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Counting lattice points in central simple algebras with a given characteristic polynomial

Jiaqi Xie

TL;DR

This work extends the Eskin–Mozes–Shah asymptotic count of integral matrices with a fixed irreducible characteristic polynomial to elements of maximal orders in central simple algebras of degree $n$ over $\mathbb{Q}$, via their reduced characteristic polynomials. It develops a quantitative local-global framework using Tamagawa measures and Brauer–Manin obstructions, representing counts as products of local solutions and a real-volume factor, with nontrivial Brauer characters contributing vanishing terms. Local orbit computations at $p$-adic places yield explicit orbit counts depending on whether a place splits or is ramified, notably giving $e_p^{-1}n$ orbits in the division-algebra case. By stitching together these local counts with real-volume evaluations and Dedekind zeta data, the paper derives an explicit asymptotic formula for $N_{\mathfrak{o}_{\mathcal{A}}}(p(\lambda),T)$, including a ramification-dependent Euler-type factor and global invariants like class-number, regulator, and Tamagawa constants, thereby generalizing lattice-point counting in homogeneous spaces to central simple algebras.

Abstract

We extend the asymptotic formula for counting integral matrices with a given irreducible characteristic polynomial by Eskin, Mozes and Shah in 1996 to the case of counting elements in a maximal order of certain central simple algebra with a given irreducible characteristic polynomial.

Counting lattice points in central simple algebras with a given characteristic polynomial

TL;DR

This work extends the Eskin–Mozes–Shah asymptotic count of integral matrices with a fixed irreducible characteristic polynomial to elements of maximal orders in central simple algebras of degree over , via their reduced characteristic polynomials. It develops a quantitative local-global framework using Tamagawa measures and Brauer–Manin obstructions, representing counts as products of local solutions and a real-volume factor, with nontrivial Brauer characters contributing vanishing terms. Local orbit computations at -adic places yield explicit orbit counts depending on whether a place splits or is ramified, notably giving orbits in the division-algebra case. By stitching together these local counts with real-volume evaluations and Dedekind zeta data, the paper derives an explicit asymptotic formula for , including a ramification-dependent Euler-type factor and global invariants like class-number, regulator, and Tamagawa constants, thereby generalizing lattice-point counting in homogeneous spaces to central simple algebras.

Abstract

We extend the asymptotic formula for counting integral matrices with a given irreducible characteristic polynomial by Eskin, Mozes and Shah in 1996 to the case of counting elements in a maximal order of certain central simple algebra with a given irreducible characteristic polynomial.

Paper Structure

This paper contains 4 sections, 6 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Quantitive local-global principle
  3. Local computation
  4. The asymptotic formula

Key Result

Theorem 1.1

When $\mathcal{A}_{p}$ is a matrix algebra or a division algebra for all primes $p$ and the ring $\mathbb{Z}[\lambda]/(p(\lambda))$ is integrally closed, then as $T\rightarrow \infty$, where $S=\{p:\mathcal{A}_{p}\ is\ a\ division\ algebra\ over\ \mathbb{Q}_{p}\}$ and $e_p$ is ramification index of extension $\Bbb Q_p[\lambda]/(p(\lambda))$ over $\Bbb Q_p$.

Theorems & Definitions (15)

  • Theorem 1.1
  • Definition 2.1
  • Theorem 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 5 more