Most hyperelliptic curves over Q have no rational points

Abstract

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^1 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) via an equation of the form C : z^2 = f(x,y) = f_0 x^n + f_1 x^{n-1} y + ... + f_n y^n where n=2g+2, the coefficients f_i lie in Z, and f factors into distinct linear factors over Q-bar. Define the height H(C) of C by H(C):=max{|f_i|}, and order all hyperelliptic curves over Q of genus g by height. Then we prove that, as g tends to infinity: 1) a density approaching 100% of hyperelliptic curves of genus g have no rational points; 2) a density approaching 100% of those hyperelliptic curves of genus g that have points everywhere locally fail the Hasse principle; and 3) a density approaching 100% of hyperelliptic curves of genus g have empty Brauer set, i.e., have a Brauer-Manin obstruction to having a rational point. We also prove positive proportion results of this type for individual genera, including g = 1.