The average number of integral points on elliptic curves is bounded

Abstract

We prove that, when elliptic curves $E/\mathbb{Q}$ are ordered by height, the average number of integral points $\#|E(\mathbb{Z})|$ is bounded, and in fact is less than $66$ (and at most $\frac{8}{9}$ on the minimalist conjecture). By "$E(\mathbb{Z})$" we mean the integral points on the corresponding quasiminimal Weierstrass model $E_{A,B}:y^2=x^3+Ax+B$ with which one computes the na\"ıve height. The methods combine ideas from work of Silverman, Helfgott, and Helfgott-Venkatesh with work of Bhargava-Shankar and a careful analysis of local heights for "most" elliptic curves. The same methods work to bound integral points on average over the families $y^2=x^3+B$, $y^2=x^3+Ax$, and $y^2=x^3-D^{2}x$.

45 pages, Mathematica file included. [v2: fixed a misattribution and a minus sign, added some citations. v3: improved result for rank 1 curves, fixed an error in the optimization, added some citations.]

References in corpus (3)

• The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point
• A majority of elliptic curves over $\mathbb{Q}$ satisfy the Birch and Swinnerton-Dyer conjecture
• Small rational points on elliptic curves over number fields