Twists of Superelliptic Curves Without Rational Points. (24th December 2016)

• Record Type: Journal Article
• Authors: Legrand, François
• Is Part Of: International mathematics research notices. Volume 2018:Number 4(2018)
• Page Start: 1153
• Page End: 1176
• Publication Date: 2016-12-24
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnw270
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: Let $n\geq 2$ be an integer, $F$ a number field, $O_F$ the integral closure of $\mathbb{Z}$ in $F$ and $N$ a positive multiple of $n$ . The article deals with degree $N$ polynomials $P(T) \in O_F[T]$ such that the superelliptic curve $Y^n=P(T)$ has twists $Y^n=d\cdot P(T)$ 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 $P(T) \in O_F[T]$ with height bounded by $H$ and such that the associated curve satisfies the desired condition tends to 1 as $H$ tends to $\infty$ . Secondly, we connect the problem with the recent notion of non-parametric extensions and give new examples of such extensions with cyclic Galois groups.