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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.
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참고문헌 (5)
참고문헌 신청
Barry Mazur / 2007 / Finding large Selmer rank via an arithmetic theory of local constants / Annals of Mathematics 166(2):579~612
B. Mazur / 2010 / Ranks of twists of elliptic curves and Hilbert’s tenth problem / Inventiones mathematicae 181(3):541~575
B. Mazur / 2007 / Twisting commutative algebraic groups / Journal of Algebra 314(1):419~438
A. Silverberg / 2008 / Applications to cryptography of twisting commutative algebraic groups / Discrete Applied Mathematics 156(16):3122~3138
A. Weil / 1982 / Progress in Mathematics, 23 / Birkh¨auser
B. Mazur / 2010 / Ranks of twists of elliptic curves and Hilbert’s tenth problem / Inventiones mathematicae 181(3):541~575
B. Mazur / 2007 / Twisting commutative algebraic groups / Journal of Algebra 314(1):419~438
A. Silverberg / 2008 / Applications to cryptography of twisting commutative algebraic groups / Discrete Applied Mathematics 156(16):3122~3138
A. Weil / 1982 / Progress in Mathematics, 23 / Birkh¨auser
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