Frobenius finds non-monogenic division fields of abelian varieties

Abstract

Let [Formula: see text] be an abelian variety over a finite field [Formula: see text] with [Formula: see text]. Let [Formula: see text] denote the Frobenius and let [Formula: see text] denote Verschiebung. Suppose the Weil [Formula: see text]-polynomial of [Formula: see text] is irreducible. When [Formula: see text], we construct a matrix which describes the action of [Formula: see text] on the prime-to-[Formula: see text]-torsion points of [Formula: see text]. We employ this matrix in an algorithm that detects when [Formula: see text] is an obstruction to the monogenicity of division fields of certain abelian varieties.