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 $\mathrm{Res}^{L}_{K}E$ decomposes (up to isogeny) into abelian varieties over $K$ $$ \mathrm{Res}^{L}_{K}E \sim \bigoplus_{F \in S}A_F, $$ 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, $L=K(\root 3\of D)$ is the cyclic cubic extension of $K$ for some $D\in K^{\times}/(K^{\times})^3$, $E=E_a: y^2=x^3+a$ 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 $E_a^D \times E_a^{D^2}$ 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.