A criterion to rule out torsion groups for elliptic curves over number fields

Peter Bruin,Filip Najman

doi:10.1007/s40993-015-0031-5

Abstract

We present a criterion for proving that certain groups of the form \(\mathbb {Z}/m\mathbb {Z}\oplus \mathbb {Z}/n\mathbb {Z}\) do not occur as the torsion subgroup of any elliptic curve over suitable (families of) number fields. We apply this criterion to eliminate certain groups as torsion groups of elliptic curves over cubic and quartic fields. We also use this criterion to give the list of all torsion groups of elliptic curves occurring over a specific cubic field and over a specific quartic field.

Highlights

A fundamental result in the theory of elliptic curves, the Mordell–Weil theorem, states that the Abelian group of points of an elliptic curve E over a number field K is finitely generated
We present a criterion for proving that certain groups of the form Z/mZ ⊕ Z/nZ do not occur as the torsion subgroup of any elliptic curve over suitable number fields
In this paper we develop a criterion, based on a careful study of the cusps of modular curves X1(m, n), which can tell us that certain groups do not occur as torsion groups of elliptic curves over a number field K

Summary

Introduction

A fundamental result in the theory of elliptic curves, the Mordell–Weil theorem, states that the Abelian group of points of an elliptic curve (or more generally an Abelian variety) E over a number field K is finitely generated. We know that if a point on an elliptic curve has prime order p, p ≤ 13 by results of Parent [31, 32].

Results

Conclusion