Frobenius finds non-monogenic division fields of abelian varieties

Let A be an abelian variety over a finite field k with vertical bar k vertical bar = q = p(m). Let pi is an element of End(k)(A) denote the Ftobenius and let v = q pi(-1) denote Verschiebung. Suppose the Weil q-polynomial of A is irreducible. When End(k)(A) = Z[pi, v], we construct a matrix which describes the action of pi on the prime-to-p-torsion points of A. We employ this matrix in an algorithm that detects when p is an obstruction to the monogenicity of division fields of certain abelian varieties.