Restriction of scalars and cubic twists of elliptic curves

J. Korean Math. Soc. Published online June 1, 2020

Dongho Byeon, Keunyoung Jeong, and Nayoung Kim
Seoul National University, UNIST

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.

Keywords : restriction of scalars; cubic twist; elliptic curve