Pencils of quadrics and the arithmetic of hyperelliptic curves

In this article, for any fixed genus g ≥ 1, we prove that a positive proportion of hyperelliptic curves over Q of genus g have points over R and over Qp for all p, but have no points globally over any extension of Q of odd degree. By a hyperelliptic curve over Q, we mean a smooth, geometrically irreducible, complete curve C over Q equipped with a fixed map of degree 2 to P defined over Q. Thus any hyperelliptic curve C over Q of genus g can be embedded in weighted projective space P(1, 1, g + 1) and expressed by an equation of the form C : z = f(x, y) = f0x n + f1x n−1y + · · ·+ fny (1)