# The average size of the kernel of a matrix and orbits of linear groups

## TL;DR

This paper investigates the average kernel sizes of matrices over certain rings, proves their generating functions are rational, and connects these to counting orbits and conjugacy classes in linear pro-p groups.

## Contribution

It introduces explicit formulas for generating functions of kernel sizes over finite quotients and links these to orbit enumeration in pro-p groups using p-adic Lie theory.

## Key findings

- Generated functions are rational and have fundamental properties.
- Explicit formulas are provided for natural families of modules.
- Connections established between kernel sizes and orbit counts in pro-p groups.

## Abstract

Let $\mathfrak{O}$ be a compact discrete valuation ring of characteristic zero. Given a module $M$ of matrices over $\mathfrak{O}$, we study the generating function encoding the average sizes of the kernels of the elements of $M$ over finite quotients of $\mathfrak{O}$. We prove rationality and establish fundamental properties of these generating functions and determine them explicitly for various natural families of modules $M$. Using $p$-adic Lie theory, we then show that special cases of these generating functions enumerate orbits and conjugacy classes of suitable linear pro-$p$ groups.