The average number of integral points on elliptic curves is bounded

We prove that, when elliptic curves $E\mathrm{/}\mathbb{Q}$ are ordered by height, the average number of integral points $\mathrm{\#}\mathrm{\mid }E(\mathbb{Z})\mathrm{\mid }$ 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"{\i}ve 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 $y^2=x^3-D^{2}x$.