论文标题
$ l $ -Series值的六曲线曲线的偏曲线上方$ \ mathbb {q} [\ sqrt {-3}] $
$L$-series values for sextic twists of elliptic curves over $\mathbb{Q}[\sqrt{-3}]$
论文作者
论文摘要
我们证明了$ l $ -function $ l（e_ {d，α}，1）$的核心价值的新公式，与六跨家族相对应，超过$ \ mathbb {q} [\ sqrt {\ sqrt {-3} $ $α\ in \ mathbb {q} [\ sqrt {-3}] $。该公式概括了Rodriguez-Villegas和Zagier的Cutic Twist的结果，以$ \ MATHBB {Q} $，以$ d \ equiv 1（9）$和Rosu的一般$ d $。对于$α$ prime和所有整数$ d $，我们还表明，Tate-Shafarevich Group订单的桦木和Swinnerton-Dyer的预期价值在某些情况下是一个整数广场，总体上最高的整数是$ 2^{2A} 3^{2B} $。
We prove a new formula for the central value of the $L$-function $L(E_{D, α}, 1)$ corresponding to the family of sextic twists over $\mathbb{Q}[\sqrt{-3}]$ of elliptic curves $E_{D, α}: y^2=x^3+16D^2α^3$ for $D$ an integer and $α\in \mathbb{Q}[\sqrt{-3}]$. The formula generalizes the result of cubic twists over $\mathbb{Q}$ of Rodriguez-Villegas and Zagier for a prime $D \equiv 1 (9)$ and of Rosu for general $D$. For $α$ prime and all integers $D$, we also show that the expected value from the Birch and Swinnerton-Dyer conjecture of the order of the Tate-Shafarevich group is an integer square in certain cases, and an integer square up to a factor $2^{2a}3^{2b}$ in general.
