Number of integral points on quadratic twists of elliptic curves

Abstract

We study integral points on the quadratic twists $E_D : y^2 = x^3+D^2Ax+D^3B$ of a fixed elliptic curve $E : y^2 = x^3+Ax+B$ over $\overline{Q}$. For sufficiently large squarefree positive integers $D$, we prove that the number of integral points on $E_D$ admits the upper bound $\ll 4^r$, where $r$ denotes the Mordell-Weil rank of $E_D$. The implied constant is absolute and effectively computable. The proof combines gap principles, bounds for spherical codes, and Diophantine approximation. As an application, we prove that the average number of integral points on the quadratic twist family is bounded.