Speaker
Description
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 ${H\, :\, y^2=f(x)}$
$H_q \, :\, qy^2=f(x)$.
Let $L$ denotes the set of positive square-free integers $q$ for which $H_q$ is everywhere locally solvable. For a real number $x$, let ${L(x)= \#\{q\in L:\, q \leq x\}}$ be the number of elements in $L$ that are less then or equal to $x$.
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 $c_f>0$, $m$ and $\alpha$ only depending on $f$ such that $m<\alpha \leq 1+m$.
We also express the Dirichlet series $F(s)=\sum_{n \in L} n^{-s}$ associated to the set $L$ in terms of Dedekind zeta functions of certain number fields.
