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

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.

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

References in corpus (1)

• Central values of $L$-functions of cubic twists