Twists arising from torsion points

Abstract

Let $p$ be a prime number, $K$ a number field that contains the $p$-th root of unity $\zeta_p$, $d$ a $p$-power-free integer and $L=K(\sqrt[p]{d})$. Let $E/K$ be an elliptic curve with full $p$-torsion and $S,T \in E(K)[p]$ be the generators. Define the cocycle $\xi_d : \operatorname{Gal}(\overline{K}/K) \to E$ by \[ \xi_d (\sigma)= \begin{cases} O, & \text{if } \sigma(\sqrt[p]{d})=\sqrt[p]{d}, \newline kS, & \text{if } \sigma(\sqrt[p]{d})=\zeta_p^k\sqrt[p]{d}, \end{cases} \] and denote by $H_S^d$ the twist of $E$ corresponding to the cocycle $\xi_d$. In this paper we construct generators $z$ and $w$ of the function field $K(H_S^d)$ and give a model of the twist \[ H_S^d\,:\, \alpha_{1}z^p+\alpha_2z^{p-2}w+\dotso+\alpha_{\frac{p+1}{2}}zw^{\frac{p-1}{2}}+\beta w^p+\gamma=0.\] We also obtain that the twist $H_S^d$ is everywhere locally solvable only for finitely many integers $d$.