Cyclotomic torsion points in elliptic schemes

TL;DR

This paper investigates the finiteness of torsion points on elliptic curves over cyclotomic fields, proposing a family version of the Manin-Mumford conjecture and proving it under certain conditions.

Contribution

It introduces a family version of the cyclotomic torsion points conjecture and provides a proof under specific integrality assumptions.

Findings

Finiteness of torsion points over cyclotomic closures

Proposed a family version of the Manin-Mumford conjecture

Proved the conjecture under integrality conditions

Abstract

An elliptic curve defined over a number field possesses only a finite number of torsion points defined over the cyclotomic closure of its field of definition. In analogy to the relative version of the Manin-Mumford conjecture stated by Masser and Zannier, we propose a family version of the above statement and prove it under a suitable integrality condition.