Frobenius Finds Non-monogenic Division Fields of Abelian Varieties

Abstract

Let A be an abelian variety over a finite field k with |k| = q = p m . Let π ∈ End k (A) denote the Frobenius and let v = q π denote Verschiebung. Suppose the Weil q-polynomial of A is irreducible. When End k (A) = Z[π, v], we construct a matrix which describes the action of π 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 monogeneity of division fields of certain abelian varieties.