Torsion Subgroups of Elliptic Curves over Quadratic Extensions

Abstract

LetEbe an elliptic curve over Q. Observing certain relations between the torsion subgroups ofEandED, theD-quadratic twist ofE, we prove that the torsion subgroups ofEis stable for all but finitely many quadratic extensions. Moreover, using the result ofK. Ono, we classify the torsion subgroup ofEover all quadratic extensions whenEis of the formE:y2=x(x+M)(x+N), whereMandNare integers. In the special case when torsion subgroup ofEover Q is isomorphic to Z/2Z⊕Z/8Z, we prove that the torsion subgroup ofEis always stable under quadratic extensions.