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arXiv · 1704.02668

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

Tobias Rossmann
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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.

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Tobias Rossmann. 2017-04-09. The average size of the kernel of a matrix and orbits of linear groups. https://doi.org/10.1112/plms.12159

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