Points \(\Delta\)-entiers sur les courbes elliptiques. (\(\Delta\)-integer points on elliptic curves) (Q1814419)

Let \(E/\mathbb{Q}\) be the nonsingular cubic \(y^ 2=x^ 3-Ax-B\), \(A\),\(B\in\mathbb{Z}\) with discriminant \(\Delta=16(4A^ 3-27B^ 2)\neq0\). This paper gives necessary and sufficient conditions for a rational point \(P=(x,y)\) on \(E\) to be \(\Delta\)-integral. These conditions are expressed in terms of classical auxiliary polynomials in \(x\), \(y\) with coefficients depending on \(A\) and \(B\). Also points \(T+nP\) are similarly considered; here \(T\) is a rational 2-torsion point on \(E\) and \(n\in\mathbb{Z}\). As the set of all \(\Delta\)-integral points is finite, the said conditions hopefully may lead to the construction of all such points on a given curve \(E\). Many examples are given.