Twists of superelliptic curves without rational points

TL;DR

This paper investigates conditions under which superelliptic curves lack rational points over number fields, showing that most such polynomials with certain Galois properties lead to twists without rational points, and explores applications to non-parametric extensions.

Contribution

The paper establishes a criterion based on Galois group elements for superelliptic curves to lack rational points and demonstrates that this occurs for a density-one subset of polynomials, also connecting to non-parametric extensions.

Findings

Most polynomials with bounded height satisfy the no-rational-points condition.

A Galois group element fixing no root guarantees the curve's twists lack rational points.

Provides new examples of non-parametric extensions with cyclic Galois groups.

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 paper 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.