Frobenius Finds Non-monogenic Division Fields of Abelian Varieties

TL;DR

This paper develops an algorithm to identify when the prime characteristic obstructs the monogeneity of division fields of abelian varieties over finite fields, based on Frobenius endomorphism actions.

Contribution

It introduces a matrix construction and an algorithm to detect obstructions to monogeneity in division fields of abelian varieties over finite fields.

Findings

The matrix describes Frobenius action on prime-to-p torsion points.

The algorithm detects when p obstructs monogeneity of division fields.

Application to abelian varieties with irreducible Weil q-polynomial.

Abstract

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.