Twists arising from torsion points

2025/10/09 by Lukas Novak, Novak, Lukas
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2510.08486

openalex publication_date 2025/10/09 · openalex created_date 2025/10/11 · openalex updated_date 2026/07/28

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(√[p]d). Let E/K be an elliptic curve with full p-torsion and S,T ∈ E(K)[p] be the generators. Define the cocycle ξ_d : Gal(K/K) → E by ξ_d (σ)= \begincases O, amp; if σ(√[p]d)=√[p]d, \newline kS, amp; if σ(√[p]d)=ζ_p^k√[p]d, \endcases 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 : α₁z^p+α₂z^p-2w+\dotso+α_(p+1)/(2)zw^(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.

Citations

• Quadratic twists of genus one curves
• Quadratic twists of genus one curves and Diophantine quintuples
• On Polyquadratic Twists of X₀(N)

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• Local points on twists of X(p) with applications
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