Near coincidences and nilpotent division fields of elliptic curves

Abstract: A natural question about the division fields of a fixed elliptic curve $E/\mathbb{Q}$ is whether there is a coincidence between the division fields. I.e. Are there distinct integers $m \neq n$ such that the $m$-division field equals the $n$-division field. In 2023, Daniels and Lozano-Robledo gave partial answers to this question, using (among other tools) the fact that the $n$-th roots of unity often fail to lie in the $m$-division field, thereby preventing such coincidences.

Motivated by this, we consider a broader notion of \emph{near coincidences}: when does there exist $E/\mathbb{Q}$ and distinct $m,n$ such that \[ \mathbb{Q}(E[n]) = \mathbb{Q}(E[m], \zeta_n)? \] In the first part of this talk, we answer this question completely in the case where $m$ and $n$ are powers of the same prime.

In the second part, we turn to a seemingly unrelated but natural problem: classifying all elliptic curves $E/\mathbb{Q}$ and positive integers $n$ such that \[ \operatorname{Gal}(\mathbb{Q}(E[n])/\mathbb{Q}) \] is a nilpotent group. This question generalizes the classification of abelian division fields obtained by Gonz\'alez-Jim\'enez and Lozano-Robledo (2016). We present a conditionally complete classification of nilpotent division fields, under either a standard conjecture about rational points on modular curves attached to normalizers of non-split Cartan subgroups or a full classification of the Mersenne primes. This is joint work with Jeremy Rouse.

algebraic geometrynumber theory

Audience: researchers in the topic

( paper )

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