Most odd degree hyperelliptic curves have only one rational point

Abstract

Consider the smooth projective models C of curves y [superscript 2] = f(x) with f(x) ∈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→∞. 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 theorems on the average number and equidistribution of nonzero 2-Selmer group elements, we prove that these conditions are often satisfied for p=2.