4. The period of $\zeta_{L}^{*}(s)$ over $\mathfrak{p}$
  5. Solvable Lie algebras and their splitting behavior over $\mathbb{F}_p$
  6. Assumptions and Notations
  7. Computing zeta functions of $\mathbb{F}_q$-Lie algebras
  8. Explicit computations of $\zeta_{L}^{*}(s)$ for solvable $\mathbb{F}_p$-Lie algebras of dimension $n\leq3$
  9. Solvable $\mathbb{F}_q$-Lie algebras of dimension $n=3$, counting ideals
  10. Solvable $\mathbb{F}_q$-Lie algebras of dimension $n=3$, counting subalgebras
  11. Explicit computations of $\zeta_{L}^{*}(s)$ for solvable $\mathbb{F}_p$-Lie algebras of dimension 4
  12. Solvable $\mathbb{F}_q$-Lie algebras of dimension 4, counting ideals
  13. Solvable $\mathbb{F}_q$-Lie algebras of dimension $n=4$, counting subalgebras
  14. Further remarks and questions
  15. Zeta functions of solvable Lie algebras over $\mathcal{O}_{\mathfrak{p}}$
  16. ...and 1 more sections

Key Result

Lemma 1.3

For a given $\mathcal{O}$-Lie algebra $L$, there are quasi-projective varieties $U_j^{*}$ ($j\in J^{*}$, $J^{*}$ finite), defined over $K$, not necessarily irreducible, and polynomials $P_{j}^{*}(X,Y)\in\mathbb{Z}[X,Y]$ such that, for almost all $\mathfrak{p}\in\mathop{\mathrm{Spec}}\nolimits(\mathc where $q=|\mathcal{O}/\mathfrak{p}|$.

Theorems & Definitions (94)

  • Definition 1.1
  • Example 1.2
  • Lemma 1.3: Lee25, Lemma 1.3
  • Definition 1.4: Lee25, Definition 1.4
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Corollary 1.10
  • Definition 1.11
  • ...and 84 more