Height of rational points on quadratic twists of a given elliptic curve

Abstract

We prove that a positive proportion of squarefree integers are congruent numbers such that the canonical height of the lowest non-torsion rational point on the corresponding elliptic curve satisfies a strong lower bound.

A statistical view on the conjecture of Lang about the canonical height on elliptic curves

“…The author [LB16] has recently studied the analogous problem for families of quadratic twists of a fixed elliptic curve. It is important to note that for these families, the analog of Conjecture A trivially holds (see for instance [LB16, Section 2.2]).…”

Average rank in families of quadratic twists: a geometric point of view

“…Letĥ E d be the canonical height on the curve E d (see Section 2 for its definition), and let E d (Q) tors be the torsion part of the abelian group E d (Q). The author [LB14] has recently investigated the quantity η d (A, B) defined by…”

Heights of Rational Points on Mordell Curves

“…Summary. In Le Boudec's work [LB16], a lower bound on the minimal non-zero canonical height of rational points of curves in a general quadratic twist family E (d) (A, B) := dy 2 = x 3 +Ax+B [LB16, Theorem 1] is established. Stronger bounds are proven upon specialization to an analysis on the minimal non-zero canonical height of rational points of curves in the family of quadratic twists of the congruent number curve, given by dy 2 = x 3 − x for d ∈ Z ≥1 square-free.…”

“…Above, ĥE represents the canonical, or Néron-Tate, height on E. Before stating the main Theorems of [LB16,LB18], let us set a target. By the analogy between number fields and elliptic curves, we have the following conjecture [LB16, Conjecture A]:…”

“…In this direction, it would be interesting to pursue stronger bounds in the quantitative arithmetic of projective varieties (see [Bro09] for an exposition) and further improve our results. Indeed, it is this field of study which allows for the greater simplicity of the proof of Theorem 1.6 compared to that of the second bound of Theorem 1.3 given in [LB16].…”