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 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.

24 pages; to appear in Annals of Math

References in corpus (2)

• Independence of rational points on twists of a given curve
• Syzygies of torsion bundles and the geometry of the level l modular variety over M_g

Cited by in corpus (8)

• Uniform bounds for the number of rational points on curves of small Mordell--Weil rank
• Average size of the 2-Selmer group of Jacobians of monic even hyperelliptic curves
• Chabauty without the Mordell-Weil group
• Division by 2 on odd degree hyperelliptic curves and their jacobians
• $E_8$ and the average size of the 3-Selmer group of the Jacobian of a pointed genus-2 curve
• Orbit Parametrizations for K3 Surfaces
• Geometric quadratic Chabauty over number fields
• On the proportion of locally soluble superelliptic curves