  Consider the smooth projective models C of curves y^2=f(x) with f(x) in Z[x]
monic and separable of degree 2g+1. We prove that for g >= 3, a positive
fraction of these have only one rational point, the point at infinity. We prove
a lower bound on this fraction that tends to 1 as g tends to infinity. Finally,
we show that C(Q) can be algorithmically computed for such a fraction of the
curves. The method can be summarized as follows: using p-adic analysis and an
idea of McCallum, we develop a reformulation of Chabauty's method that shows
that certain computable conditions imply #C(Q)=1; on the other hand, using
further p-adic analysis, the theory of arithmetic surfaces, a new result on
torsion points on hyperelliptic curves, and crucially the Bhargava-Gross
equidistribution theorem for nonzero 2-Selmer group elements, we prove that
these conditions are often satisfied for p=2.
