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arXiv · 2006.13097

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

Eugenia Rosu
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Abstract

We prove a new formula for the central value of the $L$-function $L(E_{D, \alpha}, 1)$ corresponding to the family of sextic twists over $\mathbb{Q}[\sqrt{-3}]$ of elliptic curves $E_{D, \alpha}: y^2=x^3+16D^2\alpha^3$ for $D$ an integer and $\alpha \in \mathbb{Q}[\sqrt{-3}]$. 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 $\alpha$ 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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Eugenia Rosu. 2020-06-23. $L$-series values for sextic twists of elliptic curves over $\mathbb{Q}[\sqrt{-3}]$. https://arxiv.org/abs/2006.13097

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