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

Abstract

A hyperelliptic curve over Q \mathbb Q is called “locally soluble” if it has a point over every completion of Q \mathbb Q . In this paper, we prove that a positive proportion of hyperelliptic curves over Q \mathbb Q of genus g ≥ 1 g\geq 1 are locally soluble but have no points over any odd degree extension of Q \mathbb Q . We also obtain a number of related results. For example, we prove that for any fixed odd integer k > 0 k > 0 , the proportion of locally soluble hyperelliptic curves over Q \mathbb Q of genus g g having no points over any odd degree extension of Q \mathbb Q of degree at most k k tends to 1 1 as g 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 2 , together with suitable arguments from the geometry of numbers.