Twists arising from torsion points

TL;DR

This work investigates when twists $H_S^d$ of an elliptic curve $E$ with full $p$-torsion are everywhere locally solvable. It develops explicit generators $z,w$ in the function field of the twist to produce concrete models of the form $H_S^d:\ \alpha_{1}z^p+\alpha_2z^{p-2}w+\dots+\alpha_{\frac{p+1}{2}}zw^{\frac{p-1}{2}}+\beta w^p+\gamma=0$, and analyzes local solubility via the Kummer sequence and Cassels-type arguments. The paper treats special 2- and 3-torsion cases with explicit equations, showing that in each case $H_S^d$ is ELS only for finitely many admissible $d$, a finiteness result driven by obstructions at primes dividing $d$. The authors also outline a general framework for arbitrary $p$, using a twisted Galois action to define $K(H_S^d)$ and prove analogous finiteness results, thereby contributing to the broader understanding of local-global principles for twists of elliptic curves.

Abstract

Let $p$ be a prime number, $K$ a number field that contains the $p$-th root of unity $ζ_p$, $d$ 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 $ξ_d : \operatorname{Gal}(\overline{K}/K) \to E$ by \[ ξ_d (σ)= \begin{cases} O, & \text{if } σ(\sqrt[p]{d})=\sqrt[p]{d}, \newline kS, & \text{if } σ(\sqrt[p]{d})=ζ_p^k\sqrt[p]{d}, \end{cases} \] and denote by $H_S^d$ the twist of $E$ corresponding to the cocycle $ξ_d$. In this paper we construct generators $z$ and $w$ of the function field $K(H_S^d)$ and give a model of the twist \[ H_S^d\,:\, α_{1}z^p+α_2z^{p-2}w+\dotso+α_{\frac{p+1}{2}}zw^{\frac{p-1}{2}}+βw^p+γ=0.\] We also obtain that the twist $H_S^d$ is everywhere locally solvable only for finitely many integers $d$.

Theorems & Definitions (16)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 3.1
• proof
• Proposition 3.2
• proof