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Selmer companion curves

We say that two elliptic curves E_1, E_2 over a number field K are n-Selmer companions for a positive integer n if for every quadratic character χof K, there is an isomorphism between the n-Selmer groups Sel_n(E_1^χ/K) and Sel_n(E_2^χ/K) of the quadratic twists E_1^χ, E_2^χ. We give sufficient conditions for two elliptic curves to be n-Selmer companions, and give a number of examples of non-isogenous pairs of companions.