Density of positive rank fibers in elliptic fibrations

Munshi, Ritabrata (2007) Density of positive rank fibers in elliptic fibrations Journal of Number Theory, 125 (1). pp. 254-266. ISSN 0022-314X

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A question of Mazur asks whether for any non-constant elliptic fibration {Er}r∈ℚ, the set {r∈ℚ: rank(Er(ℚ))>0}, if infinite, is dense in ℝ (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 Mazur's question affirmatively for the general quadratic and cubic fibrations. Moreover we show that our method works when is ℚ replaced by any real number field.

Item Type:Article
Source:Copyright of this article belongs to Elsevier Science.
Keywords:Elliptic Pencil; Rational Points
ID Code:110848
Deposited On:31 Jan 2018 09:09
Last Modified:31 Jan 2018 09:09

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