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(K)(L) E decomposes (up to isogeny) into abelian varieties over K

Res(K)(L) E similar to circle plus(F is an element of 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)root D) is the cyclic cubic extension of K for some D is an element of K-x/(K-x)(3), E = E-a: y(2) = ( )x(3) + a is an elliptic curve with j-invariant 0 defined over K, and E-a(D) : 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-a(D) x E-a(D2) 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.