Density of positive rank fibers in elliptic fibrations Original Research Article

A question of Mazur asks whether for any non-constant elliptic fibration image, the set image, if infinite, is dense in image (with respect to the Euclidean topology). This has been proved to be true for the family of quadratic twists of a fixed elliptic curve by a quadratic or a cubic polynomial. Here we settle Mazurʹs question affirmatively for the general quadratic and cubic fibrations. Moreover we show that our method works when image is replaced by any real number field.