RESTRICTION OF SCALARS AND CUBIC TWISTS OF ELLIPTIC CURVES

Abstract

. Let K be a number field and L a finite abelian extension of K . Let E be an elliptic curve defined over K . The restriction of scalars Res LK E decomposes (up to isogeny) into abelian varieties over K Res LK E ∼ (cid:77) F ∈ S A F , where S is the set of cyclic extensions of K in L . It is known that if L is a quadratic extension, then A L is the quadratic twist of E . In this paper, we consider the case that K is a number field containing a primitive third root of unity, L = K ( 3 √ D ) is the cyclic cubic extension of K for some D ∈ K × / ( K × ) 3 , E = E a : y 2 = x 3 + a is an elliptic curve with j invariant 0 defined over K , and E Da : y 2 = x 3 + aD 2 is the cubic twist of E a . In this case, we prove A L is isogenous over K to E Da × E D 2 a and a property of the Selmer rank of A L , which is a cubic analogue of a theorem of Mazur and Rubin on quadratic twists.