Most odd degree hyperelliptic curves have only one rational point

TL;DR

This paper demonstrates that for hyperelliptic curves of genus at least 3, a positive and increasing fraction have only one rational point, and provides an algorithm to compute these points for many such curves.

Contribution

It introduces a new p-adic reformulation of Chabauty's method and proves that most high-genus hyperelliptic curves have only the point at infinity as a rational point.

Findings

A positive fraction of hyperelliptic curves have only one rational point.

The fraction tends to 1 as genus increases.

An algorithm can compute the rational points for many such curves.

Abstract

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.