Indivisible Points on Families of Elliptic Curves Original Research Article

In this paper we consider families of elliptic curvesEnover image arising as twists. Given rational pointsPnonEn, we ask how oftenPnis indivisible in the group of rational points ofEn, asnvaries over the positive integers. We prove, following the method of Silverman for families with non-constantj-invariant, that, under mild restrictions,Pnis indivisible almost always, in a density sense. We also prove in certain cases that, for a fixedm, the pointPnis divisible bymfor only finitely manyn.