Frobenius Finds Non-monogenic Division Fields of Abelian Varieties

2021/09/09 by Smith, Hanson
#11G10 #11R04 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2109.04262

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=\fracqπ 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.

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