Number of integral points on quadratic twists of elliptic curves

Abstract: We consider the integral points on the quadratic twists $E_D : y^2 = x^3+D^2Ax+D^3B$ of the elliptic curve $E : y^2 = x^3+Ax+B$ over $\mathbb{Q}$. For sufficiently large values of $D$, 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 $E_D$.