Counting Semilinear Endomorphisms Over Finite Fields

TL;DR

This paper provides a formula for counting semilinear endomorphisms of finite-dimensional vector spaces over finite fields with specified rank and stable rank, linking algebraic structures to finite flat group schemes.

Contribution

It introduces a novel counting method for semilinear endomorphisms with given invariants, connecting linear algebra over finite fields to group scheme classification.

Findings

Derived explicit formulas for counting semilinear endomorphisms.

Connected endomorphism counts to classification of finite flat group schemes.

Enhanced understanding of Dieudonne theory applications.

Abstract

For a finite field k and a triple of integers g \ge r \ge s \ge 0, we count the number of semilinear endomorphisms of a g-dimensional k-vector space which have rank r and stable rank s. Such endomorphisms show up naturally in the classification of finite flat group schemes of p-power order over k which are killed by p and have p-rank s, via Dieudonne theory.