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

TL;DR

This paper derives a new explicit formula for the central values of L-functions associated with sextic twists of elliptic curves over 3, extending previous results and exploring implications for the Tate-Shafarevich group.

Contribution

It generalizes known formulas for cubic twists to sextic twists over 3 and analyzes the arithmetic of the Tate-Shafarevich group in this context.

Findings

Derived a new formula for L-values of sextic twists over 3.

Showed the expected Tate-Shafarevich group order is an integer square in certain cases.

Extended previous results from cubic to sextic twists over 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.