Torsion growth of rational elliptic curves in sextic number fields

摘要

We classify the possible torsion structures of rational elliptic curves over sextic number fields. Among these possible torsion group structures, all groups except C-3 (R) C-18 are known to appear as subgroups of E(K)(tors) for some elliptic curve E/Q and for some sextic number field K. We prove that if the image of mod 2 Galois representation of E is not equal to the Borel subgroup of GL(2)(Z/2Z), then E(K)(tors) can't contain the group C-3 circle plus C-18. (c) 2020 Elsevier Inc. All rights reserved.