Frobenius Finds Non-monogenic Division Fields of Abelian Varieties

Let $A$ be an abelian variety over a finite field $k$ with $|k|=q=p^m$. Let $\pi\in \text{End}_k(A)$ denote the Frobenius and let $v=\frac{q}{\pi}$ denote Verschiebung. Suppose the Weil $q$-polynomial of $A$ is irreducible. When $\text{End}_k(A)=\mathbb{Z}[\pi,v]$, 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 monogeneity of division fields of certain abelian varieties.