Mutually annihilating matrices, and a Cohen--Lenstra series for the nodal singularity

TL;DR

This paper develops a generating function for counting pairs of mutually annihilating matrices over finite fields, extending classical series and exploring motivic aspects of module varieties related to nodal singularities.

Contribution

It introduces a new generating function for mutually annihilating matrices and analyzes its factorization and meromorphic extension, connecting to motivic and algebraic geometry.

Findings

Derived a generating function for mutually annihilating matrix pairs

Extended the generating function meromorphically to the complex plane

Counted pairs of mutually annihilating nilpotent matrices

Abstract

We give a generating function for the number of pairs of $n\times n$ matrices $(A, B)$ over a finite field that are mutually annihilating, namely, $AB=BA=0$. This generating function can be viewed as a singular analogue of a series considered by Cohen and Lenstra. We show that this generating function has a factorization that allows it to be meromorphically extended to the entire complex plane. We also use it to count pairs of mutually annihilating nilpotent matrices. This work is essentially a study of the motivic aspects about the variety of modules over $\mathbb C[u,v]/(uv)$ as well as the moduli stack of coherent sheaves over an algebraic curve with nodal singularities.