Most odd degree hyperelliptic curves have only one rational point

Abstract

Consider the smooth projective models $C$ of curves $y^2=f(x)$ with $f(x) \in \mathbb{Z}[x]$ monic and separable of degree $2g+1$. We prove that for $g \ge 3$, 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 $g \to \infty$. Finally, we show that $C(\mathbb{Q})$ 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 $\#C(\mathbb{Q})=1$; 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 $p=2$.