Quadratic twists of genus one curves

Abstract

For a given irreducible and monic polynomial $f(x) \in \mathbb{Z}[x]$ of degree $4$, we consider the quadratic twists by square-free integers $q$ of the genus one quartic ${H\, :\, y^2=f(x)}$ \[ H_q \, :\, qy^2=f(x). \] We say that a curve $C$ is everywhere locally soluble (ELS) if it has a solution in $\mathbb{R}$ and in $\mathbb{Q}_p$ for every prime $p$ (i.e. if $C(\mathbb{R})\neq \emptyset$ and $C(\mathbb{Q}_p)\neq \emptyset$ for all primes $p$). Let $L=\{q\in \mathbb{N} :\, q \text{ is square-free and } H_q \text{ is ELS}\}$ denote the set of positive square-free integers $q$ for which $H_q$ is everywhere locally soluble. For a real number $x$ let ${L(x)= \#\{q\in L:\, q<x\}}$ be the number of elements in $L$ that are less then $x$. Furthermore, let us denote with \[ F(s)=\sum_{n \in L} \frac{1}{n^s} \] the corresponding Dirichlet's series of the set $L$. In this paper, we obtain that \[ L(x) = c_f \frac{x}{(\ln{x})^{m}}+O\left(\frac{x}{(\ln{x})^\alpha}\right) \] for some constants $c_f$, $m$ and $\alpha$ only depending on $f$ such that $m<\alpha \leq 1+m$. We also express the Dirichlet's series $F(s)$ via Dedekind's zeta functions of certain number fields.