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Counting subalgebras of $\mathfrak{a}$

Aaron Blas Pereda, Diego Sulca

Abstract

Let $\mathfrak{o}$ be a compact discrete valuation ring and $n\geq 2$. We introduce a method to study the cotype zeta function of subalgebras of $\mathfrak{o}^n$. This multivariable series encodes the number of finite-index subalgebras $Λ$ of the $\mathfrak{o}$-algebra $\mathfrak{o}^n$ of a given elementary divisor type. We express this zeta function as a finite sum of $\mathfrak{o}$-adic integrals and compute these integrals in many cases. As a first application, we recover known results in a natural way from our approach. For instance, we obtain a lower bound for the abscissa of convergence of the subalgebra zeta function of $\mathfrak{o}^n$ by exhibiting an explicit pole. We also determine the number of irreducible subrings of $\mathfrak{o}^n$ of small index. As a second application, we give an explicit formula for the cotype zeta function of subalgebras of $\mathfrak{o}^4$.

Counting subalgebras of $\mathfrak{a}$

Abstract

Let be a compact discrete valuation ring and . We introduce a method to study the cotype zeta function of subalgebras of . This multivariable series encodes the number of finite-index subalgebras of the -algebra of a given elementary divisor type. We express this zeta function as a finite sum of -adic integrals and compute these integrals in many cases. As a first application, we recover known results in a natural way from our approach. For instance, we obtain a lower bound for the abscissa of convergence of the subalgebra zeta function of by exhibiting an explicit pole. We also determine the number of irreducible subrings of of small index. As a second application, we give an explicit formula for the cotype zeta function of subalgebras of .
Paper Structure (20 sections, 15 theorems, 157 equations)

This paper contains 20 sections, 15 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. General properties of subring zeta functions
  3. Connection with number fields
  4. Known explicit formulas
  5. On the abscissa of convergence
  6. Uniformity
  7. From $n$ to $n-1$
  8. Results
  9. Local zeta functions as integrals
  10. Parameterization of finite-index submodules
  11. Preliminaries on $\pi$-adic integration
  12. Representations in terms of $\pi$-adic integrals
  13. The case $\# I=1$
  14. Application I
  15. Application II
  16. ...and 5 more sections

Key Result

Theorem 1.1

If $K$ is a number field of degree $n$, then $\zeta_{\mathcal{O}_K}^{{1,<}}(s)$ and $\zeta_{\mathbb{Z}^n}^{{1,<}}(s)$ have the same abscissa of convergence.

Theorems & Definitions (33)

  • Theorem 1.1: Sulca2023
  • Theorem 1.2: Isham2022
  • Theorem 1.5: Liu2007, AKKM2021
  • Definition 1.6
  • Proposition 1.7: Liu2007; see also Isham2023
  • Proposition 2.1
  • Definition 2.2
  • Example 2.3
  • Remark 2.4
  • Proposition 2.5
  • ...and 23 more