Orders of points in families of elliptic curves

Abstract:

We use a result of P. Habegger to show that for all but finitely many integer pairs $(a,b) \in \mathbb {Z}^2$ with $4a^3 +27b^2\ne 0$ at least one of the points \begin{equation*} \left (1, \sqrt {1+a+b}\right ), \qquad \left (2, \sqrt {8+2a+b}\right ), \qquad \left (3, \sqrt {27+3a+b}\right ), \end{equation*} on the elliptic curve $Y^2 =X^3 + aX + b$ is of large order when for a large prime $p$ the curve is considered in the algebraic closure of the finite field of $p$ elements.