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 $C(\mathbb{R})\ne \varnothing$ and $C({\mathbb{Q}}_p)\ne \varnothing$ 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 \[ L(x) = c_f \frac{x}{(\ln{x})^{m}}+O\left(\frac{x}{(\ln{x})^α}\right) \] for some constants $c_f$, $m$ and $\alpha$ only depending on $f$ such that $m<\alpha \le 1+m$. We also express the Dirichlet's series $F(s)$ via Dedekind's zeta functions of certain number fields.

v1: 13 pages, comments welcome v2: 16 pages, typos corrected, extended Introduction and bibliography, added proof that $c_f>0$