The average number of integral points on elliptic curves is bounded

الأصل · EN

We prove that, when elliptic curves E/Q are ordered by height, the average number of integral points #|E(Z)| is bounded, and in fact is less than 66 (and at most 8/9 on the minimalist conjecture). By "E(Z)" we mean the integral points on the corresponding quasiminimal Weierstrass model Eₐ,B: y² = x³ + Ax + B with which one computes the naıve 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 y² = x³ + B, y² = x³ + Ax, and y² = x³ - D² x.