Non-monogenic division fields of elliptic curves

Abstract

For various positive integers n, we show the existence of infinite families of elliptic curves over Q with n-division fields that are not monogenic, i.e., such that the ring of integers does not admit a power integral basis. We parametrize some of these families explicitly. Moreover, we show that every E/Q without CM has infinitely many non-monogenic division fields. Our main technique combines a global description of the Frobenius obtained by Duke and Tóth with an algorithm based on ideas of Dedekind. As a counterpoint, we are able to use different aspects of the arithmetic of elliptic curves to exhibit a family of monogenic 2-division fields.