Non-monogenic Division Fields of Elliptic Curves

TL;DR

This paper demonstrates the existence of infinitely many elliptic curves over rationals with non-monogenic n-division fields, providing explicit parametrizations and a general result for non-CM curves using Frobenius analysis and Dedekind's ideas.

Contribution

It establishes the infinite occurrence of non-monogenic division fields for elliptic curves over ield, including explicit families and a broad non-CM case, using a novel combination of techniques.

Findings

Existence of infinite families of elliptic curves with non-monogenic division fields.

Explicit parametrizations of some non-monogenic division field families.

Every non-CM elliptic curve over ield has infinitely many non-monogenic division fields.

Abstract

For various positive integers $n$, we show the existence of infinite families of elliptic curves over $\mathbb{Q}$ with $n$-division fields, $\mathbb{Q}(E[n])$, that are not monogenic, i.e., the ring of integers does not admit a power integral basis. We parametrize some of these families explicitly. Moreover, we show that every $E/\mathbb{Q}$ without CM has infinitely many non-monogenic division fields. Our main technique combines a global description of the Frobenius obtained by Duke and T\'oth with a simple algorithm based on ideas of Dedekind.