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Open Access

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

Research in Number Theory20162:3

Received: 11 June 2015

Accepted: 19 November 2015

Published: 20 January 2016


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.


Elliptic CurveIrreducible ComponentElliptic CurfAbelian VarietyNumber Field