Orbits of pairs in abelian groups

TL;DR

This paper calculates the number of orbits of pairs in finitely generated torsion modules over discrete valuation rings, revealing polynomial formulas with coefficients depending only on elementary divisors.

Contribution

It provides explicit polynomial formulas for orbit counts in modules over valuation rings, independent of the specific ring, and conjectures non-negativity of coefficients.

Findings

Number of orbits is a polynomial in the residue field size.

Coefficients depend solely on elementary divisors.

Coefficients are conjectured to be non-negative integers.

Abstract

We compute the number of orbits of pairs in a finitely generated torsion module (more generally, a module of bounded order) over a discrete valuation ring. The answer is found to be a polynomial in the cardinality of the residue field whose coefficients are integers which depend only on the elementary divisors of the module, and not on the ring in question. The coefficients of these polynomials are conjectured to be non-negative integers.