A POSITIVE PROPORTION OF LOCALLY SOLUBLE HYPERELLIPTIC CURVES OVER Q HAVE NO POINT OVER ANY ODD DEGREE EXTENSION

A hyperelliptic curve over ℚ is called “locally soluble” if it has a point over every completion of ℚ. In this paper, we prove that a positive proportion of hyperelliptic curves over ℚ of genus g≥ 1 are locally soluble but have no points over any odd degree extension of 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 ℚ of genus g having no points over any odd degree extension of 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. © 2016 American Mathematical Society.