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arXiv 2006.13097math.NT
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$\mathbb{Q}[\sqrt{-3}]$上椭圆曲线六次扭的$L$ -级数值

$L$-series values for sextic twists of elliptic curves over $\mathbb{Q}[\sqrt{-3}]$

Eugenia Rosu

备注：
Generalized to all primes alpha the result from the previous version. Improved the expositions and fixed some minor errors
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摘要：

证明了一个新的$L$ -函数$L(E_{D, α}, 1)$的中心值公式，该中心值对应于椭圆曲线的$\mathbb{Q}[\sqrt{-3}]$上的六次扭族，其中$D$为整数，$α\in \mathbb{Q}[\sqrt{-3}]$为$E_{D, α}: y^2=x^3+16D^2α^3$.该公式推广了Rodriguez-Villegas和Zagier关于素数$D \equiv 1 (9)$和Rosu关于一般$D$的关于$\mathbb{Q}$上三次扭转的结果.对于$α$素数和所有整数$D$，我们还证明了Tate-Shafarevich群的阶的Birch和Swinnerton-Dyer猜想的期望值在某些情况下是整数平方，一般情况下是整数平方到因子$2^{2a}3^{2b}$.

英文摘要：

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.

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