Twists of superelliptic curves without rational points

2016/03/23 by Legrand, François
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1603.07171

Abstract

Let n≥ 2 be an integer, F a number field, O_F the integral closure of ℤ in F and N a positive multiple of n. The paper deals with degree N polynomials P(T) ∈ O_F[T] such that the superelliptic curve Yⁿ=P(T) has twists Yⁿ=d⋅ P(T) without F-rational points. We show that this condition holds if the Galois group of P(T) over F has an element which fixes no root of P(T). Two applications are given. Firstly, we prove that the proportion of degree N polynomials P(T) ∈ O_F[T] with height bounded by H and such that the associated curve satisfies the desired condition tends to 1 as H tends to ∞. Secondly, we connect the problem with the recent notion of non-parametric extensions and give new examples of such extensions with cyclic Galois groups.

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