Non-monogenic division fields of elliptic curves

[摘要] 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 Toth 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. (C) 2021 Elsevier Inc. All rights reserved.

[发布日期] 2021-11-01

[关键词] Division field;Torsion field;Monogenic;Power integral basis