Twists of Superelliptic Curves Without Rational Points. (24th December 2016) - Legrand, François
Abstract: Let $n≥ 2$ 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 $Y^n=d· P(T)$ 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 $∞$ . Secondly, we connect the problem with the recent notion of non-parametric extensions and give new examples of such extensions with cyclic Galois groups.