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Abstract  Let   O   be a compact discrete valuation ring of characteristic 0. Given a module   M   of matrices over   O  , we study the generating function encoding the average sizes of the kernels of the elements of   M   over finite quotients of   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.

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