A positive proportion of locally soluble hyperelliptic curves over $\mathbb Q$ have no point over any odd degree extension

Abstract

A hyperelliptic curve over $\mathbb Q$ is called "locally soluble" if it has a point over every completion of $\mathbb Q$. In this paper, we prove that a positive proportion of hyperelliptic curves over $\mathbb Q$ of genus $g\geq 1$ are locally soluble but have no points over any odd degree extension of $\mathbb Q$. We also obtain a number of related results. For example, we prove that for any fixed odd integer $k > 0$, the proportion of locally soluble hyperelliptic curves over $\mathbb Q$ of genus $g$ having no points over any odd degree extension of $\mathbb Q$ of degree at most $k$ tends to 1 as $g$ tends to infinity. We also show that the failures of the Hasse principle in these cases are explained by the Brauer-Manin obstruction. Our methods involve a detailed study of the geometry of pencils of quadrics over a general field of characteristic not equal to 2, together with suitable arguments from the geometry of numbers.