Non-monogenic Division Fields of Elliptic Curves

2020/07/24 by Smith, Hanson
#11G05 #11R04 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2007.12781

Abstract

For various positive integers n, we show the existence of infinite families of elliptic curves over ℚ with n-division fields, 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/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 a simple algorithm based on ideas of Dedekind.

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