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

50 pages; expanded version

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Cited by in corpus (7)

• Bivariate representation and conjugacy class zeta functions associated to unipotent group schemes, II: Groups of type F, G, and H
• The average size of the kernel of a matrix and orbits of linear groups, II: duality
• Bivariate representation and conjugacy class zeta functions associated to unipotent group schemes, I: Arithmetic properties
• Univariate and bivariate zeta functions of unipotent group schemes of type $G$
• Analytic properties of bivariate representation and conjugacy class zeta functions of finitely generated nilpotent groups
• Twisted conjugacy in soluble arithmetic groups
• A spectral theory for transverse tensor operators